\[
\scalebox{1.3}{
\begin{Compose}{0}{0} \setdefaultfont{\hat} \setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\crectangledmark[-1]{B}{3}{2.5}{0,0} \csymbol{V}
\thispoint{Bd}{0,-7} \thispoint{Bu}{0,7}
\jointbnoarrow[right]{Bd}{0}{B}{0} \csymbolthird{a}  \jointbnoarrow[left]{B}{0}{Bu}{0} \csymbolthird{b}
\jointbnoarrow[right]{Bd}{-2}{B}{-2} \csymbolthird{x}  \jointbnoarrow[left]{B}{2}{Bu}{2} \csymbolthird{y}
\end{Compose}
\scalebox{1.3}{
\begin{Compose}{0}{0}\setdefaultfont{} \setsecondfont{\mathsfb}
\crectangle[thin]{coef}{3.2}{2}{-12,0} \csymbol{\frac{\alpha_\mathsfb{xa}\sqrt{N_\mathsfb{xa}N_\mathsfb{by}}}{\alpha_\mathsfb{by}} }
\crectangledouble{Iup}{1.5}{1.5}{0,8}\csymbolthird{\hat{I}} \crectangledouble{Idown}{1.5}{1.5}{0,-8} \csymbolthird{\hat{I}}
\crectangledleft[-1]{A}{1.5}{2.5}{-6,0} \csymbol{V}
\relpoint{A}{0,-7}{Ad} \jointbleft[left]{Ad}{0}{A}{0} \csymbolalt[-8,0]{xa}
%\relpoint{A}{-1,-7}{Ad} \jointbleft[left]{Ad}{0}{A}{-1} \csymbolalt{x}
\relpoint{A}{0,7}{Cd} \jointbleft[left]{A}{0}{Cd}{0} \csymbolalt[-8,0]{by}
%\relpoint{A}{-1,7}{Ed} \jointbleft[left]{A}{-1}{Ed}{0} \csymbolalt{b}
\crectangledleft{XA}{0.8}{0.8}{-2,3} \crectangledleft{BY}{0.8}{0.8}{-2,-3}
\relpoint{XA}{0.8,-1.8}{XAA} \joinrrnoarrow{XA}{0}{XAA}{0}
\joinrlnoarrow[below]{A}{1.2}{XAA}{0} \csymbol{xa}
\relpoint{BY}{0.8,1.8}{BYY} \joinrrnoarrow{BY}{0}{BYY}{0}
\joinrlnoarrow[above]{A}{-1.2}{BYY}{0} \csymbol{by}
\jointbleft[left]{XA}{0}{Iup}{-1} \csymbolalt[-8,6]{xa} \jointbleft[left]{Idown}{-1}{BY}{0} \csymbolalt[-8,-6]{by}
\crectangledright[-1]{AA}{1.5}{2.5}{6,0} \csymbol{V}
\relpoint{AA}{0,-7}{Ad} \jointbright[right]{Ad}{0}{AA}{0} \csymbolalt[8,0]{xa}
%\relpoint{AA}{-1,-7}{Ad} \jointbright[right]{Ad}{0}{AA}{-1} \csymbolalt{a}
\relpoint{AA}{0,7}{Cd} \jointbright[right]{AA}{0}{Cd}{0} \csymbolalt[8,0]{by}
%\relpoint{AA}{-1,7}{Ed} \jointbright[right]{AA}{-1}{Ed}{0} \csymbolalt{b}
\crectangledright{XA}{0.8}{0.8}{2,3} \crectangledleft{BY}{0.8}{0.8}{2,-3}
\relpoint{XA}{-0.8,-1.8}{XAA} \joinllnoarrow{XA}{0}{XAA}{0}
\joinlrnoarrow[below]{AA}{1.2}{XAA}{0} \csymbol{xa}
\relpoint{BY}{-0.8,1.8}{BYY} \joinllnoarrow{BY}{0}{BYY}{0}
\joinlrnoarrow[above]{AA}{-1.2}{BYY}{0} \csymbol{by}
\jointbright[right]{XA}{0}{Iup}{1} \csymbolalt[8,6]{xa} \jointbright[right]{Idown}{1}{BY}{0} \csymbolalt[8,-6]{by}
\end{Compose}
\]

The standard operational probabilistic framework (within which we can formulate Operational Quantum Theory) is time asymmetric.  This is clear because the conditions on allowed operations are time asymmetric.  It is odd, though, because Schoedinger's equation is time symmetric and probability theory does not care about time direction.  In this work we provide a time symmetric framework for operational theories in general and for Quantum Theory in particular.

The clearest expression of the time asymmetry of standard Operational Quantum Theory is that the deterministic effect is unique - meaning there is only one way to ignore the future - while deterministic (i.e. normalised) states are not unique.  In this paper, this time asymmetry is traced back to a time asymmetric understanding of the most basic elements of an operational theory - namely the operations (or boxes) out of which circuits are built.  We modify this allowing operations to have classical incomes as well as classical outcomes on these operations.  We establish a time symmetric operational framework for circuits built out of operations. In particular, we demand that the probability associated with a circuit is the same whether we calculate it forwards in time or backwards in time. We do this by imposing various double properties.  These are properties wherein a forward in time and a backward in time version of the same property are required. In this paper we provide new causality condition which we call double causality.

To apply these operational ideas to Quantum Theory we adapt the duotensor and operator tensor approaches to the time symmetric setting.   We formulate physicality conditions consisting of (i) a condition that ensures the probability for any circuit is non-negative and (ii) the double causality condition.   Time symmetric operational Quantum Theory is then stated by the single axiom: Every physical operator has an operation corresponding to it.   We prove an extension theorem (this is a time symmetric version of the Stinespring extension theorem) enabling physical operators to be built from unitaries.

Curiously, we find that gauge parameters need to be introduced when we provide a concrete representation of the operator tensors associated with operations. They drop out when we calculate probabilities for circuits.  The particular gauge choice that is adopted in standard treatments of Quantum Theory goes naturally with a forward in time point of view.  We can, though, chose any values for these parameters whilst maintaining a time symmetric point of view.

As a future research direction, we propose the idea of a conditional frame of reference and elaborate on three such frames of reference - a time symmetric one, a forward one, and a backwards one.

§ INTRODUCTION

Quantum Theory is a probability calculus.   The usual tools of probability theory, appropriately applied, must enable us to calculate probabilities in the quantum case as in any other case.  Operational probabilistic theories concern the calculation of probabilities associated with circuits. These can be applied to Quantum Theory giving Operational Quantum Theory. In the standard formulation, Operational Quantum Theory is time asymmetric.  This is elucidated in the work of Chiribella, D'Ariano, and Perinotti [12].  Future choices should not influence probabilities in the present and  corresponding to this is the property the mathematical property that deterministic effect is unique.  However, past choices do effect probabilities in the present and so the deterministic preparation is not unique.

This is rather strange since

* Abstract probability theory concerns calculation of such objects as $\text{prob}(x|y)$ and, as such, knows nothing of time.

* Quantum Theory, at the level of the Schrödinger equation, is time symmetric.

* The Quantum Theory of measurement can be treated very simply without reference to the second law, for example with the von Neumann model of measurement.

It is worth amplifying on the last point.  Although real measurement situations involve macroscopic instruments and, therefore, invariably entropic considerations are relevant, these simple models capture all the essential properties of measurement without invoking any entropic properties.

We will see it is possible to provide a time-symmetric operational formulation of quantum theory.  We do this by revisiting the notion of an operation and make it time symmetric. Then we set up a framework whereby circuits have the same probability whether the probability is calculated forwards or backwards in time.  We obtain this by imposing, and deriving various double properties (wherein there is a forward and a backward property).  The most important of these properties is double causality. This replaces the causality postulate of Chribella, D'Ariano, and Perinotti.   We further how to formulate Quantum Theory in the operator tensor framework.

An extended introduction is given below in Sec. <ref>.

As a precursor, consider the following provocative statements:

* We cannot signal from the future to the past.

* Also, we cannot signal from the past to the future.

* But we can steer information in either direction.

The first statement is not provocative - it accords with our usual intuition.  The second statement is the time reverse of the first so it must be true in a time-symmetric formulation if the first statement is true.  It does not accord with our usual intuition, nor, indeed, our everyday experience.  The third statement uses the notion of steering <cit.> in the sense it is usually employed in entanglement theory.  The point is that we can communicate information from the future to the past only if we have information about the future available in the past.  And, similarly, we can communicate information from the past to the future only if we have information from the past available in the future.

§ PREVIOUS WORK

The projection postulate of quantum theory, if taken literally as a physical process, is apparently time asymmetric since it implies that the wavefunction collapses after measurement.  However, the work of Aharonov, Bergmann and Lebowitz [1] showed that it is possible to formulate pure state quantum theory, including the theory of measurement, in a time symmetric fashion.  For years this was thought to have dealt with the problem of time asymmetry in Quantum Theory.  The gauntlet was thrown down (again) with the work of Chribella, D'Ariano, and Perinotti [12] (see also [25]) whose causality postulate - as discussed in the introduction above - revealed a deep time asymmetry in the standard operational formalism of Quantum Theory (where we now treat mixed states).   This set the stage for much thought this issue over the last decade or so.

The present work, in its inception, was strongly influenced by a presentation by Andrea Di Biagio  on preliminary ideas on time symmetry including a time-reversal invariant subtheory" of Quantum Theory.  Subsequently,  Di Biagio, in collaboration with Donà and Rovelli [29] showed how, it is possible to formulate operational Quantum Theory so that the mathematical machinery for prediction tasks - wherein we calculate $p(\text{future}|\text{past})$ - is of the same form as the mathematical machinery for postdiction tasks - wherein we calculate $p(\text{past}|\text{future})$.  This shows that there is a time reversal symmetry between prediction and postdiction.  The approach in [29] is similar to and clearly consistent with the present work.  There are some key differences however.   First, much of the discussion here is at the level of general operational theories.  Second, here we consider joint probabilities over past and future quantities of the form $p(\text{future},\text{past})$.   Then we set up mathematical machinery wherein we get the same value for these joint probabilities whether we calculate them in the forward time direction or the backwards time direction.  This perspective enables us to establish the double causality condition (discussed in Sec. <ref>) and the extension theorem (discussed in Sec. <ref>.  Conditional probabilities can be obtained from these joint probabilities by application of the conditional probability formula.  In fact, we can transform" to the prediction or postdiction point of view as discussed in Sec. <ref> and Sec. <ref>. Di Biagio et al. contains some interesting discussion on interpretation much of which is also pertinent to the approach in this paper. This covers the interpretation of having a unique deterministic  preparation (as well as a unique deterministic effect). They also discuss how different interpretations of Quantum Theory look in the light of their considerations.  Rovelli has other work on the issue of time symmetry.  For example, in  [81] he provides a model for memory (which plays a role in measurement) and discusses the connection with the second law.

A different take on the issue is provided in the paper Operational formulation of time reversal in quantum theory" by Oreshkov and Cerf [72].  They provide what looks, on the face of it, like an alternative time-symmetric operational formulation of Quantum Theory to that presented here or in DiBiagio et al.  It will be interesting to see if this approach is actually equivalent to the approach in the present paper.

There has been a tremendous amount of work on time-symmetry in Quantum Theory.  Indeed, it this is too vast a field to summarize here.  A few papers that are of particular interest are as follows. First, in 1964 Aharonov, Bergmann and Lebowitz [1] provided a time-symmetric formulation of pure state quantum theory in which there are pre-selections (preparations) post-selections, and intermediate measurements.  A multi-time version appears in [2].    A mixed state version of this was provided by Silva et al. [86] (though there is no discussion of time-symmetry in this paper).

The connection to taking the adjoint and time reversal is discussed in Coecke and Kissinger [21] (see also [17, 19]).
In [22] Coecke and Lal  provide a discussion of reversing the time direction of a measurement process.

A recent paper by Chribella and Liu [15] discusses the time flip operator.

In this paper we will be primarily interested in getting to a time symmetric operational formulation of Quantum Theory.   There has been much interesting work which focus not only on the operational realm, but also on ontological questions arising from time symmetry.  See for example Price [76], Wharton [91, 90], Leifer and Pusey [62],  Argaman [4].   Also see the books by Price [75] by Schulman [83].  It is hoped that the approach in this paper will provide a framework in which to further those discussions.

The approach taken here extends frameworks I have developed previously and makes them time symmetric.  These frameworks are duotensor framework in [53] and the operator tensor formulation of Quantum Theory   [51, 52] (these were the development of a perspectives from [43] and [44, 45]).  The duotensor and operator tensor approaches are built on the idea that we can regard states and processes as arising via a tomographic procedure using fiducial elements.

The duotensor and operator tensor approaches are diagrammatic and strongly inspired diagrammatic approaches due to Coecke and collaborators.  See the extraordinary book of Coecke and Kissinger [20] for a extensive discussion of this approach.  In Sec. <ref> I provide a summary of an approach to Hilbert spaces with input and output structure.  This is essentially a summary of the approach of Coecke [18] and Selinger [85] and summarized in [20].  The doubling up process described in these works provides an alternative route (to the tomographic procedure mentioned above) to what I call operator tensors.  The notation, both diagrammatic and symbolic, adopted here is different from that in Coecke and Kissinger and chosen so as to match with my previous notation for operator tensors (which are represented by double border boxes).  The symbolic notation, in particular, is tensor-like and intended to represent faithfully the graphical aspect of the diagrammatic notation.

The general probability framework approach of [43] has predecessors (for example, see [65, 64, 27, 41, 68, 3, 40, 33, 31]) and has been a topic of considerable research in recent years (for example, see [6, 5, 13, 12, 50, 47, 46, 71]).

The approach due to Chiribella, D'Ariano, and Perinotti [13, 12] was also inspired by the diagrammatic approach of Coecke and others. It is in this context that the causality postulate is introduced.  The causality postulate is also at work in the iterative conditions on quantum combs discussed in [10, 11]. One of the research directions discussed at the end of this paper concerns updating these iterative conditions in the light of the double causality principle from this paper.

A large part of my motivation for the present work (though not extensively discussed in this paper) is to deal with the challenge of indefinite causal structure which we would expect to arise in constructing a theory of Quantum Gravity.  Time asymmetry weds us more strongly to a background causal structure.  There has been much work on indefinite causal structure [44, 45, 48, 14, 73, 77, 58] (see the nice article by Natalie Wolchover [93] for an introduction).

A different strand of work has been pursued by Leifer and Spekkens on a formulating Quantum Theory as a causally neutral theory of inference [63].  Part of the idea there is to generalise the chain rule for conditional probabilities to Quantum Theory.  The duotensor approach discussed in Part <ref> also invokes this chain rule in the classical context and what might be regarded as a generalisation for general systems (that might be quantum).

A recent iteration of the inferential approach is due to Schmid, Selby, and Spekkens [82].  This adds classical inferential structure to the quantum structure.  This is also done in the present work.   The approach of Schmid et al. is more general though not specifically adapted to a time symmetric formulation.

The program of Oeckl,  which starts from more of a Quantum Gravity perspective with the general boundary formalism [69] and has moved in the direction of general probability theories with his positive formalism [70, 71], is of particular interest.  This approach does not assume up front any time or metric.

§ EXTENDED INTRODUCTION

Since this is a long paper I will provide an extended introduction summarizing the main ideas.  Throughout the paper, I will state double properties. These consist of a forward property and a backward property (and the two are the time reverse of one another).  I use several types of diagrammatic notation.   Generally wires that run up the page pertain to systems (quantum systems for example).   Wires that run across the page can pertain to results of measurements (this is classical information) or to indices that are summed over in some mathematical expression.

§.§ Time symmetric operations

In Sec. <ref> we argue that the usual notion of an operation is not fit for purpose when it comes to formulating operational theories in a time symmetric manner.  We propose modifying this so that, in addition to outcomes, there also incomes.  A time symmetric operation looks like this

\[
\begin{Compose}{0}{0}
\Crectangle{A}{3}{2}{0,0}
\thispoint{AL}{-6,0} \thispoint{AR}{6,0}
\thispoint{A1in}{-2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A2in}{0,-5} %\csymbolalt[-0,-15]{b}
\thispoint{A3in}{2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A1out}{-2,5}%\csymbolalt[0,15]{a}
\thispoint{A2out}{0,5}
\thispoint{A3out}{2,5} %\csymbolalt[0,15]{c}
\Thistexthere{inputs}{0,-7.1} \Thistexthere{outputs}{0,6.8}  %\relpoint{AL}{0,-1}{ALm} \relpoint{AR}{0,-1}{ARm}
%\joinrlthick{AL}{0}{A}{0}  \joinrlthick{A}{0}{AR}{0}
%\linebyhand[thin, dashed]{AL}{AR}  \Thistexthere{setting}{-8.3,0} \Thistexthere{setting}{8.4,0}
\joinrlthick{AL}{0}{A}{0}   \Thistexthere{income}{-8.4,0}
\joinrlthick{A}{0}{AR}{0}   \Thistexthere{outcome}{8.9,0}
\jointb{A1in}{0}{A}{-1} \jointb{A2in}{0}{A}{0} \jointb{A3in}{0}{A}{1}
\jointb{A}{-1}{A1out}{0} \jointb{A}{1}{A3out}{0}
\end{Compose}
\]

Outcomes are available after the operation has been applied but not before. Incomes are available before the operation has been applied but not afterwards (an example is an ancilla prepared in some initial state, or the initial position of a pointer).  The wires associated with incomes and outcomes are referred to as pointers types.  The inputs and outputs are for physical systems such as electrons, photons, etc.  We will refer to these as system types.  We can also accommodate settings (corresponding to some knob positions for example). In the above diagram thoughthese are suppressed for ease of presentation.

