# Entanglement, Complexity, and Causal Asymmetry in Quantum Theories

Porter Williams
Department of Philosophy
University of Southern California

###### Abstract

It is often claimed that one cannot locate a notion of causation in
fundamental physical theories. The reason most commonly given is that the
dynamics of those theories do not support any distinction between the past and
the future, and this vitiates any attempt to locate a notion of causal
asymmetry — and thus of causation – in fundamental physical theories. I argue
that this is incorrect: the ubiquitous generation of entanglement between
quantum systems grounds a relevant asymmetry in the dynamical evolution of
quantum systems. I show that by exploiting a connection between the amount of
entanglement in a quantum state and the algorithmic complexity of that state,
one can use recently developed tools for causal inference to identify a causal
asymmetry – and a notion of causation – in the dynamical evolution of quantum
systems.

##  1 Introduction

Consider the plight of Emma Flake. Dr. Flake runs a lab that works on
applications of quantum state tomography, but she has kind of checked out
lately. Her grad students, Alice and Bob, have conducted the following
experiment in her absence: a source prepares an ensemble of bipartite quantum
systems in identical pure states $\ket{\alpha}$. Alice knows the state
$\ket{\alpha}$, but Bob does not. Each system then passes through a region of
spacetime governed by a Hamiltonian $H$, ending up in some pure state
$\ket{\beta}$. Bob’s task is to reconstruct the pure state $\ket{\beta}$ by
performing a large number of measurements on different observables of the
system.

Dr. Flake, feeling guilty about her absenteeism, offers to write up the paper
on her own and tells Alice and Bob to take some time off. When she looks at
Alice and Bob’s notes, she discovers that there is no record of which state
was prepared by Alice and which was reconstructed by Bob; all she knows is the
two pure states $\ket{\alpha}$ and $\ket{\beta}$. Is there any way for Dr.
Flake to determine which was the initial state and which was the final? In
other words, can Dr. Flake determine whether $\ket{\alpha}$ was caused by
time-evolving $\ket{\beta}$ or vice versa?111One might wonder why I describe
this as a problem of inferring causal direction rather than temporal
direction. I will return to this question in section 6.

Dr. Flake is confronted with a special case of the general problem I will take
up in this paper: if one knows (i) the identity of two quantum states and (ii)
that they are related by unitary time evolution, does quantum theory provide
resources to determine which state was caused by time-evolving the other? It
is commonly said that the answer is “no”: the failure of the dynamical
equations describing microscopic physical systems to distinguish between past
and future leaves Dr. Flake with insufficient resources to solve her problem.
This failure is commonly presented as an obstacle to identifying any notion at
all of causal direction – and thus of causation – using the resources of
microscopic physics (see, for example, (Russell, 1912; Albert, 2000; Field,
2003; Woodward, 2007; Loewer, 2012)).

One consequence has been the widespread belief that any notion of causal
direction, and thus causation, must be located elsewhere. For many, like
(Albert, 2000; Woodward, 2007), that has been in macroscopic systems. For
some, it has been in our psychological experience as deliberators (Ismael,
2016; Fernandes, 2017). And for a few, it has been nowhere: causal direction
is illusory, tantamount to a preferred choice of coordinates (Price, 2007).

The notion of causation I employ in this paper is a minimal interventionist
one: X is a cause of Y if and only if there is an intervention that can be
performed on the value of X, while holding all other variables fixed, that
produces a change in the value of Y.222This generalizes straightforwardly to
probabilistic settings: X causes Y if and only if there is an intervention
that can be performed on $p_{X}(x)$, the probability distribution over the
values of X, while holding all other variables fixed, that produces a change
in $p_{Y}(y)$. See Woodward (2005) for a much richer philosophical development
of the interventionist account of causation than I will need here. In this
paper I will restrict to the case of two variables and presuming the absence
of confounders, so any causal relation will be a direct causal relation.

It is well known that without additional conditions, this minimal
interventionist notion of causation cannot use observational statistical data
alone to identify a direction of causation between two variables X and Y, even
when they are related by a known invertible function (see, e.g., (Pearl, 2009,
chapter 2)). Unitary time evolution between two quantum states is
deterministic and invertible; without information about the actual time
ordering of two states $\ket{\alpha}$ and $\ket{\beta}$, familiar causal
inference methods are unable to distinguish between
$\ket{\alpha}\longrightarrow\ket{\beta}$ and
$\ket{\beta}\longrightarrow\ket{\alpha}$.