§.§ Circuits

Operations can be wired together. If there are no open wires left over then we have a circuit. For example

\begin{equation} %\label{ABCcircuit}
\begin{Compose}{0}{-2.4} \setsecondfont{\mathsfb}
\crectangle{A}{2}{2}{0,0} \csymbol{A}  \crectangle{B}{2}{2}{5,9} \csymbol{B} \crectangle{C}{2}{2}{2,18} \csymbol{C}
\thispoint{amid}{-5,9}  \csymbolalt[-20,0]{a}
\jointbnoarrow[left]{A}{-1}{amid}{0}\jointbnoarrow{amid}{0}{C}{-1}
%\jointbnoarrow[left]{A}{-1}{C}{-1} \csymbolalt[-5,0]{a}
\jointbnoarrow[below right]{A}{1}{B}{0}  \csymbolalt[5,-5]{b}
\jointbnoarrow[above right]{B}{0}{C}{1}  \csymbolalt{a}
\GenxBoxincome{A}{2}{0}{x}{E}
\GenxBoxoutcome{B}{2}{0}{y}{F}
\GenxBoxoutcome{C}{2}{0}{z}{G}
\end{Compose}
\end{equation}

The small boxes with $x$, $y$, …are called readout boxes.  They readout the income or outcome.  We demand that the probability, $p(x,y,z)$, is given by the circuit description and does not depend on extraneous factors.  We also demand that we get the same answer for this probability whether we do the calculation forwards or backwards in time.   Boxes like $\mathsf E$, viewed forward in time, correspond to a distribution over income values.  Boxes like $\mathsf F$, viewed backwards in time correspond to a distribution over outcome values.

§.§ A simple classical situation

In Part <ref> we will see that we can view circuits as pertaining to a simple classical situation in which we have classical incomes and classical outcomes.  We can impose various double properties to this classical situation.  Given a few basic assumptions, we will see that time symmetry forces on us that distributions over incomes and outcomes must be flat.  Throughout the paper, we represent flat distributions by boxes with an $\mathsf R$ in them.   Thus, circuits will look like

\begin{equation} %\label{ABCcircuit}
\begin{Compose}{0}{-2.4} \setsecondfont{\mathsfb}
\crectangle{A}{2}{2}{0,0} \csymbol{A}  \crectangle{B}{2}{2}{5,9} \csymbol{B} \crectangle{C}{2}{2}{2,18} \csymbol{C}
\thispoint{amid}{-5,9}  \csymbolalt[-20,0]{a}
\jointbnoarrow[left]{A}{-1}{amid}{0}\jointbnoarrow{amid}{0}{C}{-1}
%\jointbnoarrow[left]{A}{-1}{C}{-1} \csymbolalt[-5,0]{a}
\jointbnoarrow[below right]{A}{1}{B}{0}  \csymbolalt[5,-5]{b}
\jointbnoarrow[above right]{B}{0}{C}{1}  \csymbolalt{a}
\RxBoxincome{A}{2}{0}{x}{E}
\RxBoxoutcome{B}{2}{0}{y}{F}
\RxBoxoutcome{C}{2}{0}{z}{G}
\end{Compose}
\end{equation}

Correspondingly, we argue that circuits should be doubly summing. Thus, if we sum over the incomes then the we get a flat distribution over the outcomes. And, if we sum over the outcomes, we get a flat distribution over the incomes.  This property is a generalization of bistochasticity.

§.§ Equivalence, linearity, and physicality

In Part <ref> we introduce a linearly extended notion of equivalence between operations. This makes it possible to regard expressions consisting of linearly weighted sums of operations as being equivalent to one another.  Armed with this mathematical machinery, we can define ignore operations and introduce the notion of physicality.

Ignore operations are written as

\begin{equation}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb}
\Crectangle{I}{1}{1}{0,3} \thispoint{p}{0,0} \csymbolalt[0,-20]{a} \joinbtnoarrow{I}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb}
\Crectangle{I}{1}{1}{0,0} \thispoint{p}{0,3} \csymbolalt[0,20]{a} \jointbnoarrow{I}{0}{p}{0}
\end{Compose}
\end{equation}

They correspond to the situation where we have no incomes or outcomes.  In the case on the left, we have no output - we call this the ignore result.  In the case on the right we have no input - we call this the ignore preparation.  We show, employing some assumptions, that these ignore operations are unique (so, for example, all ignore preparations for a given system type are equivalent).

Physicality consists of constraints on operations such that the probabilities associated with circuits (i) are positive and (ii) respect the double summing property.  Positivity of probabilities for circuits is guaranteed by imposing $T$-positivity on each operation. This is written

\begin{equation}
\begin{Compose}{0}{0}\setdefaultfont{}\setsecondfont{\mathsfb}\setthirdfont{\mathtt}
\crectangle{B}{1.5}{1.5}{0,0} \csymbol{\mathsf{B}}
\thispoint{d}{0,-3} \csymbolalt[0,-20]{a}
\thispoint{u}{0,3}  \csymbolalt[0,20]{b}
\thispoint{l}{-3,0} \csymbolthird[-20,0]{x}
\thispoint{r}{3,0}  \csymbolthird[20,0]{y}
\jointbnoarrow{d}{0}{B}{0}
\jointbnoarrow{B}{0}{u}{0}
\joinrlnoarrowthick{l}{0}{B}{0}
\joinrlnoarrowthick{B}{0}{r}{0}
\end{Compose}
\end{equation}

The double summing property is guaranteed by imposing double causality.  Double causality is written

\begin{equation}
\Funitalxyab{1}{1}{1}{1} ~~~~~~~~~~~~~~~~  \Bunitalyxba{1}{1}{1}{1}
\end{equation}

The condition on the left is forward causality. The condition on the right is backwards causality. We show how double causality is implied by more basic assumptions.  The usual causality condition [12] in time asymmetric operational theories can be understood as an application of backwards causality.

Note that often the earth symbol is used to represent objects like $\mathsf R$ and $\mathsf I$. So we might, instead, write

\begin{equation}
\begin{Compose}{0}{0}
\crectangle{A}{0.8}{0.8}{0,-2} \csymbol{I}\thispoint{p}{0,2} \jointbnoarrow{A}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}
\supterminal{A}{0,-2} \thispoint{p}{0,2} \jointbnoarrow{A}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}
\crectangle{A}{0.8}{0.8}{0,2} \csymbol{I} \thispoint{p}{0,-2} \jointbnoarrow{p}{0}{A}{0}
\end{Compose}
\begin{Compose}{0}{0}
\sdownterminal{A}{0,2} \thispoint{p}{0,-2} \jointbnoarrow{p}{0}{A}{0}
\end{Compose}
\end{equation}

\begin{equation}
\begin{Compose}{0}{0}
\crectangle{A}{0.8}{0.8}{-2,0} \csymbol{R}\thispoint{p}{2,0} \joinrlnoarrowthick{A}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}
\sleftterminal{A}{-2, 0} \thispoint{p}{2,0} \joinrlnoarrowthick{A}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}
\crectangle{A}{0.8}{0.8}{2,0} \csymbol{R} \thispoint{p}{-2,0} \joinrlnoarrowthick{p}{0}{A}{0}
\end{Compose}
\begin{Compose}{0}{0}
\srightterminal{A}{2,0} \thispoint{p}{-2, 0} \joinrlnoarrowthick{p}{0}{A}{0}
\end{Compose}
\end{equation}

This notation is not used here however.

§.§ Duotensor formalism

In Part <ref> we discuss the duotensor formalism. This acts as an intermediary between the operation picture and the operator picture.   I have discussed this in previous papers [49, 51]. However, I introduce new structure here so that both the pointer types and the system types have duotensor structure associated with them.  The idea is that we introduce fiducial elements

\begin{equation}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\sleftdatafid{F}{0,0} \blackdotsq{FL}{-2.5,0} \csymbolthird[-24,0]{x}
\relpoint{F}{2.5,0}{FR} \csymbolalt[15,0]{x}
\joinrlnoarrowthick[above]{F}{0}{FR}{0}  \joinrlnoarrow[above]{FL}{0}{F}{0}
\end{Compose}
\begin{Compose}{0}{0}
\Crectangle{R}{0.7}{0.7}{-3,0}
\crectangle{x}{0.6}{0.6}{0,0} \csymbolthird{x}
\relpoint{x}{3,0}{xR}
%\joinrlnoarrowthick[above]{T}{0}{R}{0} \csymbol[0,5]{x}
\joinrlnoarrowthick[above]{R}{0}{x}{0} \csymbol[0,5]{x}
\joinrlnoarrowthick[above]{x}{0}{xR}{0} \csymbol[0,5]{x}
\end{Compose}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\srightdatafid{F}{0,0} \relpoint{F}{-2.5,0}{FL} \csymbolalt[-18,0]{x}
\blackdotsq{FR}{2.5,0} \csymbolthird[21,0]{x}
\joinrlnoarrow[above]{F}{0}{FR}{0}  \joinrlnoarrowthick{FL}{0}{F}{0}
\end{Compose}
\begin{Compose}{0}{0}
\Crectangle{R}{0.7}{0.7}{3,0}
\crectangle{x}{0.6}{0.6}{0,0} \csymbolthird{x}
\relpoint{x}{-3,0}{xR}
%\joinrlnoarrowthick[above]{T}{0}{R}{0} \csymbol[0,5]{x}
\joinlrnoarrowthick[above]{R}{0}{x}{0} \csymbol[0,5]{x}
\joinlrnoarrowthick[above]{x}{0}{xR}{0} \csymbol[0,5]{x}
\end{Compose}
\end{equation}

for pointers and

\begin{equation}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb} \setthirdfont{\mathpzc}
\supsysfid{A}{0,0}  \thispoint{p}{0,2.5} \csymbolalt[0,20]{x} \jointbnoarrow{A}{0}{p}{0} \blackdot{d}{0,-2.5} \csymbolthird[0,-23]{x}\joinbtnoarrow{A}{0}{d}{0}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb} \setthirdfont{\mathpzc}
\sdownsysfid{A}{0,0}  \thispoint{p}{0,-2.5} \csymbolalt[0,-20]{x} \joinbtnoarrow{A}{0}{p}{0} \blackdot{d}{0,2.5} \csymbolthird[0,23]{x}\jointbnoarrow{A}{0}{d}{0}
\end{Compose}
\end{equation}

for systems.  Using linearity and equivalence, as discussed in Part <ref>, we can represent an operation as a sum over fiducial elements as follows

\begin{equation}\label{introGeneralOperationExpansion}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb} \setthirdfont{\mathpzc}  \setdefaultfont{\mathtt}
\crectangle{T}{3}{3}{0,0} \csymbol{\mathsf{C}}
\thispoint{y7}{-7,1.5}   \csymbol[-24,0]{y} \joinrlnoarrowthick[above]{y7}{0}{T}{1.5}
\thispoint{x1}{-7,-1.5}   \csymbol[-24,0]{x} \joinrlnoarrowthick[above]{x1}{0}{T}{-1.5}
\thispoint{w8}{7,1.5}   \csymbol[24,0]{v} \joinrlnoarrowthick[above]{T}{1.5}{w8}{0}
%\thispoint{w6}{7,0}   \csymbol[24,0]{w} \joinrlnoarrowthick[above]{T}{0}{w6}{0}
\thispoint{y2}{7,-1.5}   \csymbol[24,0]{u} \joinrlnoarrowthick[above]{T}{-1.5}{y2}{0}
\upwire{T}{-1.5}{c}  \upwire{T}{1.5}{d}
\downwire{T}{-1.5}{a} \downwire{T}{1.5}{b}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb} \setthirdfont{\mathpzc} \setdefaultfont{\mathtt}
\crectangle{T}{2.5}{2.5}{0,0} \csymbolfourth{C}
\srightdatafid{y7}{-7,1.5}   \joinrlnoarrowbw[above]{y7}{0}{T}{1.5}\csymbolfourth[0,10]{y}
\thispoint{yL7}{-9.5,1.5}   \csymbol[-24,0]{y}  \joinrlnoarrowthick{yL7}{0}{y7}{0}
\srightdatafid{x1}{-7,-1.5}  \joinrlnoarrowbw[above]{x1}{0}{T}{-1.5} \csymbolfourth[0,10]{x}
\thispoint{xL1}{-9.5,-1.5} \csymbol[-24,0]{x} \joinrlnoarrowthick[above]{xL1}{0}{x1}{0}
\sleftdatafid{w8}{7,1.5}  \joinrlnoarrowwb[above]{T}{1.5}{w8}{0} \csymbolfourth[0,10]{v}
\thispoint{wR8}{9.5,1.5} \csymbol[24,0]{v}  \joinrlnoarrowthick[above]{w8}{0}{wR8}{0}
%\sleftdatafid{w6}{7,0}   \joinrlnoarrowwb[above]{T}{0}{w6}{0} \csymbolthird[0,10]{w}
%\thispoint{wR6}{9.5,0} \csymbolalt[24,0]{w}  \joinrlnoarrowthick[above]{w6}{0}{wR6}{0}
\sleftdatafid{y2}{7,-1.5} \joinrlnoarrowwb[above]{T}{-1.5}{y2}{0}  \csymbolfourth[0,10]{u}
\thispoint{yR2}{9.5,-1.5} \csymbol[24,0]{u}  \joinrlnoarrowthick[above]{y2}{0}{yR2}{0}
\fiddyupwire{T}{-1.5}{c}  \fiddyupwire{T}{1.5}{d}
\fiddydownwire{T}{-1.5}{a} \fiddydownwire{T}{1.5}{b}
\end{Compose}
\end{equation}

so long as we have tomographic locality.  Whenever there is an adjacent black and white dot we sum over the corresponding variable.  In this equation we have expanded the operation in terms of fiducials using a duotensor with all white dots.   Using the ideas of linearity and equivalence discussed in Part <ref> we see that we can represent an operation as a sum over fiducial elements.

§.§ The operator tensor formalism

In Part <ref> we discuss the operator tensor formalism [51, 52].  As with duotensors, I introduce new structure to allow both pointer and system types to be represented.    An operator tensor can be represented diagrammatically as follows

\begin{equation}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb} \setthirdfont{\mathpzc}  \setdefaultfont{\mathtt}
\crectangledouble{T}{3}{3}{0,0} \csymbolfourth{\hat{C}}
\thispoint{y7}{-7,1.5}   \csymbol[-24,0]{y} \joinrlnoarrowthick[above]{y7}{0}{T}{1.5}
\thispoint{x1}{-7,-1.5}   \csymbol[-24,0]{x} \joinrlnoarrowthick[above]{x1}{0}{T}{-1.5}
\thispoint{w8}{7,1.5}   \csymbol[24,0]{v} \joinrlnoarrowthick[above]{T}{1.5}{w8}{0}
%\thispoint{w6}{7,0}   \csymbol[24,0]{w} \joinrlnoarrowthick[above]{T}{0}{w6}{0}
\thispoint{y2}{7,-1.5}   \csymbol[24,0]{u} \joinrlnoarrowthick[above]{T}{-1.5}{y2}{0}
\upwire{T}{-1.5}{c}  \upwire{T}{1.5}{d}
\downwire{T}{-1.5}{a} \downwire{T}{1.5}{b}
\end{Compose}
\end{equation}

Operator tensors play the role of density operators and completely positive maps in more standard approaches (though they do so in such a way that a calculation for the probability associated with a circuit is represented by a picture that looks the same as that circuit).  Before coming to operator tensors we provide a treatment of Hilbert spaces equipped with input and output structure which is, pretty much, equivalent to that presented in Coecke and Kissinger [20] though presented in new notation so it melds with previously adopted notation for operator tensors.  For example, an example of an operator tensor is

\begin{equation}
\begin{Compose}{0}{0}\setdefaultfont{} \setsecondfont{\mathsfb}
\crectangledouble{A}{2.5}{2.3}{0,0} \csymbol{\hat{B}}
\relpoint{A}{0,-4.5}{Ad} \jointbnoarrow[right]{Ad}{0}{A}{0} \csymbolalt{a}
\relpoint{A}{1,4.5}{Cd} \jointbnoarrow[right]{A}{1}{Cd}{0} \csymbolalt{a}
\relpoint{A}{-1,4.5}{Ed} \jointbnoarrow[right]{A}{-1}{Ed}{0} \csymbolalt{b}
\end{Compose}
\begin{Compose}{0}{0}\setdefaultfont{} \setsecondfont{\mathsfb}
\crectangledleft{A}{1.5}{2.3}{-3,0} \csymbol{B}
\relpoint{A}{0,-4.5}{Ad} \jointbleft[right]{Ad}{0}{A}{0} \csymbolalt{a}
\relpoint{A}{1,4.5}{Cd} \jointbleft[right]{A}{1}{Cd}{0} \csymbolalt{a}
\relpoint{A}{-1,4.5}{Ed} \jointbleft[right]{A}{-1}{Ed}{0} \csymbolalt{b}
\crectangledright{AA}{1.5}{2.3}{3,0} \csymbol{B}
\relpoint{AA}{0,-4.5}{Ad} \jointbright[right]{Ad}{0}{AA}{0} \csymbolalt{a}
\relpoint{AA}{1,4.5}{Cd} \jointbright[right]{AA}{1}{Cd}{0} \csymbolalt{b}
\relpoint{AA}{-1,4.5}{Ed} \jointbright[right]{AA}{-1}{Ed}{0} \csymbolalt{a}
\joinrlnoarrow[above]{A}{1.7}{AA}{1.7}\csymbol{a}
\joinrlnoarrow[above]{A}{0}{AA}{0}\csymbol[0,3]{b}
\joinrlnoarrow[above]{A}{-1.7}{AA}{-1.7}\csymbol{a}
\end{Compose}
\end{equation}

Each of the two halves are Hilbert space objects.  The horizontal wires connecting the two halves on the right represent summation over the associated indices.
This input output Hilbert space machinery is very useful as it allows us to see under the hood" and prove theorems.

We can restrict to the case of Hermitian operator tensors for which there exists an expansion in terms of fiducial operators having the form

\begin{equation}\label{introGeneralOperatorExpansion}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb} \setthirdfont{\mathpzc}  \setdefaultfont{\mathtt}
\crectangledouble{T}{3}{3}{0,0} \csymbolfourth{\hat{C}}
\thispoint{y7}{-7,1.5}   \csymbol[-24,0]{y} \joinrlnoarrowthick[above]{y7}{0}{T}{1.5}
\thispoint{x1}{-7,-1.5}   \csymbol[-24,0]{x} \joinrlnoarrowthick[above]{x1}{0}{T}{-1.5}
\thispoint{w8}{7,1.5}   \csymbol[24,0]{v} \joinrlnoarrowthick[above]{T}{1.5}{w8}{0}
%\thispoint{w6}{7,0}   \csymbol[24,0]{w} \joinrlnoarrowthick[above]{T}{0}{w6}{0}
\thispoint{y2}{7,-1.5}   \csymbol[24,0]{u} \joinrlnoarrowthick[above]{T}{-1.5}{y2}{0}
\upwire{T}{-1.5}{c}  \upwire{T}{1.5}{d}
\downwire{T}{-1.5}{a} \downwire{T}{1.5}{b}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb} \setthirdfont{\mathpzc} \setdefaultfont{\mathtt}
\crectangle{T}{2.5}{2.5}{0,0} \csymbolfourth{C}
\srightvecfid{y7}{-7,1.5}   \joinrlnoarrowbw[above]{y7}{0}{T}{1.5}\csymbolfourth[0,10]{y}
\thispoint{yL7}{-9.5,1.5}   \csymbol[-24,0]{y}  \joinrlnoarrowthick{yL7}{0}{y7}{0}
\srightvecfid{x1}{-7,-1.5}  \joinrlnoarrowbw[above]{x1}{0}{T}{-1.5} \csymbolfourth[0,10]{x}
\thispoint{xL1}{-9.5,-1.5} \csymbol[-24,0]{x} \joinrlnoarrowthick[above]{xL1}{0}{x1}{0}
\sleftvecfid{w8}{7,1.5}  \joinrlnoarrowwb[above]{T}{1.5}{w8}{0} \csymbolfourth[0,10]{v}
\thispoint{wR8}{9.5,1.5} \csymbol[24,0]{v}  \joinrlnoarrowthick[above]{w8}{0}{wR8}{0}
\sleftvecfid{y2}{7,-1.5} \joinrlnoarrowwb[above]{T}{-1.5}{y2}{0}  \csymbolfourth[0,10]{u}
\thispoint{yR2}{9.5,-1.5} \csymbol[24,0]{u}  \joinrlnoarrowthick[above]{y2}{0}{yR2}{0}
\fidopupwire{T}{-1.5}{c}  \fidopupwire{T}{1.5}{d}
\fidopdownwire{T}{-1.5}{a} \fidopdownwire{T}{1.5}{b}
\end{Compose}
\end{equation}

Thus, duotensors act as a bridge between operations and operator tensors in the following sense. From the expansion in (<ref>) we obtain a duotensor.  Then, we can use this duotensor to form an operator tensor using (<ref>).   As long as certain conditions are met on the fiducials from the operation theory and the fiducials from the operator theory, we are guaranteed that the circuits built from operator tensors will give the correct probabilities.

In Part <ref> we see that there are gauge parameters, $\alpha_\mathsf{x}$ associated with providing a concrete representation of operator tensors.  For example, the ignore operators can be written

\begin{equation}
\begin{Compose}{0}{-0.07}\setdefaultfont{\hat}\setsecondfont{\mathtt}\setthirdfont{\mathsfb}
\Crectangledouble{I}{1}{1}{0,0}
\thispoint{p}{0,3.5}  \csymbolthird[0,20]{x} \jointbnoarrow{I}{0}{p}{0}
\end{Compose}
~=~ \frac{\alpha_\mathsfb{x}}{\sqrt{N_\mathsfb{x}}} \hat{\mathds{1}}^{\mathsfb{x}_1}
\begin{Compose}{0}{-0.07}\setdefaultfont{\hat}\setsecondfont{\mathtt}\setthirdfont{\mathsfb}
\Crectangledouble{I}{1}{1}{0,0}
\thispoint{p}{0,-3.5}  \csymbolthird[0,-20]{x} \joinbtnoarrow{I}{0}{p}{0}
\end{Compose}
~=~ \frac{1}{\alpha_\mathsfb{x}\sqrt{N_\mathsfb{x}}} \hat{\mathds{1}}_{\mathsfb{x}_1}
\end{equation}

We are free to choose any value for $\alpha_\mathsf{x}$.  The standard choice in the usual (time asymmetric) formulation of Quantum Theory is $\alpha_\mathsfb{a}=\frac{1}{\sqrt{N_\mathsfb{a}}}$.  This corresponds to a forwards point of view.  Other choices are discussed in Sec. <ref>.  This story corresponds to physical systems.   There is a similar story corresponding to pointer types and the associated gauge parameter is $\beta_\mathtt{x}$.

§.§ Time symmetric operational Quantum Theory

In Part <ref> we finally come to the time symmetric operational formulation of Quantum Theory given by a single mathematical quantum axiom.  To state this axiom we require two notions (a) correspondence and (b) physicality.  We say we have correspondence from operations to operators if there is an operator associated with every operation such that if we replace operations by operators in a circuit then this operator circuit gives the probability for the circuit.  We prove the correspondence theorem that if

\begin{equation}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\blackdotsq{FL}{-2.5,0} \csymbolthird[-24,0]{x}
\sleftdatafid{F}{0,0}
\srightdatafid{rF}{3,0}
\blackdotsq{rFR}{5.5,0} \csymbolthird[24,0]{x}
\joinrlnoarrow[above]{FL}{0}{F}{0}
\joinrlnoarrowthick[above]{F}{0}{rF}{0} \csymbolalt{x}
\joinrlnoarrow[above]{rF}{0}{rFR}{0}
\end{Compose}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\blackdotsq{FL}{-2.5,0} \csymbolthird[-24,0]{x}
\sleftvecfid{F}{0,0}
\srightvecfid{rF}{3,0}
\blackdotsq{rFR}{5.5,0} \csymbolthird[24,0]{x}
\joinrlnoarrow[above]{FL}{0}{F}{0}
\joinrlnoarrowthick[above]{F}{0}{rF}{0} \csymbolalt{x}
\joinrlnoarrow[above]{rF}{0}{rFR}{0}
\end{Compose}
~:=~ \begin{Compose}{0}{0}
\bbmatrixnoarrowsq{h}{0,0} \csymbolthird{x}
\end{Compose}
\end{equation}

\begin{equation}
\begin{Compose}{0}{0} \setsecondfont{\mathsfb} \setfourthfont{\mathpzc}
\blackdot{u}{0,3.5} \csymbolfourth[0,20]{x}\sdownsysfid{up}{0,1.5} \supsysfid{do}{0,-1.5} \blackdot{d}{0,-3.5}\csymbolfourth[0,-20]{x}
\jointbnoarrow[left]{d}{0}{do}{0} \jointbnoarrow[left]{do}{0}{up}{0} \csymbolalt{x} \jointbnoarrow[left]{up}{0}{u}{0}
\end{Compose}
\begin{Compose}{0}{0} \setsecondfont{\mathsfb} \setfourthfont{\mathpzc}
\blackdot{u}{0,3.5} \csymbolfourth[0,20]{x}\sdownopfid{up}{0,1.5} \supopfid{do}{0,-1.5} \blackdot{d}{0,-3.5}\csymbolfourth[0,-20]{x}
\jointbnoarrow[left]{d}{0}{do}{0} \jointbnoarrow[left]{do}{0}{up}{0} \csymbolalt{x} \jointbnoarrow[left]{up}{0}{u}{0}
\end{Compose}
\begin{Compose}{0}{0} \setfourthfont{\mathpzc}
\blackdot{u}{0,1.5}\blackdot{d}{0,-1.5} \jointbnoarrow[left]{d}{0}{u}{0}\csymbolfourth{x}
\end{Compose}
\end{equation}

then operations correspond to operators whose expansion is in terms of the same duotensor (compare (<ref>) with (<ref>)).

Next we translate physicality to the operator tensor setting.  Physicality now demands that we have $T$ positivity for operator tensors

\begin{equation}
\begin{Compose}{0}{0}\setdefaultfont{}\setsecondfont{\mathsfb}\setthirdfont{\mathtt}
\crectangledouble{B}{1.5}{1.5}{0,0} \csymbol{\hat{B}}
\thispoint{d}{0,-3} \csymbolalt[0,-20]{a}
\thispoint{u}{0,3}  \csymbolalt[0,20]{b}
\thispoint{l}{-3,0} \csymbolthird[-20,0]{x}
\thispoint{r}{3,0}  \csymbolthird[20,0]{y}
\jointbnoarrow{d}{0}{B}{0}
\jointbnoarrow{B}{0}{u}{0}
\joinrlnoarrowthick{l}{0}{B}{0}
\joinrlnoarrowthick{B}{0}{r}{0}
\end{Compose}
\end{equation}

and double causality

\begin{equation}
\Funitaldoublexyab{1}{1}{1}{1} ~~~~~~~~~~~~  \Bunitaldoubleyxba{1}{1}{1}{1}
\end{equation}

The $T$ positivity condition is the same as demanding that $\hat{B}$ is positive after we take the transpose on the input (or output) space.

Once we have the notions of correspondence and physicality we can state a mathematical axiom for Quantum Theory. This is as follows

Quantum Axiom.
Every physical operator has an operation corresponding to it.

This provides a time symmetric operational formulation of Quantum Theory.  We provide a proof of this in Sec. <ref>.

In the standard formalism of Quantum Theory an important result is the Stinespring extension theorem.  This shows how any permissible operator corresponding to an operation can be modeled as a unitary acting on the original system and an ancilla prepared in some initial state and then traced out afterwards.  This theorem is not applicable here since the initial ancilla would have to be prepared in a totally mixed state to respect double causality.  We prove an extension theorem that works in this time symmetric setting.  This says that physical operators can be modeled as follows

\begin{equation}\label{extensiontheoremequationintro}
\begin{Compose}{0}{0}\setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\crectangledouble{U}{2}{2}{0,0} \csymbol{B} \thispoint{AL}{-4,0} \csymbolalt[-20,0]{x} \thispoint{AR}{4,0} \csymbolalt[20,0]{y}
\joinlrnoarrowthick{U}{0}{AL}{0} \joinrlnoarrowthick{U}{0}{AR}{0}
\thispoint{a}{0,-4} \csymbolthird[0,-20]{a} \jointbnoarrow{a}{0}{U}{0}
\thispoint{d}{0,4} \csymbolthird[0,20]{b} \joinbtnoarrow{d}{0}{U}{0}
\end{Compose}
\begin{Compose}{0}{0} \setdefaultfont{}\setsecondfont{\mathsfb}\setthirdfont{\mathtt}
\crectangledouble{U}{3}{2}{0,0} \csymbol{\hat{U}_\mathsf{B}}
\relpoint{U}{-0.3,4.5}{Iupos}  \crectangledouble{Iu}{0.9}{0.9}{Iupos} \csymbol{\hat{I}}  \jointbnoarrow[left]{U}{-0.3}{Iu}{0}  \csymbolalt{z}
\relpoint{U}{0.3,-4.5}{Idpos}  \crectangledouble{Id}{0.9}{0.9}{Idpos} \csymbol{\hat{I}} \jointbnoarrow[left]{Id}{0}{U}{0.3} \csymbolalt[-3,0]{w}
\relpoint{U}{2.3,4.5}{Ypos}  \crectangledouble{Y}{0.9}{0.9}{Ypos} \csymbol{\hat{Y}}  \jointbnoarrow[left]{U}{2.3}{Y}{0}  \csymbolalt{y}
\relpoint{Y}{4,0}{r} \joinlrnoarrowthick[above]{r}{0}{Y}{0} \csymbolthird[0,3]{y}
\relpoint{U}{-2.3,-4.5}{Xpos}  \crectangledouble{X}{0.9}{0.9}{Xpos} \csymbol{\hat{X}}  \jointbnoarrow[left]{X}{0}{U}{-2.3} \csymbolalt{x} \relpoint{X}{-4,0}{l} \joinrlnoarrowthick[above]{l}{0}{X}{0} \csymbolthird{x}
\thispoint{down}{0,-7} \thispoint{up}{0,7}
\jointbnoarrow[right]{down}{2.3}{U}{2.3} \csymbolalt{a}
\jointbnoarrow[left]{U}{-2.3}{up}{-2.3} \csymbolalt{b}
\end{Compose}
\end{equation}

where $\hat{U}$ is unitary, and $\hat{X}$ and $\hat{Y}$ are maximal.  The proof of this uses the Hilbert space machinery adapted from Coecke and Kissinger.

Finally, we show that this formalism is equivalent to the standard (time asymmetric) operational formalism in Sec. <ref>.

Basic operational considerations

§ TIME SYMMETRIC OPERATIONS

In operational theories the central object is the operation.  An operation corresponds to a single use of an apparatus.  Operations can be wired together to form circuits.

§.§ The standard notion of an operation

The standard notion of an operation in operational theories can be depicted as follows.

\[
\begin{Compose}{0}{0}
\crectangle{A}{3}{2}{0,0}
\thispoint{AL}{-6,0} \Thistexthere{setting}{-8.3,0} \thispoint{AR}{6,0} \Thistexthere{outcome}{8.9,0}
\thispoint{A1in}{-2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A2in}{0,-5} %\csymbolalt[-0,-15]{b}
\thispoint{A3in}{2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A1out}{-2,5}%\csymbolalt[0,15]{a}
\thispoint{A2out}{0,5}
\thispoint{A3out}{2,5} %\csymbolalt[0,15]{c}
\Thistexthere{inputs}{0,-7.1} \Thistexthere{outputs}{0,6.8}
\joinrlthick{AL}{0}{A}{0}  \joinrlthick{A}{0}{AR}{0}
\jointb{A1in}{0}{A}{-1.5} \jointb{A2in}{0}{A}{0} \jointb{A3in}{0}{A}{1.5}
\jointb{A}{-1.5}{A1out}{0} \jointb{A}{1.5}{A3out}{0}
\end{Compose}
\]

Settings are controlled by knobs, etc.   These are classical.  Outcomes correspond to pointer readings, detector clicks, etc.  These are also classical.
Inputs and outputs are for the physical systems (e.g. quantum systems like electrons, photons, …).

If we think about this a little more carefully, we should really depict an operation as follows.

\[
\begin{Compose}{0}{0}
\crectangle{A}{3}{2}{0,0}
\thispoint{AL}{-6,0} \thispoint{AR}{6,0}
\thispoint{A1in}{-2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A2in}{0,-5} %\csymbolalt[-0,-15]{b}
\thispoint{A3in}{2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A1out}{-2,5}%\csymbolalt[0,15]{a}
\thispoint{A2out}{0,5}
\thispoint{A3out}{2,5} %\csymbolalt[0,15]{c}
\Thistexthere{inputs}{0,-7.1} \Thistexthere{outputs}{0,6.8}\relpoint{AL}{0,-1}{ALm} \relpoint{AR}{0,-1}{ARm}
\joinrlthick{AL}{-1}{A}{-1}  \joinrlthick{A}{-1}{AR}{-1}
\linebyhand[thin, dashed]{ALm}{ARm}  \Thistexthere{setting}{-8.3,-1} \Thistexthere{setting}{8.4,-1}
\joinrlthick{A}{1}{AR}{1}   \Thistexthere{outcome}{8.9,1}
\jointb{A1in}{0}{A}{-1.5} \jointb{A2in}{0}{A}{0} \jointb{A3in}{0}{A}{1.5}
\jointb{A}{-1.5}{A1out}{0} \jointb{A}{1.5}{A3out}{0}
\end{Compose}
\]

Settings are available before and after so we now show this in the diagram.  Outcomes are available afterwards but not before.   We can now see clearly that this basic idea of an operation is not time-symmetric. We have, right at the very beginning of our theoretical development of the subject, injected a basic time-asymmetry.

§.§ Time symmetric operations

Fortunately, it is clear from the above picture what we have to do to make it time symmetric.

\[
\begin{Compose}{0}{0}
\crectangle{A}{3}{2.5}{0,0}
\thispoint{AL}{-6,0} \thispoint{AR}{6,0}
\thispoint{A1in}{-2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A2in}{0,-5} %\csymbolalt[-0,-15]{b}
\thispoint{A3in}{2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A1out}{-2,5}%\csymbolalt[0,15]{a}
\thispoint{A2out}{0,5}
\thispoint{A3out}{2,5} %\csymbolalt[0,15]{c}
\Thistexthere{inputs}{0,-7.1} \Thistexthere{outputs}{0,6.8}  %\relpoint{AL}{0,-1}{ALm} \relpoint{AR}{0,-1}{ARm}
\joinrlthick{AL}{0}{A}{0}  \joinrlthick{A}{0}{AR}{0}
\linebyhand[thin, dashed]{AL}{AR}  \Thistexthere{setting}{-8.3,0} \Thistexthere{setting}{8.4,0}
\joinrlthick{AL}{-1.5}{A}{-1.5}   \Thistexthere{income}{-8.4,-1.5}
\joinrlthick{A}{1.5}{AR}{1.5}   \Thistexthere{outcome}{8.9,1.5}
\jointb{A1in}{0}{A}{-1.5} \jointb{A2in}{0}{A}{0} \jointb{A3in}{0}{A}{1.5}
\jointb{A}{-1.5}{A1out}{0} \jointb{A}{1.5}{A3out}{0}
\end{Compose}
\]

Now we have incomes as well as outcomes.  Within the confines of an operation, an income is available before but not after (this is not to say that some communication structure extrinsic to the operation will not communicate this forward in time - this will be discussed later in Sec. <ref>).  Like outcomes, incomes are classical.  Note that we use the English word income" in a way different from its usual modern usage (which has a financial meaning).  In fact, the original usage of the word is consistent with a broader definition (of something that comes in").