In this paper, I argue that a generic physical fact about the time evolution
of quantum systems identifies a causally relevant asymmetry: interaction
between quantum systems almost always entangles those systems but almost never
disentangles them. This fact enables one to identify a causally relevant
asymmetry between quantum states related by unitary time evolution. To connect
this physical fact to formal methods of causal inference, I employ recently
developed tools whose applications include the ability to identify the causal
direction of deterministic, noiseless processes solely from observational data
(Janzing and Schölkopf, 2010; Daniušis et al., 2010; Janzing et al., 2012;
Peters et al., 2017), (Mooij et al., 2016, sections 3 and 5.2) to argue that
the generic entanglement of quantum systems by unitary time evolution allows
one to identify criteria for inferring causal direction even between states
related by time-symmetric dynamical laws. The majority of my discussion
focuses on paradigmatically microscopic physical systems: bipartite quantum
systems undergoing unitary time evolution. That one can identify causally
relevant asymmetries in such evolutions is indication that one can locate a
notion of causation in microscopic physics after all.

The structure of the paper is as follows. In Section 2, I review the sense(s)
in which quantum theories fail to distinguish between past and future and the
challenge this poses to any attempt to locate a notion of causation in
microscopic physics. In Section 3, I review relevant properties of
entanglement in quantum theories: the ubiquity of entangled states, the
generic creation of entanglement by interactions between systems, and
quantitative measures of how entangled a quantum state is. I argue that
already, on physical grounds, these features ground a (fairly weak) causal
asymmetry between quantum states related by unitary evolution. Section 4
reviews the causal inference methods I later use to argue that one can
identify a direction of causation even between states related by time-
symmetric dynamical laws, along with some necessary concepts from algorithmic
information theory. Section 5 introduces a generalization of algorithmic
information to quantum theories and demonstrates that in the restricted but
important setting of a system of N qubits, one can derive a stronger version
of the asymmetry from section 3 as a theorem of the causal inference methods
introduced in section 4. In section 6 I conclude with some remarks about the
relationship between more familiar temporal asymmetries in physics and the
causal asymmetry argued for here, and situate my discussion within a more
general understanding of the relationship between the epistemology of
causation and its metaphysics.

I want to emphasize three things before proceeding.333Well, four things. The
fourth thing is that throughout the paper $\log=\log_{2}$ and I adopt units
where $\hbar=c=1$. The first thing is that my primary interest here is the
epistemology of causal direction.444See (Eberhardt, 2009) for an overview. For
a clarifying sketch of several types of projects one might pursue when
thinking about causation, see (Woodward, 2014). One can view sections 4 and 5
in particular as an exercise in causal epistemology: taking a procedure for
inferring causal direction based on observational data that has proved
successful in other domains and applying it to quantum theories. I think of
this as akin to expanding an experimental technique that has proved successful
at detecting a certain object or property in one department of nature to a
novel domain, where the presence of that object or property is not settled,
and seeing what the experiment turns up. In section 6 I return to a discussion
of how this primarily epistemological exercise bears on questions about the
physical ground of the causal asymmetry I identify.

The second thing is that unless otherwise stated, I will be considering only
unitary time evolution. Once departures from unitary evolution are allowed,
either through the tracing out of the environment associated with decoherence
(Wallace, 2012, chapter 9), spontaneous collapse of GRW or CSL type (Albert,
2000, chapter 7), or old-fashioned measurement-induced collapse, then the time
evolution distinguishes between past and future and an asymmetry between cause
and effect is introduced.555This doesn’t mean that all causal inference
puzzles are solved once one allows departures from unitary evolution. Far from
it: it is only after considering such departures that one encounters the most
vexing causal inference problem posed by quantum theories: the explanation of
EPR-type correlations.

The third thing concerns the discussion of entanglement. Entanglement of
quantum systems by interaction is a generic physical fact. In this paper I
employ a particular measure of entanglement, the Schmidt measure, for reasons
discussed in section 3. The reader should bear in mind that this is only one
of multiple ways to quantify entanglement and the physical fact that
interactions generically entangle but almost never disentangle – the physical
foundation for the causal inference strategy developed in this paper – does
not depend on this particular choice of entanglement measure nor the
particular causal inference methods used. Indeed, the idea that the generation
of entanglement by dynamical evolution is connected to a closely related
asymmetry, the asymmetry of time, has been explored by a number of authors
(Popescu et al., 2006; Reimann, 2008; Linden et al., 2009; Jennings and
Rudolph, 2010; Short and Farrelly, 2012; Goldstein et al., 2013; Malabarba et
al., 2014; Goldstein et al., 2015a).