In this paper we will be primarily concerned with incomes and outcomes so we will suppress the settings

\[
\begin{Compose}{0}{0}
\Crectangle{A}{3}{2}{0,0}
\thispoint{AL}{-6,0} \thispoint{AR}{6,0}
\thispoint{A1in}{-2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A2in}{0,-5} %\csymbolalt[-0,-15]{b}
\thispoint{A3in}{2,-5} %\csymbolalt[-0,-18]{a}
\thispoint{A1out}{-2,5}%\csymbolalt[0,15]{a}
\thispoint{A2out}{0,5}
\thispoint{A3out}{2,5} %\csymbolalt[0,15]{c}
\Thistexthere{inputs}{0,-7.1} \Thistexthere{outputs}{0,6.8}  %\relpoint{AL}{0,-1}{ALm} \relpoint{AR}{0,-1}{ARm}
%\joinrlthick{AL}{0}{A}{0}  \joinrlthick{A}{0}{AR}{0}
%\linebyhand[thin, dashed]{AL}{AR}  \Thistexthere{setting}{-8.3,0} \Thistexthere{setting}{8.4,0}
\joinrlthick{AL}{0}{A}{0}   \Thistexthere{income}{-8.4,0}
\joinrlthick{A}{0}{AR}{0}   \Thistexthere{outcome}{8.9,0}
\jointb{A1in}{0}{A}{-1} \jointb{A2in}{0}{A}{0} \jointb{A3in}{0}{A}{1}
\jointb{A}{-1}{A1out}{0} \jointb{A}{1}{A3out}{0}
\end{Compose}
\]

The setting is now taken to be implicit in the symbol $\mathsf A$ (so if we have a different setting we would put a different symbol here).

We will associate symbols with the incomes, outcomes, inputs, and outputs as shown in this example

\[
\begin{Compose}{0}{-0.1}\setsecondfont{\mathtt}\setthirdfont{\mathsfb}
\Crectangle{A}{3}{2}{0,0}
\thispoint{AL}{-6,0} \csymbolalt[-15,0]{x} \thispoint{AR}{6,0} \csymbolalt[15,0]{y}
\thispoint{A1in}{-2,-5} \csymbolthird[-0,-18]{a} \thispoint{A2in}{0,-5} \csymbolthird[-0,-15]{b}\thispoint{A3in}{2,-5} \csymbolthird[-0,-18]{a}
\thispoint{A1out}{-2,5}\csymbolthird[0,15]{a} \thispoint{A2out}{0,5} \thispoint{A3out}{2,5} \csymbol[0,15]{c}
%\Thistexthere{inputs}{0,-7.2} \Thistexthere{outputs}{0,7}
\joinrlnoarrowthick{AL}{0}{A}{0}  \joinrlnoarrowthick{A}{0}{AR}{0}
\jointbnoarrow{A1in}{0}{A}{-1.5} \jointbnoarrow{A2in}{0}{A}{0} \jointbnoarrow{A3in}{0}{A}{1.5}
\jointbnoarrow{A}{-1.5}{A1out}{0} \jointbnoarrow{A}{1.5}{A3out}{0}
\end{Compose}
\mathsf{A}_{\mathtt{x}_1\mathsfb{a}_3\mathsfb{b}_4\mathsfb{a}_5}^{\mathtt{y}_2\mathsfb{a}_6\mathsfb{c}_7}
\]

We use $\mathtt x$, $\mathtt y$, etc. to denote income/outcome types (this is LaTeX  font).  For the sake of having a name, we will call these pointer types.  They are classical and take values

\[
x=1, 2, \dots N_\mathtt{x}  ~~~~~~~~
y=1, 2, \dots N_\mathtt{y}
\]

We call the types associated with inputs and outputs system types.
We use $\mathsfb{a}$, $\mathsfb{b}$, etc. to denote these (this is LaTeX  font).  These physical systems could be quantum systems.  We use LaTeX font  to denote the operation ($\mathsf A$ in the above example).   We give the diagrammatic notation for the operation on the left. On the right we give the symbolic notation wherein incomes and inputs are represented by subscripts and outcomes and outputs are represented by superscripts.  In the symbolic notation we also include integer labels on these subscripts and superscripts.  These label the wires as will be explained in Sec. <ref>.

A special type pointer type is the null pointer type, which we will denote by $\mathtt{0}$.  The null pointer type has $N_\mathtt{0}=1$ so it only has one value associated with it (namely $x=1$).   The null type can be useful.  For example, if we have an operation with no incomes, then we can append the null income without changing anything.  Then we can apply mathematical conditions that refer to incomes.

§.§ Terminology for incomes and outcomes in standard Quantum Theory

In the usual terminology of Quantum Theory we have examples of incomes and outcomes:

Incomes. Here are examples of incomes taken from the usual way of thinking

* Initial pointer states (from maximal set)

* Initial ancilla state (from maximal set)

Outcomes. Here are examples of outcomes taken from the usual way of thinking

* Final pointer state (from maximal set)

* Postselection on ancilla (onto element of a maximal set).

Note that, usually, we take the initial pointer state to be in some fixed zero state.  However, we may more correctly need to take it to be one of a set of income states.   These examples are the reason for adopting the terminology pointer types.  However, the notion of incomes and outcomes is a more general idea (or at least, it can represent other things not normally regarded as pointers).

§ JOINING OPERATIONS TOGETHER

§.§ Joining rules

When we wire operations together there are two rules:

* We must match types.  Outputs must be matched with inputs of the same type. Outcomes must be matched with incomes of the same type.

* There must be no closed loops.  Thus, if we start on any particular operation, it must be impossible to get back to the same operation by following wires forward in time (whether they are pointer or system wires).

The second rule means that we have a directed acyclic graph.

§.§ Notation

If we wire together a bunch of operations we can notate it diagrammatically or symbolically as shown in this example.

\begin{equation}\label{fragmentexample1}
\begin{Compose}{0}{0}\setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{A}{1.5}{1.5}{0,0}  \relpoint{A}{-4,0}{AL}\csymbolalt[-20,0]{x} \relpoint{A}{0,-4}{AD}\csymbolthird[0,-20]{b}
\joinbtnoarrow{A}{0}{AD}{0} \joinlrnoarrowthick{A}{0}{AL}{0}
\Crectangle{B}{1.5}{1.5}{-3,6} \relpoint{B}{-4,0}{BL} \csymbolalt[-20,0]{z}
\joinlrnoarrowthick{B}{0}{BL}{0}
\Crectangle{C}{1.5}{1.5}{4,8}  \relpoint{C}{4,0}{CR} \csymbolalt[20,0]{y}  \relpoint{C}{0,4}{CU}\csymbolthird[0,20]{a}
\jointbnoarrow{C}{0}{CU}{0} \joinrlnoarrowthick{C}{0}{CR}{0}
\jointbnoarrow[below left]{A}{-1}{B}{0} \csymbolthird{a}
\jointbnoarrow[below right]{A}{1}{C}{0} \csymbolthird{b}
\joinrlnoarrowthick[above left]{B}{0}{C}{0}  \csymbolalt{x}
\end{Compose}
\mathsf{A}_{\mathtt{x_1}\mathsfb{b}_5}^{~\mathsfb{a}_6\mathsfb{b}_7} \mathsf{B}_{\mathtt{z_3}\mathsfb{a}_6}^{\mathtt{x_2}\mathsfb{c}_8}
\mathsf{C}_{\mathtt{x_2}\mathsfb{b}_7}^{\mathtt{y_4}\mathsf{a}_9}
\end{equation}

In the symbolic notation on the left we have the same integer label for any given wire.  For example $\mathsfb{b}_7$ leaves $\mathsf A$ and enters $\mathsf{C}$ and so is repeated in the symbolic notation.  The integers are only labels. The diagram has the same meaning if we relabel them with different integers (as long as distinct labels remain distinct).

§.§ Circuits and readout boxes

A circuit is where we wire together operations and have no open wires left over.  For example,

\begin{equation} %\label{ABCcircuit}
\begin{Compose}{0}{-2.4} \setsecondfont{\mathsfb}
\crectangle{A}{2}{2}{0,0} \csymbol{A}  \crectangle{B}{2}{2}{5,9} \csymbol{B} \crectangle{C}{2}{2}{2,18} \csymbol{C}
\thispoint{amid}{-5,9}  \csymbolalt[-20,0]{a}
\jointbnoarrow[left]{A}{-1}{amid}{0}\jointbnoarrow{amid}{0}{C}{-1}
%\jointbnoarrow[left]{A}{-1}{C}{-1} \csymbolalt[-5,0]{a}
\jointbnoarrow[below right]{A}{1}{B}{0}  \csymbolalt[5,-5]{b}
\jointbnoarrow[above right]{B}{0}{C}{1}  \csymbolalt{a}
\GenxBoxincome{A}{2}{0}{x}{E}
\GenxBoxoutcome{B}{2}{0}{y}{F}
\GenxBoxoutcome{C}{2}{0}{z}{G}
\end{Compose}
\end{equation}

The small boxes with an $x$, $y$, and $z$ in this circuit are readout boxes.  They can be thought of as reading the income/outcome they are attached to.  They have the property

\begin{equation}\label{xboxprop}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\crectangle{AL}{0.7}{0.8}{-6,0} \csymbolthird{x} \relpoint{AL}{-2.5,0}{ALL} \csymbolalt[-18,0]{x}
\crectangle{AR}{0.7}{0.8}{-2,0} \csymbolthird{x'}
\relpoint{AR}{2.5,0}{ARR} \csymbolalt[15,0]{x}
\joinrlnoarrowthick{ALL}{0}{AL}{0}
\joinrlnoarrowthick[above]{AL}{0}{AR}{0}   \csymbolalt[0,4]{x}
\joinrlnoarrowthick{AR}{0}{ARR}{0}
\end{Compose}
\equiv
\left\{
\begin{array}{ll}
\begin{Compose}{0}{-0.1}\setsecondfont{\mathtt}
\crectangle{A}{0.6}{0.7}{0,0} \csymbolthird{x} \relpoint{A}{-2.5,0}{AL} \csymbolalt[-18,0]{x}
\relpoint{A}{2.5,0}{AR} \csymbolalt[15,0]{x}
\joinlrnoarrowthick{A}{0}{AL}{0}
\joinrlnoarrowthick{A}{0}{AR}{0}
\end{Compose}  &
\text{if} ~~x=x'  \\
\begin{Compose}{0}{-0.1}\setsecondfont{\mathtt}
\crectangle{A}{0.6}{0.7}{0,0} \csymbolthird{0} \relpoint{A}{-2.5,0}{AL} \csymbolalt[-18,0]{x}
\relpoint{A}{2.5,0}{AR} \csymbolalt[15,0]{x}
\joinlrnoarrowthick{A}{0}{AL}{0}
\joinrlnoarrowthick{A}{0}{AR}{0}
\end{Compose} &
\text{if} ~~x\not=x'
\end{array}
\right.
\end{equation}

since, if reading out the value twice is the same as reading it out once.  Here, the null box,

\begin{equation}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\crectangle{A}{0.6}{0.7}{0,0} \csymbolthird{0} \relpoint{A}{-2.5,0}{AL} \csymbolalt[-18,0]{x}
\relpoint{A}{2.5,0}{AR} \csymbolalt[15,0]{x}
\joinlrnoarrowthick{A}{0}{AL}{0}
\joinrlnoarrowthick{A}{0}{AR}{0}
\end{Compose}
\end{equation}

has the property that, if it is added to any circuit, that circuit has probability equal to 0.

§.§ Fragments and their causal structure

A fragment is where we have some wires left open (fragment" here means fragment of a circuit). An example of a fragment is shown in (<ref>).

Any fragment, $\mathsf F$, has a certain causal structure indicating the temporal relationships between the open wires which determines the set, $S_\mathsf{F}$,  of fragments, $\mathsf E$, with complementary causal structure such that we can join $\mathsf F$ and any $\mathsf{E}\in S_\mathsf{F}$ such that we obtain a circuit.  This is a useful notion we will make use of later.    We say two fragments, $\mathsf A$ and $\mathsf B$, have compatible causal structure if $S_\mathsf{A}\cap S_\mathsf{B}\not=\emptyset$.  This means there exist fragments that complete both $\mathsf A$ and $\mathsf B$ into circuits.

§.§ The circuit probability assumption

We make the following assumption

Circuit probability assumption:  Every circuit (wherein all inputs and outputs and all incomes and outcomes are closed) has a probability associated with it that depends only on the specification of that circuit.

A similar assumption was used in earlier work [51].  When we have time symmetry, this assumption is more intriguing because we demand we get the same answer whether we calculate the probability forward in time or backwards in time.

We denote a circuit by $\mathsf{A}$, $\mathsf{B}$, …(since these are simply operations without any open wires).  Let us denote the readouts associated with circuit $\mathsf A$ by $x_\mathsf{A}$ and the specification of circuit $\mathsf A$ by $s_\mathsf{A}$.  The specification, $s_\mathsf{A}$, details the operations used and the arrangement of wires between these operations.  In this notation we denote the probability associated with circuit $\mathsf A$ by

\begin{equation}
\text{prob}(\mathsf{A}) = \text{prob}(x_\mathsf{A}|s_\mathsf{A})
\end{equation}

The circuit probability assumption can now be written as

\begin{equation}\label{extraneous}
\text{prob}(x_\mathsf{A}|s_\mathsf{A}, \text{extraneous conditions})= \text{prob}(x_\mathsf{A}|s_\mathsf{A})
\end{equation}

where $\text{extraneous conditions}$ are conditions that do not pertain to circuit $\mathsf A$.

§.§ Probabilities for circuits factorize

If we have a circuit formed from two separate circuits then it is easy to show that the probabilities factorise

\begin{align*}
\text{prob}(\mathsf{AB}) & =  \text{prob}(x_\mathsf{A},x_\mathsf{B}|s_\mathsf{A},s_\mathsf{B})  \\
&= \text{prob}(x_\mathsf{A}|x_\mathsf{B}s_\mathsf{A},s_\mathsf{B})\text{prob}(x_\mathsf{B}|s_\mathsf{A},s_\mathsf{B}) \\
&= \text{prob}(x_\mathsf{A}|s_\mathsf{A})\text{prob}(x_\mathsf{B}|s_\mathsf{B}) \\
&= \text{prob}(\mathsf{A})\text{prob}(\mathsf{B})
\end{align*}

In going from the second to the third line of the proof we have used the circuit probability assumption - see (<ref>).
This factorization property is very important when we come to set up the duotensor approach in Part <ref>

§.§ Equivalence

We will say that two fragments, $\mathsf A$ and $\mathsf B$, are equivalent if they have compatible causal structure and give rise to the same probabilities when completed into a circuit by a fragment with complementary causal structure

\begin{equation}
\mathsf{A}\equiv \mathsf{B} ~~~\text{iff} ~~~ \text{prob}(\mathsf{AE}) = \text{prob}(\mathsf{BE}) ~~ \forall ~~ \mathsf{E} \in S_\mathsf{A}\cap S_\mathsf{B}
\end{equation}

We are suppressing subscripts and superscripts here for ease of expression.  Later, in Sec. <ref>, we will introduce a more general notion of equivalence which linearly extends the present definition to weighted sums of fragments.

§.§ Equivalence concerning readout boxes

A useful property of readout boxes is the following equivalence

\begin{equation}\label{sumoverx}
\sum_x
\begin{Compose}{0}{-0.1}\setsecondfont{\mathtt}
\crectangle{A}{0.6}{0.7}{0,0} \csymbolthird{x} \relpoint{A}{-2.5,0}{AL} \csymbolalt[-18,0]{x}
\relpoint{A}{2.5,0}{AR} \csymbolalt[15,0]{x}
\joinlrnoarrowthick{A}{0}{AL}{0}
\joinrlnoarrowthick{A}{0}{AR}{0}
\end{Compose}
\equiv
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\thispoint{X}{-4,0}
\thispoint{Y}{4,0}
\joinrlnoarrowthick[above]{X}{0}{Y}{0}\csymbolalt[0,5]{x}
\end{Compose}
\end{equation}

This is true because, if we sum over the $x$ variables, we are taking the marginal of a joint probability.

§.§ Definition of time symmetric operational theory

We will say an operational theory is time symmetric if,

* Every allowed operation, $\mathsf A$, can be mapped to a time reversed" operation, $\utilde{\mathsf A}$, which is also allowed, wherein input and output types are swapped as are income and outcome types.    For example,

\begin{equation}
\begin{Compose}{0}{-0.1}\setsecondfont{\mathtt}\setthirdfont{\mathsfb}
\Crectangle{A}{3}{2}{0,0}
\thispoint{AL}{-6,0} \csymbolalt[-15,0]{x} \thispoint{AR}{6,0} \csymbolalt[15,0]{y}
\thispoint{A1in}{-2,-5} \csymbolthird[-0,-18]{a} \thispoint{A2in}{0,-5} \csymbolthird[-0,-15]{b}\thispoint{A3in}{2,-5} \csymbolthird[-0,-18]{a}
\thispoint{A1out}{-2,5}\csymbolthird[0,15]{a} \thispoint{A2out}{0,5} \thispoint{A3out}{2,5} \csymbolthird[0,15]{c}
%\Thistexthere{inputs}{0,-7.2} \Thistexthere{outputs}{0,7}
\joinrlnoarrowthick{AL}{0}{A}{0}  \joinrlnoarrowthick{A}{0}{AR}{0}
\jointbnoarrow{A1in}{0}{A}{-1.5} \jointbnoarrow{A2in}{0}{A}{0} \jointbnoarrow{A3in}{0}{A}{1.5}
\jointbnoarrow{A}{-1.5}{A1out}{0} \jointbnoarrow{A}{1.5}{A3out}{0}
\end{Compose}
~~~ \extleftrightarrow[\text{reverse}]{\text{time}} ~~~
\begin{Compose}{0}{-0.1}\setsecondfont{\mathtt}\setthirdfont{\mathsfb}
\crectangle{A}{3}{2}{0,0} \csymbol{\utilde{A}}
\thispoint{AL}{-6,0} \csymbolalt[-15,0]{y} \thispoint{AR}{6,0} \csymbolalt[15,0]{x}
\thispoint{A1in}{-2,-5} \csymbolthird[-0,-18]{a} \thispoint{A2in}{0,5} \csymbolthird[0,15]{b}\thispoint{A3in}{2,-5} \csymbolthird[-0,-18]{c}
\thispoint{A1out}{-2,5}\csymbolthird[0,15]{a} \thispoint{A2out}{0,5} \thispoint{A3out}{2,5} \csymbolthird[0,15]{a}
%\Thistexthere{inputs}{0,-7.2} \Thistexthere{outputs}{0,7}
\joinrlnoarrowthick{AL}{0}{A}{0}  \joinrlnoarrowthick{A}{0}{AR}{0}
\jointbnoarrow{A1in}{0}{A}{-1.5} \joinbtnoarrow{A2in}{0}{A}{0} \jointbnoarrow{A3in}{0}{A}{1.5}
\jointbnoarrow{A}{-1.5}{A1out}{0} \jointbnoarrow{A}{1.5}{A3out}{0}
\end{Compose}
\end{equation}

Note readout boxes are already time symmetric

\[
\begin{Compose}{0}{-0.1}\setsecondfont{\mathtt}
\crectangle{A}{0.6}{0.7}{0,0} \csymbolthird{x} \relpoint{A}{-2.5,0}{AL} \csymbolalt[-18,0]{x}
\relpoint{A}{2.5,0}{AR} \csymbolalt[15,0]{x}
\joinlrnoarrowthick{A}{0}{AL}{0}
\joinrlnoarrowthick{A}{0}{AR}{0}
\end{Compose}
~~~ \extleftrightarrow[\text{reverse}]{\text{time}} ~~~
\begin{Compose}{0}{-0.1}\setsecondfont{\mathtt}
\crectangle{A}{0.6}{0.7}{0,0} \csymbolthird{x} \relpoint{A}{-2.5,0}{AL} \csymbolalt[-18,0]{x}
\relpoint{A}{2.5,0}{AR} \csymbolalt[15,0]{x}
\joinlrnoarrowthick{A}{0}{AL}{0}
\joinrlnoarrowthick{A}{0}{AR}{0}
\end{Compose}
\]

This follows from their definition.

* Given any circuit, $\mathsf E$, we can obtain the time reversed circuit, $\utilde{\mathsf E}$, by inverting the graph replacing operations with their time reversed counterparts. We require that $\text{prob}(\mathsf{E})= \text{prob}(\utilde{\mathsf{E}})$.  For example,

\[
\text{prob}\left(
\scalebox{0.75}{
\begin{Compose}{0}{-2.4} \setsecondfont{\mathsfb}
\crectangle{A}{2}{2}{0,0} \csymbol{A}  \crectangle{B}{2}{2}{5,9} \csymbol{B} \crectangle{C}{2}{2}{2,18} \csymbol{C}
\thispoint{amid}{-5,9}  \csymbolalt[-20,0]{a}
\jointbnoarrow[left]{A}{-1}{amid}{0}\jointbnoarrow{amid}{0}{C}{-1}
%\jointbnoarrow[left]{A}{-1}{C}{-1} \csymbolalt[-5,0]{a}
\jointbnoarrow[below right]{A}{1}{B}{0}  \csymbolalt[5,-5]{b}
\jointbnoarrow[above right]{B}{0}{C}{1}  \csymbolalt{a}
\GenxBoxincome{A}{2}{0}{x}{E}
\GenxBoxoutcome{B}{2}{0}{y}{F}
\GenxBoxoutcome{C}{2}{0}{z}{G}
\end{Compose}  }
\right)
\text{prob}\left( \hspace{-3mm}
\scalebox{0.75}{
\begin{Compose}{0}{2.4} \setsecondfont{\mathsfb}
\crectangle{A}{2}{2}{0,0} \csymbol{\utilde{A}}  \crectangle{B}{2}{2}{5,-9} \csymbol{\utilde{B}} \crectangle{C}{2}{2}{2,-18} \csymbol{\utilde{C}}
\thispoint{amid}{-5,-9}  \csymbolalt[-20,0]{a}
\joinbtnoarrow[left]{A}{-1}{amid}{0}\joinbtnoarrow{amid}{0}{C}{-1}
\joinbtnoarrow[above right]{A}{1}{B}{0}  \csymbolalt[5,-5]{b}
\joinbtnoarrow[below right]{B}{0}{C}{1}  \csymbolalt{a}
\GenxBoxoutcome{A}{2}{0}{x}{\utilde{E}}
\GenxBoxincome{B}{2}{0}{y}{\utilde{F}}
\GenxBoxincome{C}{2}{0}{z}{\utilde{G}}
\end{Compose} }
\right)
\]

The circuit on the right is the time reverse of the circuit on the left and has the same probability.

This means that, for every process described in the forward direction, there is a corresponding process described in the backwards direction with the same predictions.

This definition is different from that of Leifer and Pusey in [62] which concerned conditional probabilities rather than joint probabilities.

A simple classical situation

§ A CIRCUIT AS A SIMPLE CLASSICAL SITUATION

Consider wiring together some operations so that there are no physical systems left open.  For example,

\begin{equation}\label{circuitexample}
\begin{Compose}{0}{-2.5}\setdefaultfont{\mathsf}\setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{A}{1.5}{1.2}{0,0}
\relpoint{A}{-4,0}{AL}\csymbolalt[-20,0]{x}  \joinrlnoarrowthick[above]{AL}{0}{A}{0}
\relpoint{A}{4,0}{AR}  \csymbolalt[20,0]{y}\joinrlnoarrowthick[above]{A}{0}{AR}{0}
\Crectangle{D}{1.5}{1.2}{4,18}
\relpoint{D}{-4,0}{DL}  \csymbolalt[-20,0]{y}\joinrlnoarrowthick[above]{DL}{0}{D}{0}
\relpoint{D}{4,0}{DR} \csymbolalt[20,0]{w} \joinrlnoarrowthick[above]{D}{0}{DR}{0}
\Crectangle{B}{1.5}{1.2}{-6, 7.5}
\relpoint{B}{-4,0}{BL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick[above]{BL}{0}{B}{0}
\relpoint{B}{4,0}{BR} \csymbolalt[20,0]{x} \joinrlnoarrowthick[above]{B}{0}{BR}{0}
\Crectangle{C}{1.5}{1.2}{10,11}
\relpoint{C}{-4,0}{CL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick[above]{CL}{0}{C}{0}
\relpoint{C}{4,0}{CR} \csymbolalt[20,0]{w}\joinrlnoarrowthick[above]{C}{0}{CR}{0}
%\joinrlnoarrowthick[above left]{B}{0}{C}{0} \csymbolalt{x}
\jointbnoarrow[above left]{B}{0}{D}{-1} \csymbolthird[-4,0]{c} \jointbnoarrow[above right]{C}{0}{D}{1} \csymbolthird[5,0]{b}
\jointbnoarrow[below left]{A}{-1}{B}{0} \csymbolthird[-5,0]{a}
\jointbnoarrow[below right]{A}{1}{C}{0} \csymbolthird[4,0]{c}
\end{Compose}
\end{equation}

This is equivalent to

\begin{equation}\label{Mmanyports}
\begin{Compose}{0}{0}\setdefaultfont{\mathsf}\setsecondfont{\mathsf}
\crectangle{T}{3}{4}{0,0} \csymbol{M}
\thispoint{y7}{-5,3}   \csymbol[-20,0]{y} \joinrlnoarrowthick[above]{y7}{0}{T}{3}
\thispoint{x4}{-5,1}  \csymbol[-20,0]{x} \joinrlnoarrowthick[above]{x4}{0}{T}{1}
\thispoint{x3}{-5,-1}   \csymbol[-20,0]{x} \joinrlnoarrowthick[above]{x3}{0}{T}{-1}
\thispoint{x1}{-5,-3}   \csymbol[-20,0]{x} \joinrlnoarrowthick[above]{x1}{0}{T}{-3}
\thispoint{w8}{5,3}   \csymbol[20,0]{w} \joinrlnoarrowthick[above]{T}{3}{w8}{0}
\thispoint{w6}{5,1}   \csymbol[20,0]{w} \joinrlnoarrowthick[above]{T}{1}{w6}{0}
\thispoint{x5}{5,-1} \csymbol[20,0]{x} \joinrlnoarrowthick[above]{T}{-1}{x5}{0}
\thispoint{y2}{5,-3}   \csymbol[20,0]{y} \joinrlnoarrowthick[above]{T}{-3}{y2}{0}
\end{Compose}
\end{equation}

Here $\mathsf M$ is has classical incomes and classical outcomes. However, it has physics (possibly quantum) hidden inside.
We can simplify this situation further as

\begin{equation}\label{simpleclassicalsituation}
\begin{Compose}{0}{0}
\crectangle{M}{3}{2}{0,0} \csymbol{M}
\thispoint{u}{-6,0} \csymbol[-20,0]{u}
\thispoint{v}{6,0} \csymbol[20,0]{v}
\joinrlnoarrowthick[above]{u}{0}{M}{0}
\joinrlnoarrowthick[above]{M}{0}{v}{0}
\end{Compose}
\end{equation}

where we define the composite pointer types $\mathtt{u}=\mathtt{xxxy}$ and $\mathtt{v}=\mathtt{yxww}$.

The incredibly simple classical situation shown in (<ref>) is the key to understanding time-symmetry in operational theories (including Quantum Theory).  To this end we will, in this Part of the paper, describe a number of double" properties pertaining to this situation.
Included are the following five related double properties

Double determinism. Probabilities add up to one in both directions.

Double distribution uniqueness. Income/outcome distribution pairs are unique.

Double pointer causality. We cannot send information in either time direction.

Double flatness. A flat distribution in gives flat distribution out in either time direction.

Double summation.  This is a mathematical condition resulting from double determinism and double flatness.

We will explain these in more detail in what follows.  We use the word double" since, in each case, the double property is comprised of a forward" and a backward" property which are the time reverse of one another.

§ DOUBLE DETERMINISM

For the simple situation in (<ref>) we define

Forward determinism is the condition that

\begin{equation} \sum_v p_\mathsf{M}(v|u) = 1  \end{equation}

This says that, for each $u$ on the income, the probabilities for the different outcomes add to one.

Backward determinism is the condition that

\begin{equation} \sum_u p_\mathsf{M}(u|v) = 1  \end{equation}

This says that, for each $v$ on the outcome, the probabilities for the different incomes add to one.

We say $\mathsf M$ is doubly deterministic if it is both forward and backward deterministic.   Any operation, $\mathsf M$, that has no readout boxes (explicit or hidden), is taken to be doubly deterministic.

If $\mathsf M$ has only an outcome we call it a pointer preparation. This case can be treated by making $\mathtt u$ the null type (discussed in Sec. <ref>).  This means $N_\mathtt{u}=1$. Then we must necessarily have $u=1$ and so we can drop the sum over $u$ in the backwards causality condition.  Then the conditions for double determinism are $\sum_v p_\mathsf{M}(v|1) = 1$ and $p_\mathsf{M}(1|v)=1$.  We must necessarily have $p_\mathsf{M}(1|v)=1$ since there are no other incomes.  Hence, a pointer preparation is doubly deterministic if and only if it is forward deterministic.

If $\mathsf M$ has only an income we call it a pointer result (we use the word result" in the time reversed sense to the use of the word preparation"). By similar reasoning,  a pointer result is doubly deterministic if and only if it is backwards deterministic.

We have defined the notion of double determinism for operations having only pointer types (but no system types).   We can extend the notion to general operations (that may have system types).  We will say that an operation which has no readouts (either explicit or implicit) is doubly deterministic.  We assume that, when we wire together such operations so that there are no system types left open, as illustrated in (<ref>), then we obtain a pointer type operation that is doubly deterministic in the sense defined above.  We will assume that operations are doubly deterministic unless otherwise stated.   Note that the readout box is not doubly deterministic.

§ POINTER PREPARATIONS AND RESULTS

§.§ Pointer type closing operations

For the operation $\mathsf M$ shown in (<ref>), we can close the income, the outcome.   To do this we need pointer type closing operations having the form

\begin{equation}\label{pointerclosings}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{R}{0.7}{0.9}{0,0} \csymbol{E}\thispoint{p}{4,0} \csymbolalt[20,0]{u} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\crectangle{R}{0.7}{0.9}{0,0}  \csymbol{F} \thispoint{p}{-4,0} \csymbolalt[-20,0]{v} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\end{equation}

The operation on the left is a pointer preparation.   The operation on the right is a pointer result.

Closing $\mathsf M$ with a pointer preparation gives

\[
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{M}{3}{2}{0,0} \csymbol{M}
\crectangle{u}{0.7}{0.9}{-6,0} \csymbol{E}
\thispoint{v}{6,0} \csymbolalt[20,0]{v}
\joinrlnoarrowthick[above]{u}{0}{M}{0}\csymbolalt{u}
\joinrlnoarrowthick[above]{M}{0}{v}{0}
\end{Compose}
\]

which is another pointer preparation.  Closing $\mathsf M$ with a pointer result gives

\[
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\crectangle{M}{3}{2}{0,0} \csymbol{M}
\thispoint{u}{-6,0} \csymbolalt[-20,0]{u}
\crectangle{v}{0.7}{0.9}{6,0} \csymbol{F}
\joinrlnoarrowthick[above]{u}{0}{M}{0}
\joinrlnoarrowthick[above]{M}{0}{v}{0}\csymbolalt{v}
\end{Compose}
\]

which is another pointer result.

As a consequence of assumptions we introduce, we will see that the only allowed pointer preparations correspond to the flat distribution over the outcome variables (when regarded forward in time).  Similarly, we will see that the only allowed pointer results correspond to the flat distribution over the income variables (when regarded backward in time).

§.§ Double pointer tomography

We say that a set of probabilities, pertaining to some given operation, are tomographically complete if all operations that are equivalent to the given operation have the same values for the probabilities in this set.  In this case, we will say that the set of probabilities fully specify the given operation.

Consider a circuit,

\begin{equation}
\begin{Compose}{0}{-0.06} \setsecondfont{\mathtt}
\crectangle{RR}{0.7}{0.9}{0,0}\csymbol{E}
\GenxBoxoutcome{RR}{0.9}{0}{x}{F}
\end{Compose}
\end{equation}

Forward pointer tomography is the property that any pointer preparation, $\mathsf{E}$, is fully specified by the $N_\mathtt{x}$ probabilities

\begin{equation}\label{Nxprobabilities}
p(x) = \left(
\begin{Compose}{0}{-0.06} \setsecondfont{\mathtt}
\crectangle{RR}{0.7}{0.9}{0,0}\csymbol{E}
\GenxBoxoutcome{RR}{0.9}{0}{x}{F}
\end{Compose}
\right)
\end{equation}

if $\mathsf F$ is doubly deterministic (which, following the discussion in  Sec. <ref> is the same as demanding that $\mathsf F$ is backwards deterministic since it has no outcomes).  Backward pointer tomography is the property that any pointer result, $\mathsf F$, is fully specified by the $N_\mathtt{x}$ probabilities in (<ref>) if $\mathsf E$ is doubly deterministic (which is the same as demanding that $\mathsf E$ is forward deterministic).  Double pointer tomography is the property that both forward and backward pointer tomography hold.

It is noteworthy that, if both $\mathsf E$ and $\mathsf F$ are doubly deterministic, then they are both fully specified by the same probability distribution.  This issue will be explored in greater generality in Sec. <ref> below.

§.§ Circuits

We can close all incomes and outcomes in our example shown in (<ref>) as follows.

\begin{equation}\label{circuitexampleclosed}
\begin{Compose}{0}{-2.5}\setdefaultfont{\mathsf}\setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{A}{1.5}{1.2}{0,0}
\GenxdifBoxincome{A}{1.5}{0}{x}{x}{E}   %\relpoint{A}{-4,0}{AL}\csymbolalt[-20,0]{x}  \joinrlnoarrowthick[above]{AL}{0}{A}{0}
\GenxdifBoxoutcome{A}{1.5}{0}{y}{y}{F}  %\relpoint{A}{4,0}{AR}  \csymbolalt[20,0]{y}\joinrlnoarrowthick[above]{A}{0}{AR}{0}
\Crectangle{D}{1.5}{1.2}{4,18}
\GenxdifBoxincome{D}{1.5}{0}{y}{y'}{G}  %\relpoint{D}{-4,0}{DL}  \csymbolalt[-20,0]{y}\joinrlnoarrowthick[above]{DL}{0}{D}{0}
\GenxdifBoxoutcome[1.2]{D}{1.5}{0}{w}{w'}{H}   %\relpoint{D}{4,0}{DR} \csymbolalt[20,0]{w} \joinrlnoarrowthick[above]{D}{0}{DR}{0}
\Crectangle{B}{1.5}{1.2}{-6, 7.5}
\GenxdifBoxincome{B}{1.5}{0}{x}{x'}{E}  %\relpoint{B}{-4,0}{BL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick[above]{BL}{0}{B}{0}
\GenxdifBoxoutcome[1.2]{B}{1.5}{0}{x}{x''}{K}%\relpoint{B}{4,0}{BR} \csymbolalt[20,0]{x} \joinrlnoarrowthick[above]{B}{0}{BR}{0}
\Crectangle{C}{1.5}{1.2}{10,11}
\GenxdifBoxincome[1.45]{C}{1.5}{0}{x}{x'''}{Q}%\relpoint{C}{-4,0}{CL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick[above]{CL}{0}{C}{0}
\GenxdifBoxoutcome{C}{1.5}{0}{w}{w}{M}%\relpoint{C}{4,0}{CR} \csymbolalt[20,0]{w}\joinrlnoarrowthick[above]{C}{0}{CR}{0}
%\joinrlnoarrowthick[above left]{B}{0}{C}{0} \csymbolalt{x}
\jointbnoarrow[above left]{B}{0}{D}{-1} \csymbolthird[-4,0]{c} \jointbnoarrow[above right]{C}{0}{D}{1} \csymbolthird[5,0]{b}
\jointbnoarrow[below left]{A}{-1}{B}{0} \csymbolthird[-5,0]{a}
\jointbnoarrow[below right]{A}{1}{C}{0} \csymbolthird[4,0]{c}
\end{Compose}
\end{equation}

We can write this circuit as

\[
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{E}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{F}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose}
\]

where we have followed steps similar to those in Sec. <ref>.  The form of circuit shown in (<ref>) is not the most general because we can allow an outcome from one operation to feed into the income of another operation.  It turns out that such examples can be modeled by circuits of the form in (<ref>) if we include appropriate multiplicative factors. This is discussed in Sec. <ref>.