Finally, this paper contributes to a large and rapidly growing literature on
causal structure and causal inference in quantum theories (Leifer and
Spekkens, 2013; Barrett et al., 2019; Chiribella et al., 2009; Oreshkov et
al., 2012; Rubino et al., 2017; Costa and Shrapnel, 2016; Ried et al., 2015;
Allen et al., 2017; Wood and Spekkens, 2015; Chaves et al., 2015, 2014). A
number of mathematical and conceptual frameworks for analyzing causal
structure have been developed in this literature. For example, (Leifer and
Spekkens, 2013) develop a generalization of Bayesian inference and an
associated concept of a conditional quantum state, while the process matrix
formalism (Oreshkov et al., 2012; Barrett et al., 2019) introduces the
“process matrix” which generalizes the standard notions of a quantum state and
a quantum channel and has proven especially valuable for analyzing quantum
processes with indeterminate causal order. And some of this work has taken up
a similar project as I undertake here: locating causal or temporal asymmetries
in the structure of quantum theory (Thompson et al., 2018; Schmid et al.,
2020; Hardy, 2021; Di Biagio et al., 2021). In the course of this paper, I
introduce an additional set of tools into this literature by taking the
initial steps toward generalizing the algorithmic-information-theoretic
framework for causal inference developed by (Janzing and Schölkopf, 2010) to
quantum theories.

##  2 Time Evolution in Quantum Theories

The time evolution of a quantum system over an interval of duration $t$ is
described by the unitary operator U(t) = $\displaystyle e^{-iHt}$ where $H$ is
the Hamiltonian governing the system.666This presumes time-translation
invariance. Time-translation invariance will always be assumed in this paper,
a reflection of the fact that I am considering closed quantum systems. These
unitary operators form a group: Stone’s theorem ensures that as long as the
Hamiltonian $H$ is self-adjoint then it defines a strongly continous one-
parameter group of unitary operators U(t) (see, e.g. (Hall, 2013, chapter
10.2)). These unitary operators describe time evolution.

Quantum theories can fail to distinguish between past and future in each of
the following two senses:777See (Earman, 2002, section 4) or (Farr and
Reutlinger, 2013) for a discussion of the importance of distinguishing between
the two when considering how the time-symmetry of a theory bears on our
ability to distinguish cause and effect in microphysics.

  1. 1.

Given a state $\ket{\psi(t)}$, the unitary dynamics determine both the earlier
state $\ket{\psi(t-\tau)}$ and the later state $\ket{\psi(t+\tau)}$ for
arbitary $\tau$.

Quantum theories always fail to distinguish between past and future in this
sense. It is true whether or not the theory is time-reversal invariant. It is
a consequence of the fundamental fact secured by Stone’s theorem: the time
evolution operators U(t) form a one-parameter group, which ensures that every
U(t) has an inverse U(t)-1. Given a quantum state $\ket{\psi(t=0)}$, the
operator U(t)= $\displaystyle e^{-iHt}$ uniquely determines the future state
$\ket{\psi(t)}$ and its inverse U(t)${}^{-1}=\displaystyle e^{iHt}$ uniquely
determines the past state $\ket{\psi(-t)}$. A consequence of this is that
given any two quantum states related by unitary time evolution, we have
$\ket{\alpha}=\text{U(t)}\ket{\beta}$ and
$\ket{\beta}=\text{U(t)}^{-1}\ket{\alpha}$. If one does not have any
additional information about their temporal ordering then one cannot determine
whether $\ket{\alpha}$ causes $\ket{\beta}$ or vice-versa.888The distinct but
related question of whether the two unitaries U(t)and U(t)-1 are equally
implementable in practice is complicated (Janzing et al., 2002; Janzing, 2019;
Janzing and Wocjan, 2018).

It is worth pausing here to address a natural question: isn’t it obvious that
whichever evolution is generated by U(t) is the true one, with U(t)-1
generating the backward-in-time, or acausal, evolution? The reason this does
not work is that without antecedent information about the direction of time,
one is really considering two candidate time variables: t and T, related by
$T=-t$. Whether one considers U(t) as generating forward-in-time evolution or
backward-in-time evolution depends on the time variable one chooses. The
notation U(t)-1 is thus a bit misleading; writing the same unitary operator as
U(T) makes the symmetry between t and T manifest.