§.§ Double uniqueness

Consider again the circuit

\[
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{E}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{F}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose}
\]

We are interested in calculating

\begin{equation}\label{probuv}
\text{prob}(u, v)
\end{equation}

How do we actually calculate the probability associated with this circuit?    We will consider two ways of doing this. We can do the calculation forwards in time" by starting off with a distribution over the initial $u$ variables and evolve it forwards in time.  Or we can do the calculation backwards in time" by starting off with a distribution over the final variables and evolve this backwards in time.   By the circuit assumption we must get the same answer whether we do the calculation forward or backward in time. We will see that, if we are not careful, we run into problems with this.
Doing the calculation forwards we obtain

\[p(u,v) = p_\mathsf{E}(u) p_\mathsf{M}(v|u)\]

Doing the calculation backwards we obtain

\[p(u,v) = p_\mathsf{F}(v) p_\mathsf{M}(u|v)\]

\begin{equation}\label{EMFequation}
p_\mathsf{E}(u) p_\mathsf{M}(v|u) = p_\mathsf{F}(v) p_\mathsf{M}(u|v)
\end{equation}

Consider, instead, the situation

\[
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{E'}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{F'}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose}
\]

By similar reasoning we obtain

\begin{equation}\label{EpMFpequation}
p_\mathsf{E'}(u) p_\mathsf{M}(v|u) = p_\mathsf{F'}(v) p_\mathsf{M}(u|v)
\end{equation}

Dividing (<ref>) by (<ref>) gives

\[ \frac{p_\mathsf{E}(u)}{p_\mathsf{E'}(u)} = \frac{p_\mathsf{F}(v)}{p_\mathsf{F'}(v)}  \]

unless $p_\mathsf{M}(u|v)=0$ or $p_\mathsf{M}(v|u)=0$.
Using double determinism we obtain that

\begin{equation} \label{agreeuniqueness}
p_\mathsf{E}(u) = p_\mathsf{E'}(u) ~~~~\text{and}~~~~ p_\mathsf{F}(v) = p_\mathsf{F'}(v)
\end{equation}

Hence, for $u$ and $v$ having non-zero conditional probabilities, there is a unique distribution pair $p_\mathsf{E}(u)$ and $p_\mathsf{F}$ that agree with $\mathsf{M}$ given that we demand we get the same joint probability $p(u,v)$ whether we do the calculation forwards or backwards in time.  We will call this property double uniqueness since we have a unique distribution coming in from the past and a unique distribution coming in from the future.

There is another property related to double uniqueness.   To see this let us write

\[p_\text{forwards}(u,v) = p_\mathsf{E}(u) p_\mathsf{M}(v|u)\]

for the forwards calculation and

\[p_\text{backwards}(u,v) = p_\mathsf{F}(v) p_\mathsf{M}(u|v)\]

for the backwards calculation.  According to the circuit probability assumption we have

\begin{equation}
p_\text{forwards}(u,v) = p_\text{backwards}(u,v)
\end{equation}

If we sum this over $u$ and use backwards determinism, we obtain

\[ p_\mathsf{F}(v) = p(v|\mathsf{E,M}) ~~~~~\text{where}~~~~~~  p(v|\mathsf{E,M})= \sum_u p_\mathsf{E}(u) p_\mathsf{M}(v|u) \]

Similarly if we sum over $v$ and use forwards determinism, we obtain

\[ p_\mathsf{E}(u) = p(u|\mathsf{M, F}) ~~~~~\text{where}~~~~~~ p(u|\mathsf{M, F})= \sum_v p_\mathsf{F}(v) p_\mathsf{M}(u|v)   \]

In the first case we see that the forward evolved distribution agrees with the final distribution. In the second case we see that the backward evolved distribution agrees with the initial distribution.  We call this double agreement.  This accords with the remarks at the end of Sec. <ref>.

§.§ Disagreements and secrets

Double uniqueness and double agreement appear, on the face of it, to be obviously wrong. Assume, for the sake of argument, that $p_\mathsf{M}(u|v)$ and $p_\mathsf{M}(v|u)$ are strictly positive for all $u$ and $v$ so we have a unique matching pair. Now we can simply send in a distribution, $p_\mathsf{G}(u)$, that is not part of this matching pair.   We must, surely, obtain some distribution, $p_\mathsf{H}(v)$, on the other side.  Now, by construction, we have a pair of probability distributions where, at least, $p_\mathsf{G}(u)$ is not in the unique matching pair.  What has gone wrong?

To disentangle this, first we will sharpen the apparent contradiction. We have

\[p_\text{forwards}(u,v) = p_\mathsf{E}(u) p_\mathsf{M}(v|u)\]

for the forwards calculation and

\[p_\text{backwards}(u,v) = p_\mathsf{F}(v) p_\mathsf{M}(u|v)\]

for the backwards calculation.  According to the circuit probability assumption we have

\begin{equation}
p_\text{forwards}(u,v) = p_\text{backwards}(u,v)
\end{equation}

Consequently, we obtain

\[ p_\mathsf{F}(v) = p(v|\mathsf{E,M})  := \sum_u p_\mathsf{E}(u) p_\mathsf{M}(v|u) \]

\[ p_\mathsf{E}(u) = p(u|\mathsf{M, F}) := \sum_v p_\mathsf{F}(v) p_\mathsf{M}(u|v)   \]

Now we will look at an explicit example.   Consider the quantum circuit

\begin{equation}
\begin{Compose}{0}{0} \setsecondfont{\mathsfb}\setthirdfont{\mathtt}
\Crectangle{U}{0.8}{0.8}{0,-3} \Crectangle{V}{0.8}{0.8}{0,3}
\jointbnoarrow[right]{U}{0}{V}{0} \csymbolalt{a}
\GenxBoxincome{U}{0.8}{0}{u}{E}
\GenxBoxoutcome{V}{0.8}{0}{v}{F}
\end{Compose}
\end{equation}

where $\mathsf U$ prepares quantum system $\mathsfb s$ in spin up state ($u=+1$) and spin down state($u=-1$) along the $\mathbf u$ direction and $\mathsf{V}$ prepares spin up state ($v=+1$) and spin down state ($v=-1$) along the $\mathbf v$ axis. Here $\mathbf u$ and $\mathbf v$ are taken to be unit vectors.   We can collapse this down to the circuit

\[
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{E}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{F}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose}
\]

We can now easily calculate that

\begin{equation}
p_\mathsf{M}(v|u) = \frac{1}{2}\left(1+uv\mathbf{u}\cdot \mathbf{v}\right) = p_\mathsf{M}(u|v)
\end{equation}

Consider the case where $\mathbf{u}\cdot \mathbf{v}=0.75$.  We have

\begin{equation} \arraycolsep=1.4pt\def\arraystretch{1.3}
p_\mathsf{M}(v|u) = \left( \begin{array}{cc}
\frac{7}{8} & \frac{1}{8} \\
\frac{1}{8} & \frac{7}{8}
\end{array}  \right)
= p_\mathsf{M}(u|v)
\end{equation}

Note there are no zero entries in $p_\mathsf{M}(v|u)$ and $p_\mathsf{M}(u|v)$ so there exists a unique matching pair.

First consider the special case where $p_\mathsf{E}(+1)=1$ and $p_\mathsf{E}(-1)=0$ which we write as

\begin{equation}\label{pEuexample}
p_\mathsf{E}(u) =  \left(\begin{array}{c}
1 \\
\end{array}  \right)
\end{equation}

Then, evolving forward, we obtain

\begin{equation} \arraycolsep=1.4pt\def\arraystretch{1.3}
p(v|\mathsf{E,M}) =\sum_u p_\mathsf{M}(v|u) p_\mathsf{E}(u) =\left(\begin{array}{c}
\frac{7}{8} \\
\frac{1}{8}
\end{array}  \right)
\end{equation}

where we can use matrix multiplication to obtain this result.  As noted above, we require $p(v|\mathsf{E,M}) = p_\mathsf{F}(v)$
Hence, we can put

\begin{equation} \arraycolsep=1.4pt\def\arraystretch{1.3}
p_\mathsf{F}(v) =\sum_u p_\mathsf{M}(v|u) p_\mathsf{E}(u) =\left(\begin{array}{c}
\frac{7}{8} \\
\frac{1}{8}
\end{array}  \right)
\end{equation}

We can now evolve this backwards in time and we obtain

\begin{equation} \arraycolsep=1.4pt\def\arraystretch{1.3}
p(u|\mathsf{M,F}) =\sum_v p_\mathsf{M}(u|v) p_\mathsf{F}(v) =\left(\begin{array}{c}
\frac{50}{64} \\
\frac{14}{64}
\end{array}  \right)
\end{equation}

But, this contradicts the requirement that $p_\mathsf{E}(u)=p(u|\mathsf{E,M})$.  This is the disagreement.

One way to resolve this disagreement by allowing the preparation, $\mathsf E$, to have a secret income denoted by $\mathtt s$:

\begin{equation}
\begin{Compose}{0}{0} \setsecondfont{\mathsfb}\setthirdfont{\mathtt}
\Crectangle{E}{0.8}{0.8}{0,0}
\thispoint{l}{-3,0}\thispoint{r}{3,0}
\joinrlnoarrowthick[above]{l}{0}{E}{0} \csymbolthird{s}
\joinrlnoarrowthick[above]{E}{0}{r}{0} \csymbolthird{u}
\end{Compose}
\end{equation}

where we suppose that the income takes values $s=0$ and $s=1$.  Furthermore, we suppose that,

\begin{equation}\label{psEuexample}
p_\mathsf{E}(u|s=1) =  \left(\begin{array}{c}
1 \\
\end{array}  \right)
\end{equation}

so that the preparation we are interested in is actually only implemented when we have $s=1$.   In this case we are interested in $p(u,v|s=1)$. We can calculate if forward in time as

\begin{equation}
p_\text{forwards}(u,v|s=1) = p_\mathsf{E}(u|s=1)p_\mathsf{M}(v|u)
\end{equation}

and backward in time as

\begin{equation}
p_\text{backwards}(u,v|s=1) = \frac{p_\text{backwards}(u,v,s=1)}{p_\mathsf{E}(s=1)}
= \frac{p_\mathsf{F}(v)p_\mathsf{M}(u|v) p_\mathsf{E}{s=1|u} }{p_\mathsf{E}(s=1)}
\end{equation}

If we now equate $p_\text{forwards}(u,v|s=1)=p_\text{backwards}(u,v|s=1)$ we do not obtain a double uniqueness theorem as before.  It is worth thinking a little bit about the physics of what is happening here.  We could think of $s=1$ as corresponding to pressing the button that prepares $u=1$ thereby preparing the spin up state along the $\mathbf u$ direction.  If we time reverse this but send in from the future a spin down state then the dynamics (now evolved in reverse) will not necessarily undo the button press and so we have a different possible income, namely $s=0$.  This discussion illustrates that it is important we correctly characterize boxes in terms of their incomes and outcomes in representing physical situations within this theoretical framework.

§ DOUBLE POINTER CAUSALITY AND DOUBLE FLATNESS

§.§ Flat pointer operations

We define two special pointer operations, the flat pointer operations, that are taken to be the time reverse of one another. These are

\begin{equation}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,0} \thispoint{p}{4,0} \csymbolalt[20,0]{x} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
~~~ \extleftrightarrow[\text{reverse}]{\text{time}} ~~~
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,0}  \thispoint{p}{-4,0} \csymbolalt[-20,0]{x} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\end{equation}

Strictly, we should use the notation $\utilde{\mathsf R}$ for one of these two operations (as per the notation introduced in Sec. <ref>). However, it is clear from the position of the wire that they are distinct so we omit the tilde.  We preserve the symbol $\mathsf R$ to refer to the flat pointer operations only.  We define the flat pointer operations by means of the following properties

\begin{equation}\label{Rflatcond}
\text{prob}\left(
\begin{Compose}{0}{-0.06} \setsecondfont{\mathtt}
\crectangle{RR}{0.7}{0.9}{0,0}\csymbol{R}
\RxBoxoutcome{RR}{0.9}{0}{x}
\end{Compose}\right)
= \frac{1}{N_\mathtt{x}}
\end{equation}

\begin{equation}\label{Rdeterministiccond}
\text{prob}\left(
\begin{Compose}{0}{-0.06} \setsecondfont{\mathtt}
\crectangle{R}{0.7}{0.9}{0,0} \csymbol{R}
\crectangle{RR}{0.7}{0.9}{4,0} \csymbol{R}
\joinrlnoarrowthick{R}{0}{RR}{0} \csymbolalt{x}
\end{Compose}\right)
\end{equation}

Using double pointer tomography and the property that probabilities over disjoint circuits factorize, we can prove the factorization properties

\begin{equation}\label{factorization}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,0} \thispoint{p}{4,0} \csymbolalt[25,0]{xy} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,-1.3} \thispoint{p}{4,-1.3} \csymbolalt[20,0]{x} \joinrlnoarrowthick{R}{0}{p}{0}
\Crectangle{R}{0.7}{0.9}{0,1.3} \thispoint{p}{4,1.3} \csymbolalt[20,0]{y} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,0}  \thispoint{p}{-4,0} \csymbolalt[-25,0]{xy} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,-1.3}  \thispoint{p}{-4,-1.3} \csymbolalt[-20,0]{x} \joinlrnoarrowthick{R}{0}{p}{0}
\Crectangle{R}{0.7}{0.9}{0,1.3}  \thispoint{p}{-4,1.3} \csymbolalt[-20,0]{x} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\end{equation}

This is clear since, if we form the operations,

\begin{equation}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{R}{0.7}{0.9}{0,0} \csymbol{Q} \thispoint{p}{4,0} \csymbolalt[25,0]{xy} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,-1.3} \thispoint{p}{4,-1.3} \csymbolalt[20,0]{x} \joinrlnoarrowthick{R}{0}{p}{0}
\Crectangle{R}{0.7}{0.9}{0,1.3} \thispoint{p}{4,1.3} \csymbolalt[20,0]{y} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\crectangle{R}{0.7}{0.9}{0,0} \csymbol{Q} \thispoint{p}{-4,0} \csymbolalt[-25,0]{xy} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,-1.3}  \thispoint{p}{-4,-1.3} \csymbolalt[-20,0]{x} \joinlrnoarrowthick{R}{0}{p}{0}
\Crectangle{R}{0.7}{0.9}{0,1.3}  \thispoint{p}{-4,1.3} \csymbolalt[-20,0]{x} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\end{equation}

then using the property discussed in Sec. <ref> that, when circuits are disjoint, the probabilities multiply, we have

\begin{equation}\label{Qflatxy}
\text{prob}\left(
\begin{Compose}{0}{-0.06} \setsecondfont{\mathtt}
\crectangle{RR}{0.7}{0.9}{0,0}\csymbol{Q}  \crectangle{R}{0.7}{0.9}{7,0}\csymbol{Q}
\GenxdifBoxincome[1.3]{RR}{0.7}{0}{xy}{xy}{Q}
\end{Compose}\right)
= \frac{1}{N_\mathtt{xy}}
\text{prob}\left(
\begin{Compose}{0}{-0.06} \setsecondfont{\mathtt}
\crectangle{R}{0.7}{0.9}{0,0} \csymbol{Q}
\crectangle{RR}{0.7}{0.9}{4,0} \csymbol{Q}
\joinrlnoarrow{R}{0}{RR}{0} \csymbolalt{xy}
\end{Compose}\right)
\end{equation}

where $N_\mathtt{xy}=N_\mathtt{x} N_\mathtt{y}$.
Hence we see that the flat and deterministic properties are automatically satisfied.  We see that $\mathsf{Q}^{\mathtt{x}_1\mathtt{y}_2}$ and $\mathsf{Q}_{\mathtt{x}_1\mathtt{y}_2}$ are doubly deterministic since they satisfy the conditions given in Sec. <ref>.  Hence, according to the assumption of double pointer tomography, $\mathsf{Q}^{\mathtt{x}_1\mathtt{y}_2}$, is fully specified by the probabilities in the equation on the left of (<ref>) and so we must have $\mathsf{Q}^{\mathtt{x}_1\mathtt{y}_2}=\mathsf{R}^{\mathtt{x}_1\mathtt{y}_2}$.  Similarly, we obtain $\mathsf{Q}_{\mathtt{x}_1\mathtt{y}_2}=\mathsf{R}_{\mathtt{x}_1\mathtt{y}_2}$.  This proves the factorisation properties in (<ref>).

Given these properties, we can interpret these operations.  The operation

\begin{equation}\label{flatforwardop}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,0} \thispoint{p}{4,0} \csymbolalt[20,0]{x} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
\end{equation}

gives the flat distribution

\begin{equation}
p_R(x) = \frac{1}{N_\mathtt{x}}
\end{equation}

if we think in the forward in time point of view.
The operation

\begin{equation}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,0}  \thispoint{p}{-4,0} \csymbolalt[-20,0]{x} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\end{equation}

gives the flat probability distribution,

\begin{equation}
p_R(x) = \frac{1}{N_\mathtt{x}}
\end{equation}

if we think in the backwards in time point of view.

§.§ Double maximality

Some physical theories (including Quantum Theory) we consider will have the following property

Double maximality is the property that, for each pointer type, $\mathtt x$, we can associate a physical type $\mathsfb x$ such that there exist operations

\begin{equation}\label{maximaloperations}
\begin{Compose}{0}{0}\setthirdfont{\mathsfb}\setsecondfont{\mathtt}
\Crectangle{X}{0.8}{0.8}{0,0}
\thispoint{p}{0,3} \csymbolthird[0,20]{x} \jointbnoarrow[left]{X}{0}{p}{0}
\thispoint{pp}{-3,0} \csymbolalt[-20,0]{x} \joinlrnoarrowthick{X}{0}{pp}{0}
\end{Compose}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}\setthirdfont{\mathsfb}
\Crectangle{X}{0.8}{0.8}{0,3}
\thispoint{p}{0,0} \csymbolthird[0,-20]{x} \jointbnoarrow[left]{p}{0}{X}{0}
\thispoint{pp}{3,3} \csymbolalt[20,0]{x} \joinlrnoarrowthick[left]{pp}{0}{X}{0}
\end{Compose}
\end{equation}

for which we have the equivalence

\begin{equation}\label{doublemaximality}
\begin{Compose}{0}{0}\setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{X}{0.8}{0.8}{0,-2} \relpoint{X}{-4,0}{XL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick{XL}{0}{X}{0}
\crectangle{Y}{0.8}{0.8}{0,2}\csymbol{X} \relpoint{Y}{4,0}{YR} \csymbolalt[20,0]{x}\joinlrnoarrowthick{YR}{0}{Y}{0}
\jointbnoarrow[left]{X}{0}{Y}{0}\csymbolthird[0,5]{x}
\end{Compose}
~~~~\equiv ~~~~~~~
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\thispoint{X}{-4,0}
\thispoint{Y}{4,0}
\joinrlnoarrowthick[above]{X}{0}{Y}{0}\csymbolalt[0,5]{x}
\end{Compose}
\end{equation}

Furthermore, this is maximal in that there does not exist a similar equivalence for the same physical type, $\mathsfb x$, but for another pointer type, $\mathtt z$, having $N_\mathtt{z} > N_\mathtt{x}$.  The right hand side of (<ref>) is just the identity for the type $\mathtt x$.

This means that if we have the value $x$ on the income (in the diagram on the left hand side above) then we will have the same value $x$ on the outcome (forward maximality) and if we have the value $x$ on the outcome then we will have the same value $x$ on the income (backward maximality).  We will call the operations in (<ref>) maximal operations.  The maximal operation on the left in (<ref>) is called a maximal preparation and the maximal operation on the right is called a maximal measurement. Note that forward maximality does not imply backwards maximality.

Note that the operations shown in (<ref>) are the time reverse of one another (in the sense discussed in Sec. <ref>).  We should, strictly indicate this by putting a tilde under the $\mathsf X$ for one of the two operations. However, we omit this since it is clear whether we have a preparation or measurement from the positions of the wires.

§.§ Double pointer causality

Causality plays an important role in physics in general.  In the standard operational formalism it is a time asymmetric notion - namely that choices in the future should not influence probabilities in the present (this is the Pavia causality condition [12]).  We need a time symmetric notion of causality.  In Sec. <ref> we will present such a condition which we call double causality.  In the meantime, in this section, we present an intermediate notion which we call double pointer causality.  We will see in Sec. <ref> that double pointer causality is a consequence of double causality.

We define the following property:

Double pointer causality. Consider an operation, $\mathsf M$, having incomes and outcomes but no open inputs or outputs (for systems).

\[
\begin{Compose}{0}{0}
\crectangle{M}{3}{2}{0,0} \csymbol{M}
\thispoint{u}{-6,0} \csymbol[-20,0]{u}
\thispoint{v}{6,0} \csymbol[20,0]{v}
\joinrlnoarrowthick[above]{u}{0}{M}{0}
\joinrlnoarrowthick[above]{M}{0}{v}{0}
\end{Compose}
\]

The property of double pointer causality imposes that we have both

Forwards pointer causality is the property that there is no access to information about $\mathsf{M}$ from the outcome without some knowledge of the income.

Backward pointer causality is the property that there is no access to information about $\mathsf{M}$ from the income without some knowledge of the outcome.

No knowledge means the flat distribution

We can spell this out in a little more detail.   Forward pointer causality is the property that the probability distribution over $v$ associated with

\[
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\crectangle{M}{3}{2}{0,0} \csymbol{M}
\crectangle{u}{0.7}{0.9}{-6,0} \csymbol{R}
\thispoint{v}{6,0} \csymbolalt[20,0]{v}
\joinrlnoarrowthick[above]{u}{0}{M}{0}\csymbolalt{u}
\joinrlnoarrowthick[above]{M}{0}{v}{0}
\end{Compose}
\]

is independent of $\mathsf M$.  Backward pointer causality is the property that the probability distribution over $u$ associated with

\[
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{M}{3}{2}{0,0} \csymbol{M}
\thispoint{u}{-6,0} \csymbolalt[-20,0]{u}
\crectangle{v}{0.7}{0.9}{6,0} \csymbol{R}
\joinrlnoarrowthick[above]{u}{0}{M}{0}
\joinrlnoarrowthick[above]{M}{0}{v}{0}\csymbolalt{v}
\end{Compose}
\]

is independent of $\mathsf M$.

§.§ Double flatness

We are now in a position to define the following property.

Double flatness.  Consider an operation, $\mathsf M$, having incomes and outcomes but no open inputs or outputs (for systems). The property of double flatness imposes we have both

Forward flatness: if we have a flat probability distribution over the incomes and evolve this forward in time through $\mathsf M$, then we obtain a flat probability distribution over the outcomes, and

Backwards flatness: if we have a flat probability distribution over outcomes and evolve this backwards in time through $\mathsf M$, then we obtain a flat probability over the incomes.

In pictures the property of forward flatness says that

\begin{equation}\label{forwardflatnesspic}
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{M}{3}{2}{0,0} \csymbol{M}
\crectangle{u}{0.7}{0.9}{-6,0} \csymbol{R}
\thispoint{v}{6,0} \csymbolalt[20,0]{v}
\joinrlnoarrowthick[above]{u}{0}{M}{0}\csymbolalt{u}
\joinrlnoarrowthick[above]{M}{0}{v}{0}
\end{Compose}
~~~~~\equiv ~~~~~
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,0} \thispoint{p}{4,0} \csymbolalt[20,0]{v} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
\end{equation}

If we have a flat distribution on the incomes $p(u)=\frac{1}{N_\mathsf{u}}$ then according to forward flatness

\begin{equation}
p(v) = \sum_u \frac{1}{N_\mathsf{u}} p(v|u) = \frac{1}{N_\mathsf{v}}
\end{equation}

and hence

\begin{equation}
\sum_u p(v|u) = \frac{N_\mathsf{u}}{N_\mathsf{v}}
\end{equation}

Similarly, the property of backward flatness says

\begin{equation}\label{backwardflatnesspic}
\begin{Compose}{0}{0}\setsecondfont{\mathtt}
\crectangle{M}{3}{2}{0,0} \csymbol{M}
\thispoint{u}{-6,0} \csymbolalt[-20,0]{u}
\crectangle{v}{0.7}{0.9}{6,0} \csymbol{R}
\joinrlnoarrowthick[above]{u}{0}{M}{0}
\joinrlnoarrowthick[above]{M}{0}{v}{0}\csymbolalt{v}
\end{Compose}
~~~~~ \equiv ~~~~~
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\Crectangle{R}{0.7}{0.9}{0,0} \thispoint{p}{-4,0} \csymbolalt[-20,0]{u} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\end{equation}

If we have $p(v)=\frac{1}{N_\mathsf{v}}$ then according to the principle

\begin{equation}
p(u) = \sum_v \frac{1}{N_\mathsf{v}} p(u|v) = \frac{1}{N_\mathsf{u}}
\end{equation}

and hence

\begin{equation}
\sum_v p(u|v) = \frac{N_\mathsf{v}}{N_\mathsf{u}}
\end{equation}

§.§ Double pointer causality and double flatness

It is clear that

\begin{equation}\text{double flatness} \Rightarrow \text{double pointer causality} \end{equation}

We can also argue for the converse

\begin{equation}\label{causalityimpliesflattness} \text{double pointer causality} \Rightarrow \text{double flatness} \end{equation}

under some reasonableness assumptions.  By causality we know that the flat distribution on the input, $p_\mathsf{E}(u)=\frac{1}{N_\mathtt{u}}$, evolves to some distribution, $p_\mathsf{F}(v)$, and that this is independent of $\mathsf M$.  Thus, it is reasonable to assume $p_\mathsf{E}(u)=\frac{1}{N_\mathtt{u}}$ and $p_\mathsf{F}(v)$ are a matching pair.  Second, we assume that there exists some $\mathsf{M}$ such that both $p_\mathsf{M}(u|v)$ and $p_\mathsf{M}(v|u)$ are nonzero for all $u$ and $v$ (in the simple example discussed in Sec. <ref> we saw that this was the case).  Hence, this matching pair is unique.  We can also think backwards in time.  Thus, we similarly argue that some $p_\mathsf{E}(u)$ and $p_\mathsf{F}(v)=\frac{1}{N_\mathtt{v}}$ are the unique matching pair.  It follows from uniqueness that the matching pair is $p_\mathsf{E}(u)=\frac{1}{N_\mathtt{u}}$ and $p_\mathsf{F}(v)=\frac{1}{N_\mathtt{v}}$.  In other words, we have double flatness.

The reason we have been able to argue for  (<ref>) in this way is because, when we defined double pointer causality, we defined the zero knowledge situation as corresponding to the flat distribution.  We could imagine changing this so that the zero knowledge distribution corresponds to some other distribution.

§.§ Example of double pointer causality/double flatness

It is illustrative to consider a simple example of double pointer causality/double flatness.

Backwards flatness implies

\[
\begin{Compose}{0}{0} \setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{X}{0.8}{0.8}{0,0} \Crectangle{W}{1}{1}{0,4} \Crectangle{Y}{1}{1}{0,8}
\jointbnoarrow[right]{X}{0}{W}{0} \csymbolthird{x}
\jointbnoarrow[right]{W}{0}{Y}{0} \csymbolthird{x}
\thispoint{XL}{-4,0}
\crectangle{YR}{0.8}{0.8}{4,8} \csymbol{R}
\joinrlnoarrowthick{XL}{0}{X}{0} \csymbolalt[0,5]{x}
\joinrlnoarrowthick{Y}{0}{YR}{0} \csymbolalt[0,5]{y}
\end{Compose}
\begin{Compose}{0}{0} \setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{R}{0.7}{0.9}{0,0} \thispoint{p}{-4,0} \csymbolalt[-20,0]{x} \joinlrnoarrowthick{R}{0}{p}{0}
\end{Compose}
\]

We have no information about the choice of $\mathsf W$ as demanded by backward pointer causality.

Forwards flatness implies

\[
\begin{Compose}{0}{-2} \setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{X}{0.8}{0.8}{0,0} \Crectangle{W}{1}{1}{0,4} \Crectangle{Y}{1}{1}{0,8}
\jointbnoarrow[right]{X}{0}{W}{0} \csymbolthird{x}
\jointbnoarrow[right]{W}{0}{Y}{0} \csymbolthird{y}
\crectangle{XL}{0.8}{0.8}{-4,0}\csymbol{R}
\thispoint{YR}{4,8}
\joinrlnoarrowthick{XL}{0}{X}{0} \csymbolalt[0,5]{x}
\joinrlnoarrowthick{Y}{0}{YR}{0} \csymbolalt[0,5]{y}
\end{Compose}
\begin{Compose}{0}{0} \setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{R}{0.7}{0.9}{0,0} \thispoint{p}{4,0} \csymbolalt[20,0]{y} \joinrlnoarrowthick{R}{0}{p}{0}
\end{Compose}
\]

We have no information about $\mathsf W$ as demanded by forward pointer causality.

§.§ Upshot - only flat distributions

Given these assumptions, the upshot of all this is that we can only consider the flat distribution over incomes and outcomes (otherwise we get different answers for the probability of a circuit depending on whether we do the calculation forwards or backwards in time).

That is, we are interested in circuits that reduce to

\[
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose}
\]

We can omit either or both of the $u$ box and the $v$ box.  We can omit some of the readout boxes. To elaborate this point further,  recall that $\mathtt u$ and $\mathtt v$ are composites as explained in Sec. <ref> and so we can omit the readout boxes on some of the components on either side of $\mathsf M$.

§.§ Double summation property

An alternative way of stating the double flatness property is what we will call the double summation property for certain matrices.
By forward flatness (<ref>) and the defining property in (<ref>) for flat pointer operations (also applying (<ref>)), we obtain the marginal probability

\begin{equation}\label{RMvR}
\sum_u \text{prob}\left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose} \right)
\text{prob} \left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-4,0} \csymbol{R}
%\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{M}{0} \csymbolalt[0,5]{u}
%\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose}  \right)  = \frac{1}{N_\mathsf{v}}
\end{equation}

We call this property forward summation.

Similarly, by backwards flatness (using (<ref>) and (<ref>)) we can obtain the marginal probability

\begin{equation}\label{RuMR}
\sum_v \text{prob}\left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose} \right)
\text{prob} \left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{4,0} \csymbol{R}
%\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{M}{0} \csymbolalt[0,5]{v}
%\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose} \right)  = \frac{1}{N_\mathsf{u}}
\end{equation}

We call this property backward summation.

Double summation is when we have both forward summation and backward summation.  If we form the matrix with entries equal to

\begin{equation}
\text{prob}\left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose}\right)
\end{equation}

then this matrix is an example what we will call a doubly summing matrix.  They have been studied a little in the literature as a generalisation of doubly stochastic matrices [9].  Doubly stochastic matrices (sometimes called bistochastic matrices) are matrices whose entries are nonnegative and whose columns add to 1 and whose rows add to 1.  They are necessarily square.  We define doubly summing matrices to be matrices having the property that the sum, $s_\text{row}$,  of the entries in a row is the same for every row, and the sum, $s_\text{column}$, of the entries in a column is the same for every column (we omit the nonnegativity constraint here choosing, instead, to regard that separately - see Sec. <ref>).  Furthermore, we impose the normalisation condition that the columns sum to 1 (so $s_\text{column}=1$).  We note the following

* For doubly summing matrices we have

\begin{equation}
\frac{s_\text{row}}{s_\text{column}} = \frac{N_\text{row}}{N_\text{column}}
\end{equation}

where $N_\text{row}$ is the number of entries in each row (i.e. the number of columns) and $N_\text{column}$ is the number of entries in each column (i.e. the number of rows). This equation follows from the fact that the sum of all the entries in the matrix can be written as $s_\text{column}N_\text{row}$ or as $s_\text{row}N_\text{column}$.

* The case where $N_\text{row}=N_\text{column}$ is a square matrix.  Imposing  normalisation (that $s_\text{column}=1$ gives us a bistochastic matrix in the usual sense if we also impose that the entries are non-negative.

* It can be proven that doubly summing matrices compose (in parallel and in sequence) to give doubly summing matrices.

It is convenient, pedagogically, for us to impose non-negativity as a separate requirement to the doubly summing property since, for our application, they correspond to different constraints on operations (see Sec. <ref>).

We prove in Sec. <ref>that quantum theory is doubly summing.  This proof makes use of the extension theorem whereby the operator tensors associated with quantum operations can be written in terms of unitaries.

§.§ Midcomes

Till now we have been considering circuits of the form

\begin{equation}
\begin{Compose}{0}{-2.5}\setdefaultfont{\mathsf}\setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{A}{1.5}{1.2}{0,0}
\GenxdifBoxincome{A}{1.5}{0}{x}{x}{R}   %\relpoint{A}{-4,0}{AL}\csymbolalt[-20,0]{x}  \joinrlnoarrowthick[above]{AL}{0}{A}{0}
\GenxdifBoxoutcome{A}{1.5}{0}{y}{y}{R}  %\relpoint{A}{4,0}{AR}  \csymbolalt[20,0]{y}\joinrlnoarrowthick[above]{A}{0}{AR}{0}
\Crectangle{D}{1.5}{1.2}{4,18}
\GenxdifBoxincome{D}{1.5}{0}{y}{y'}{R}  %\relpoint{D}{-4,0}{DL}  \csymbolalt[-20,0]{y}\joinrlnoarrowthick[above]{DL}{0}{D}{0}
\GenxdifBoxoutcome[1.2]{D}{1.5}{0}{w}{w'}{R}   %\relpoint{D}{4,0}{DR} \csymbolalt[20,0]{w} \joinrlnoarrowthick[above]{D}{0}{DR}{0}
\Crectangle{B}{1.5}{1.2}{-6, 7.5}
\GenxdifBoxincome{B}{1.5}{0}{x}{x'}{R}  %\relpoint{B}{-4,0}{BL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick[above]{BL}{0}{B}{0}
\GenxdifBoxoutcome[1.2]{B}{1.5}{0}{x}{x''}{R}%\relpoint{B}{4,0}{BR} \csymbolalt[20,0]{x} \joinrlnoarrowthick[above]{B}{0}{BR}{0}
\Crectangle{C}{1.5}{1.2}{10,11}
\GenxdifBoxincome[1.45]{C}{1.5}{0}{x}{x'''}{R}%\relpoint{C}{-4,0}{CL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick[above]{CL}{0}{C}{0}
\GenxdifBoxoutcome{C}{1.5}{0}{w}{w}{R}%\relpoint{C}{4,0}{CR} \csymbolalt[20,0]{w}\joinrlnoarrowthick[above]{C}{0}{CR}{0}
%\joinrlnoarrowthick[above left]{B}{0}{C}{0} \csymbolalt{x}
\jointbnoarrow[above left]{B}{0}{D}{-1} \csymbolthird[-4,0]{c} \jointbnoarrow[above right]{C}{0}{D}{1} \csymbolthird[5,0]{b}
\jointbnoarrow[below left]{A}{-1}{B}{0} \csymbolthird[-5,0]{a}
\jointbnoarrow[below right]{A}{1}{C}{0} \csymbolthird[4,0]{c}
\end{Compose}
\end{equation}