To illustrate this by the example above, if one chooses t as the time variable
then given an initial state $\ket{\psi(t=0)}$, the operator U(t)=
$\displaystyle e^{-iHt}$ uniquely determines the future state $\ket{\psi(t)}$
and its inverse U(t)${}^{-1}=\displaystyle e^{iHt}$ uniquely determines the
past state $\ket{\psi(-t)}$. Using T, the temporal ordering is inverted:
$\displaystyle e^{-iHT}$ uniquely determines the past state $\ket{\psi(-T)}$
and its inverse U(T)${}^{-1}=\displaystyle e^{iHT}$ uniquely determines the
future state $\ket{\psi(T)}$. Any attempt to determine the causal direction
that requires first choosing between t or T would beg the question. (See
(Donoghue and Menezes, 2019, 2020) for a pedagogical presentation and
additional discussion of the facts above. In particular, see their interesting
discussion of the fact that a choice of the time variable is fixed by the
choice of a sign convention for the canonical commutation relations
$\left[x,\,p\right]=\pm i\hbar$.)

In addition to the invertibility of the dynamics, a quantum theory may also be
time-reversal invariant:999The appropriate understanding of time-reversal in
quantum mechanics has received a fair amount of philosophical attention in
recent years (Albert, 2000; Callender, 2000; Earman, 2002; Roberts, 2017;
Allori, 2019; Farr, 2020; Donoghue and Menezes, 2019, 2020; Callender, 2020;
Struyve, 2020). The initial stimulation for much of this work were the
arguments for a non-standard definition of time reversal by (Albert, 2000) and
(Callender, 2000). For reasons compactly summarized in (Roberts, 2019), I
remain partial to the traditional account and will adopt it throughout this
paper.

  1. 2.

There exists a time-reversal operator $\mathcal{T}$ such that

    1. (a)

$\mathcal{T}$ commutes with the Hamiltonian of the theory,
$[\mathcal{T},\,H]=0$, and

    2. (b)

If the state $\ket{\psi_{0}(t=0)}$ evolves under H as $\ket{\psi_{0}(t=0)}$,
$\ket{\psi_{1}(t_{1})}$, $\ldots$, $\ket{\psi_{n}(t_{n})}$, then the state
$\mathcal{T}\ket{\psi_{n}(-t_{n})}$ evolves under
$\mathcal{T}H\mathcal{T}^{-1}=H$ as $\mathcal{T}\ket{\psi_{n}(-t_{n})}$,
$\mathcal{T}\ket{\psi_{n-1}(-t_{n-1})}$ $\ldots$,
$\mathcal{T}\ket{\psi_{1}(-t_{1})}$, $\mathcal{T}\ket{\psi_{0}(t=0)}$.

It is important to note that the state $\ket{\psi_{0}(t=0)}$ and the time-
reversed state $\mathcal{T}\ket{\psi_{0}(t=0)}$ will not, in general, be the
same state. Momentum eigenstates (with $p\neq 0$) provide a simple example:
the time-reverse of the state $\ket{p}$ is $\mathcal{T}\ket{p}=\ket{-p}$ (up
to a phase), so the state and its time-reverse are orthogonal:
$\expectationvalue{\mathcal{T}}{p}=0$. Eigenstates of angular momentum
observables for systems with non-integer spin exhibit the same behavior – both
the spin observables $\sigma_{x},\,\sigma_{y},\,\sigma_{z}$ and orbital
angular momentum observables $L_{x},\,L_{y},\,L_{z}$.101010This is a special
case of the fact that the state of any degree of freedom that is odd under
time-reversal – i.e. $\mathcal{T}^{2}\ket{\psi}=-\ket{\psi}$ – is orthogonal
to its time-reversed state (Sachs, 1987, section 3.2). One can specify
conditions that ensure that a quantum state and its time-reverse are the same
state (for example (Earman, 2002, section 4) or (Sakurai and Napolitano, 2011,
theorem 4.12)), but such cases are the exception, not the rule.