Though this form is quite general, it misses out an important possibility.  We can also feed outcomes into incomes as in the following example

\begin{equation}\label{circuitexamplemidcomes}
\begin{Compose}{0}{-2.5}\setdefaultfont{\mathsf}\setsecondfont{\mathtt} \setthirdfont{\mathsfb}
\Crectangle{A}{1.5}{1.2}{0,0}
\GenxdifBoxincome{A}{1.5}{0}{x}{x}{R}   %\relpoint{A}{-4,0}{AL}\csymbolalt[-20,0]{x}  \joinrlnoarrowthick[above]{AL}{0}{A}{0}
\GenxdifBoxoutcome{A}{1.5}{0}{y}{y}{R}  %\relpoint{A}{4,0}{AR}  \csymbolalt[20,0]{y}\joinrlnoarrowthick[above]{A}{0}{AR}{0}
\Crectangle{D}{1.5}{1.2}{4,18}
\GenxdifBoxincome{D}{1.5}{0}{y}{y'}{R}  %\relpoint{D}{-4,0}{DL}  \csymbolalt[-20,0]{y}\joinrlnoarrowthick[above]{DL}{0}{D}{0}
\GenxdifBoxoutcome[1.2]{D}{1.5}{0}{w}{w'}{R}   %\relpoint{D}{4,0}{DR} \csymbolalt[20,0]{w} \joinrlnoarrowthick[above]{D}{0}{DR}{0}
\Crectangle{B}{1.5}{1.2}{-6, 7.5}
\GenxdifBoxincome{B}{1.5}{0}{x}{x'}{R}  %\relpoint{B}{-4,0}{BL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick[above]{BL}{0}{B}{0}
%\GenxdifBoxoutcome[1.2]{B}{1.5}{0}{x}{x''}{K}%\relpoint{B}{4,0}{BR} \csymbolalt[20,0]{x} \joinrlnoarrowthick[above]{B}{0}{BR}{0}
\Crectangle{C}{1.5}{1.2}{10,11}
%\GenxdifBoxincome[1.45]{C}{1.5}{0}{x}{x'''}{Q}%\relpoint{C}{-4,0}{CL} \csymbolalt[-20,0]{x} \joinrlnoarrowthick[above]{CL}{0}{C}{0}
\GenxdifBoxoutcome{C}{1.5}{0}{w}{w}{R}%\relpoint{C}{4,0}{CR} \csymbolalt[20,0]{w}\joinrlnoarrowthick[above]{C}{0}{CR}{0}
\crectangle{xm}{0.9}{0.8}{2, 9.25} \csymbolfourth{x''}
\joinrlnoarrowthick[above left]{B}{0}{xm}{0} \csymbolalt{x} \joinrlnoarrowthick[above left]{xm}{0}{C}{0} \csymbolalt{x}
%\joinrlnoarrowthick[above left]{B}{0}{C}{0} \csymbolalt{x}
\jointbnoarrow[above left]{B}{0}{D}{-1} \csymbolthird[-4,0]{c} \jointbnoarrow[above right]{C}{0}{D}{1} \csymbolthird[5,0]{b}
\jointbnoarrow[below left]{A}{-1}{B}{0} \csymbolthird[-5,0]{a}
\jointbnoarrow[below right]{A}{1}{C}{0} \csymbolthird[4,0]{c}
\end{Compose}
\end{equation}

We will call the $x''$ readout a midcome.   Circuits with midcomes can be reduced to circuits without midcomes employing the following equivalence

\begin{equation}\label{midcomeequivalence}
\begin{Compose}{0}{-0.1} \setdefaultfont{} \setfourthfont{\mathtt}
\thispoint{l}{-4,0} \thispoint{r}{4,0}
\Crectangle{x}{0.7}{0.8}{0,0}
\joinrlnoarrowthick[above]{l}{0}{x}{0} \csymbolfourth{x}
\joinrlnoarrowthick[above]{x}{0}{r}{0} \csymbolfourth{x}
\end{Compose}
~~ \equiv ~~
\begin{Compose}{0}{-0.1}
\thispoint{l}{-7.6,0} \thispoint{r}{7.6,0}
\RxBoxoutcome{l}{0}{0}{x}\RxBoxincome{r}{0}{0}{x}
\crectangle[thin]{number}{1}{1}{0,2.3} \csymbolthird{N_\mathtt{x}}
\end{Compose}
\end{equation}

The $N_\mathtt{x}$ inside the rectangular box is an over all factor.  Including such factors goes a little beyond the definition of equivalence introduced in Sec. <ref>.  It is, however, a simple application of the notion of the linearly extended notion of equivalence introduced in Sec. <ref>.    It is clear (<ref>) is true.  If we think forwards in time then there is a $\frac{1}{N_\mathtt{x}}$ probability that the $x$ in the second readout box will equal the $x$ in the first so this accounts for the $N_\mathtt{x}$ factor (similar remarks apply if we think in the backwards time direction).   This equivalence means that we can always reduce circuits with midcomes to ones without midcomes by including the appropriate $N_\mathtt{x}$ factors.

§.§ Calculating some probabilities

We can use these result to calculate conditional probabilites.  Here is a simple example:

\begin{equation}
p_\mathtt{M}(u|v) ~~=~~
\frac{\text{prob} \left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose} \right)}
{ \text{prob} \left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
%\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{M}{0} \csymbolalt[0,5]{v}
%\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose} \right) }
\end{equation}

Using (<ref>) we obtain

\begin{equation}\label{pugvpic}
p_\mathtt{M}(u|v) ~~=~~
N_\mathtt{u} ~
\text{prob} \left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose} \right)
\end{equation}

\begin{equation}\label{pvgupic}
p_\mathtt{M}(v|u) ~~=~~
N_\mathtt{v} ~
\text{prob} \left(
\begin{Compose}{0}{0} \setsecondfont{\mathtt}
\crectangle{T}{0.7}{1}{-7,0} \csymbol{R}
\crectangle{x}{0.7}{0.7}{-4,0} \csymbolthird{u}
\Crectangle{M}{1}{1}{0,0}
\joinrlnoarrowthick[above]{T}{0}{x}{0} \csymbolalt[0,5]{u}
\joinrlnoarrowthick[above]{x}{0}{M}{0} \csymbolalt[0,5]{u}
\crectangle{Tr}{0.7}{1}{7,0} \csymbol{R}
\crectangle{xr}{0.7}{0.7}{4,0} \csymbolthird{v}
\joinlrnoarrowthick[above]{Tr}{0}{xr}{0} \csymbolalt[0,5]{v}
\joinlrnoarrowthick[above]{xr}{0}{M}{0} \csymbolalt[0,5]{v}
\end{Compose} \right)
\end{equation}

We see here that

\begin{equation}\label{bayesinversion}
\frac{p_\mathtt{M}(u|v) }{N_\mathtt{u}} =  \frac{p_\mathtt{M}(v|u) }{N_\mathtt{v}}
\end{equation}

This is an example of Bayesian inversion.

Equivalence, Linearity, and Physicality

§ EXTENDED EQUIVALENCE

§.§ The $p(\cdot)$ function

We wish to set up a notion of equivalence between different fragments of a circuit which linearly extends the simple notion introduced in Sec. <ref>.  To do this we define the $p(\cdot)$ function.  Consider various circuits, $\mathsf A$, $\mathsf B$, …. We know, by the circuit probability assumption (see Sec. <ref>) that the probabilities $\text{prob}(\mathsf{A})$, $\text{prob}(\mathsf{B})$, …are defined.  Hence we can define

\begin{equation}
p(\alpha \mathsf{A} + \beta\mathsf{B} + \dots ) = \alpha \text{prob}(\mathsf{A}) + \beta \text{prob}(\mathsf{B}) + \dots
\end{equation}

where $\alpha$, $\beta$, …are real numbers (they can be positive, zero, or negative).
Here we have done something a little odd - we have formed a weighted linear sum of circuits.  The weights are mathematical while the circuits live in the physical world.

§.§ Definition of equivalence

A fragment of a circuit (which we will simply called a fragment - see Sec. <ref>) is a collection of operations wired together that could be completed into a circuit by other operations.  Consider expressions given by

\begin{equation}
\text{expression}=\alpha + \beta \mathsf{C} + \gamma \mathsf{D}+ \dots
\end{equation}

where $\mathsf{C}$, $\mathsf{D}$, …are fragments of circuits (we have suppressed the subscripts and superscripts which would normally appear for the sake of generality).   We will say that

\begin{equation}
\text{expression}_1 \equiv \text{expression}_2
\end{equation}

if all the terms in both expressions can be completed into a circuit by a fragment and, furthermore,

\begin{equation}
p(\text{expression}_1 \mathsf{E}) = p(\text{expression}_2 \mathsf{E})
\end{equation}

for all such fragments, $\mathsf{E}$, which have the property that they convert the expressions into a linear sum of circuits.  In order that fragments $\mathsf E$ exist, it is necessary that all the terms in both $\text{expression}_1$ and $\text{expression}_2$ have compatible causal structure (see Sec. <ref>).

§.§ Examples of equivalence

Here are two examples of equivalence.  First, we have

\begin{equation}\label{equivexample1}
\alpha\mathsf{A}_{\mathtt{x}_1}^{\mathsfb{x}} + \beta\mathsf{B}_{\mathtt{x}_1}^{\mathsfb{x}} \equiv
\gamma \mathsf{C}_{\mathtt{x}_1}^{\mathsfb{x}} + \delta \mathsf{D}_{\mathtt{x}_1}^{\mathsfb{x}}
\end{equation}

\begin{equation}
p(\alpha\mathsf{A}_{\mathtt{x}_1}^{\mathsfb{x}_2} \mathsf{E}^{\mathtt{x}_1}_{\mathsfb{x}_2} + \beta\mathsf{B}_{\mathtt{x}_1}^{\mathsfb{x}_2}\mathsf{E}^{\mathtt{x}_1}_{\mathsfb{x}_2} )
\equiv
p(\gamma \mathsf{C}_{\mathtt{x}_1}^{\mathsfb{x}_2}\mathsf{E}^{\mathtt{x}_1}_{\mathsfb{x}_2} +
\delta\mathsf{D}_{\mathtt{x}_1}^{\mathsfb{x}_2}\mathsf{E}^{\mathtt{x}_1}_{\mathsfb{x}_2})  ~~~~~ \forall ~~ \mathsf{E}^{\mathtt{x}_1}_{\mathsfb{x}_2}
\end{equation}

Second, for a circuit, $\mathsf F$

\begin{equation}\label{circuitequivprob}
\mathsf{F} \equiv \text{prob}(\mathsf{F})
\end{equation}

That is, a circuit is equivalent to its own probability.  The proof of this is simple.  The only fragments that can complete $\mathsf{F}$  into a circuit is another circuit.  We have

\begin{equation}
p(\mathsf{F}\mathsf{E})= p(\mathsf{F})p(\mathsf{E}) = \text{prob}(\mathsf{F})p(\mathsf{E})  ~~~\forall ~\mathsf{E}
\end{equation}

where we use the fact that probabilities factorise for circuits (see Sec. <ref>) and the linearity of the $p(\cdot)$ function on the right.  This proves (<ref>).  This second example illustrates that equivalence is a different notion from equality.  Clearly a circuit (which lives in the physical world) is not equal to a number (which is a mathematical object) but the two are equivalent under the definition used here.

§.§ Spanning sets and equivalence

Consider a set, $S$, of fragments. A subset, $S_\text{spanning}\subseteq S$, is a spanning set for $S$ if every fragment in $S$ can be written as being equivalent to a weighted linear sum of the fragments in $S_\text{spanning}$.

Here we will provide a useful test for whether two fragments $\mathsf A$ and $\mathsf B$ are equivalent (here we omit the type indices for the sake of providing a general discussion).  First, recall from Sec. <ref> that a necessary condition for two fragments to be equivalent is that they have compatible causal structures (as defined in Sec. <ref>). This means  $S_\mathsf{A}\cap S_\mathsf{B}\not= \emptyset$ where $S_\mathsf{A}$ is the set of fragments that can complete $\mathsf A$ into a circuit (and similarly for $S_\mathsf{B}$).

From the definition of a spanning set above, it is clear that, if $S_\mathsf{A}\cap S_\mathsf{B}\not= \emptyset$, then

\begin{equation}
\mathsf{A}\equiv \mathsf{B} ~~\text{iff}~~ \text{prob}(\mathsf{AE})= \text{prob}(\mathsf{BE})~~ \text{for all}~~ \mathsf{E}\in [S_\mathsf{A}\cap S_\mathsf{B}]_\text{spanning}
\end{equation}

where $\mathsf{AE}$ and $\mathsf{BE}$ are circuits (the wiring is not shown explicitly for generality) and $[S_\mathsf{A}\cap S_\mathsf{B}]_\text{spanning}$ is any spanning set for $S_\mathsf{A}\cap S_\mathsf{B}$

§ SPANNING PROPERTIES AND IDENTITY OPERATIONS

§.§ System preparations and results

A general system preparation is a fragment having only a output left (or, possibly, multiple outputs) open.  If there are no midcomes (see Sec. <ref>) then a general system preparation can always be put in the form

\begin{equation}\label{generalpreparation}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb}
\Crectangle{A}{1.5}{1.5}{0,0}
\RxBoxincome{A}{1.5}{0}{x} \RxBoxoutcome{A}{1.5}{0}{s}
\thispoint{p}{0,4} \csymbolalt[0,20]{a} \jointbnoarrow{A}{0}{p}{0}
\end{Compose}
\end{equation}

If there are midcomes we can still put the fragment in this form though we accumulate appropriate $N_\mathtt{x}$ factors through use of the equivalence in (<ref>) as described in Sec. <ref>.
We can also consider income only system preparations. These are preparations of the form

\begin{equation}\label{incomeonlyprep}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb}
\crectangle{A}{1.5}{1.5}{0,0} \csymbol{E}
\RxBoxincome{A}{1.5}{0}{x} %\RxBoxoutcome{A}{1.5}{0}{s}
\thispoint{p}{0,4} \csymbolalt[0,20]{a} \jointbnoarrow{A}{0}{p}{0}
\end{Compose}
\end{equation}

where $\mathsf E$ is doubly deterministic.  This is more restricted than the preparation shown in (<ref>).

Similarly, a general system result is a fragment having only an input (or inputs) left open.  A general system result can be put in the form
\begin{equation}\label{generalresult}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb}
\crectangle{A}{1.5}{1.5}{0,0} \csymbol{B}
\RxBoxincome{A}{1.5}{0}{w} \RxBoxoutcome{A}{1.5}{0}{y}
\thispoint{p}{0,-4} \csymbolalt[0,-20]{a} \joinbtnoarrow{A}{0}{p}{0}
\end{Compose}
\end{equation}

(where there maybe additional $N_\mathtt{x}$ factors if there are midcomes in the fragment).  An outcome only results is of the form

\begin{equation}\label{outcomeonlyresult}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb}
\crectangle{A}{1.5}{1.5}{0,0} \csymbol{F}
\RxBoxoutcome{A}{1.5}{0}{y}
\thispoint{p}{0,-4} \csymbolalt[0,-20]{a} \joinbtnoarrow{A}{0}{p}{0}
\end{Compose}
\end{equation}

where $\mathsf F$ is doubly deterministic.

§.§ Pure and homogeneous system preparations and results

[dashed] (0,0) arc (170:10:2cm and 0.4cm)coordinate[pos=0] (a);
[thick] (0,0) arc (-170:-10:2cm and 0.4cm)coordinate (b);
[thick] (a) – ([yshift=-4cm]$(a)!0.5!(b)$) – (b);
[thick] (a) – ([yshift=6cm]$(a)!0.5!(b)$) – (b);
([yshift=-4cm]$(a)!0.5!(b)$) – ($(b)+(-1,-0.27)$);
n2,-4 [0,-10]null
I2,6 [0,10]ignore
pp6,-1.5 [22,0.5]pure pp0p-60
hh6,-3 [125,1]homogeneous (pure parallel)  hh0h-30

Different types of preparation (or result).  The notions of pure and pure parallel (or homogeneous) are illustrated.

It is useful to introduce some distinctions concerning types of system preparations and results.  First, we will say two system preparations (or results) are parallel if one is equivalent to the other times a positive number.  Now we define the following cases:

Mixed. A system preparation (or result) is mixed if it can be written as being equivalent to a convex sum of distinct preparations (or results).

Extremal. A system preparation (or result) is extremal if it is not mixed.

Heterogeneous. A system preparation (or result) is heterogeneous if it can be written as being equivalent to a sum, with non-negative weights, of non-parallel preparations (or results).

Homogeneous (pure parallel). A system preparation (or result) is homogeneous if it is not heterogeneous.  This means that if it is written as being equivalent to a sum of other preparations (or results), all these preparations(or results) must be parallel.

Pure.  A system preparation (or result) is pure if it is homogeneous and extremal.

The pure and homogenous cases are illustrated by the example shown in Fig. <ref>.   We will sometimes call homogeneous preparations or results pure parallel since the concept of purity is more familiar. In this example we have a unique deterministic preparation (result) which we label ignore".  In this example, the ignore preparation (result) is extremal. Thus, the ignore preparation (or result) is not mixed - however it is heterogeneous (as is clear from looking at the figure).   This example illustrates the distinction between the concepts of being mixed and being heterogenous.  The null preparation (result) is equivalent to any preparation (result) weighted by zero.  If it is included in a circuit, the circuit will have probability equal to zero.

The classification shown here is subtlety different from that standardly used.  In particular, the pure case does not cover all extremal non-null cases.  The figure is just an example (though illustrative of what happens in time-symmetric Quantum Theory).  We could have a case where the top is cut off so there is not a unique ignore preparation or result (though, in the approach taken in this paper, the ignore operation will be seen to be unique both for preparations and results - see Sec. <ref>).

A little more nomenclature.  For linguistic brevity, we will say that operations of the following sort

\begin{equation}\label{DEininoutout}
\begin{Compose}{0}{0}\setsecondfont{\mathsfb}