Time-reversal invariance of the laws of microphysics is often identified as an
obstacle to interpreting microphysics causally, but I think it is a red
herring.111111For two arguments for the same conclusion in classical and
quantum statistical mechanics that have points of contact with the argument I
offer here, but which are ultimately distinct, see (Maudlin, 2007, chapter 4)
or (Myrvold, 2020). Time-reversal invariance secures the following: if a
sequence of states $\ket{\psi_{a}},\,\ket{\psi_{b}},\,\ldots,\,\ket{\psi_{z}}$
is dynamically allowed by unitary evolution under a Hamiltonian $H$, then a
sequence of generally distinct states $\mathcal{T}\ket{\psi_{z}}$, $\ldots$,
$\mathcal{T}\ket{\psi_{b}}$, $\mathcal{T}\ket{\psi_{a}}$ is also allowed by
unitary evolution governed by the time-reversed Hamiltonian
$\mathcal{T}H\mathcal{T}^{-1}=H$.

For this to be an obstacle to interpreting microphysics causally it would have
to be the case that given two states $\ket{\psi_{a}}$ and $\ket{\psi_{z}}$
related by some unitary evolution U(t), the time-reversal invariance makes it
impossible to determine whether $\ket{\psi_{a}}$ is the cause of
$\ket{\psi_{z}}$ or vice-versa. Time-reversal invariance is unecessary for
this; as I discussed above, this obstacle arises from the invertibility of the
time-evolution operators U(t) and that follows from Stone’s theorem, whether
the theory is time-reversal invariant or not. Time-reversal invariance is also
in general insufficient for this, except in the trivial sense that if quantum
theory is time-reversal invariant, that entails that one was able to define a
Hamiltonian for the system and that, in turn, gets us back to the real
obstacle to determining whether $\ket{\psi_{a}}$ is the cause of
$\ket{\psi_{z}}$ or vice-versa: the invertibility of U(t) ensured by Stone’s
theorem.121212For a related point about the relationship between time-reversal
invariance and determinism to the past and future and additional discussion,
see (Earman, 2002, section 4). If, on whatever basis, I were to claim to have
good reason to believe that $\ket{\psi_{a}}$ is the cause of $\ket{\psi_{z}}$,
it would do nothing to undermine my belief to tell me that the theory also
allows for a physically distinct state $\mathcal{T}\ket{\psi_{z}}$ to be the
cause of another physically distinct state $\mathcal{T}\ket{\psi_{a}}$.

This is especially true for the situation I am considering in this paper: I am
imagining that one knows the two states $\ket{\psi_{a}}$ and $\ket{\psi_{z}}$.
It is difficult to understand the belief that the difficulty with identifying
whether $\ket{\psi_{a}}$ causes $\ket{\psi_{z}}$ or vice-versa stems from a
fact about dynamically allowed sequences of states that are, in general,
physically distinct and observationally distinguishable from $\ket{\psi_{a}}$
and $\ket{\psi_{z}}$. The real difficulty with identifying cause and effect in
quantum theories stems from the invertibility of the dynamics: one cannot
distinguish between the two possibilities
$\ket{\psi_{a}}=\text{U(t)}\ket{\psi_{z}}$ and
$\ket{\psi_{z}}=\text{U(t)}^{-1}\ket{\psi_{a}}$ in the absence of information
about the temporal ordering of $\ket{\psi_{a}}$ and $\ket{\psi_{z}}$.

##  3 Interactions Generate Entanglement

Use of the Schmidt decomposition of quantum states is ubiquitous and useful
when discussing entanglement. I will rely on it throughout the paper so what
follows is a brief reminder about some of its relevant properties.131313See
(Nielsen and Chuang, 2010, section 2.5) for a pedagogical presentation.

Any pure state $\ket{\Psi}$ of a bipartite quantum system can be written as
its Schmidt decomposition

$\ket{\Psi}=\sum_{i}\lambda_{i}\ket{a_{i}}_{A}\otimes\ket{b_{i}}_{B}$

where $\\{\ket{a_{i}}_{A}\\}$ and $\\{\ket{b_{i}}_{B}\\}$ form orthonormal
bases for the Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$,
respectively. The coefficients $\lambda_{i}$ are called Schmidt coefficients
and are non-negative real numbers that satisfy $\sum\lambda_{i}^{2}=1$. The
Schmidt rank of $\ket{\Psi}$ is the number of non-zero Schmidt coefficients
$\lambda_{i}$; a state is entangled iff it has a Schmidt rank greater than 1
and it is “fully” entangled – in a sense I will elaborate on in a moment – if
its Schmidt rank is equal to
$\min\\{\dim(\mathcal{H}_{A}),\,\dim(\mathcal{H}_{B})\\}$. For simplicity I
will consider systems where $\dim(\mathcal{H}_{A})=\dim(\mathcal{H}_{B})$, but
nothing in the paper depends on this.

If the Schmidt coefficients are non-degenerate, then the basis vectors
$\ket{a_{i}}_{A}$ and $\ket{b_{i}}_{B}$ are unique (up to a phase) so the
Schmidt decomposition of a state $\ket{\Psi}$ is itself unique (up to a
phase). If any of the Schmidt coefficients are degenerate then there is more
freedom in the choice of basis vectors, but both the Schmidt rank and the
values of the Schmidt coefficients are the same for any allowed choice of
basis. A familiar example of this latter case is the EPR state. It can be
Schmidt decomposed as

$\ket{\text{EPR}}=\frac{1}{\sqrt{2}}\ket{0}_{A}\ket{0}_{B}+\frac{1}{\sqrt{2}}\ket{1}_{A}\ket{1}_{B}$

where the Schmidt coefficients are
$\lambda_{1}=\lambda_{2}=\frac{1}{\sqrt{2}}$. One could equally well have
written the state in the Schmidt decomposition

$\ket{\text{EPR}}=\frac{1}{\sqrt{2}}\ket{+}_{A}\ket{+}_{B}+\frac{1}{\sqrt{2}}\ket{-}_{A}\ket{-}_{B}$

where $\ket{\pm}=\frac{1}{\sqrt{2}}\ket{0}\pm\frac{1}{\sqrt{2}}\ket{1}$. The
values of the Schmidt coefficients are unchanged, while the non-uniqueness of
the basis reflects the degeneracy of those coefficients.

This highlights a useful feature of the Schmidt decomposition: the Schmidt
coefficients are invariant under unitary operations $U_{A}\otimes U_{B}$ that
act on Alice and Bob’s subsystems alone and, as an obvious corollary, the
Schmidt rank is invariant too. This reflects the physical fact that
entanglement between separated systems cannot be created by unitary operations
performed locally on each system.141414It also reflects the mathematical fact
that the amount of entanglement between two systems is independent of a
unitary change of basis. As a result, the set of Schmidt coefficients
$\lambda_{i}$ completely characterizes the entanglement of a bipartite system
in a pure state and is sometimes referred to as the entanglement spectrum.

This brings me to the sense in which a bipartite pure state with full Schmidt
rank can be considered “fully” entangled. Consider an entangled bipartite
system $\ket{\Psi}\in\mathcal{H}_{AB}$ with
$\dim(\mathcal{H_{A}})=\dim(\mathcal{H_{B}})=N$. If $\ket{\Psi}$ has Schmidt
rank 1 then it is separable and Alice and Bob’s measurement results will be
uncorrelated for any measurements they perform. If $\ket{\Psi}$ has Schmidt
rank N, however, Alice and Bob’s results will be correlated for every
measurement they perform in the same Schmidt basis on $\ket{\Psi}$: if Alice
finds her system in state $\ket{a_{k}}$ then Bob will find his in state
$\ket{b_{k}}$, and so on. This is the sense in which the two subystems are as
entangled as they could be.

The Schmidt rank may seem like a rather coarse entanglement measure. It does
not distinguish states that an apparently more fine-grained measure of
entanglement, like the commonly adopted entanglement entropy, would
distinguish. Consider the density operators $\rho_{1}$ and $\rho_{2}$
corresponding to the entangled states

$\ket{\Psi_{1}}=\frac{1}{\sqrt{2}}\ket{11}+\frac{1}{\sqrt{2}}\ket{00}$

and

$\ket{\Psi_{2}}=\sqrt{1-\varepsilon}\ket{11}+\sqrt{\varepsilon}\ket{00}$

defined on $\mathcal{H}_{AB}$ with
$\dim(\mathcal{H}_{A})=\dim(\mathcal{H}_{B})=2$. Both $\rho_{1}$ and
$\rho_{2}$ have the full Schmidt rank 2. However, the entanglement entropy of
a density operator

$EE(\sigma)=-\Tr(\sigma\log\sigma)=-\sum_{i}\lambda^{2}_{i}\log\lambda_{i}^{2}$

distinguishes the two: $EE(\rho_{1})=1$ while $EE(\rho_{2})$ is
$\order{\varepsilon}$.151515The squares of the Schmidt coefficients
$\lambda_{i}^{2}$ in the Schmidt decomposition of a bipartite pure state
$\ket{\Psi}$ are the eigenvalues of the density operator
$\sigma=\outerproduct{\Psi}{\Psi}$. A convenient fact about the Schmidt
decomposition is that the $\lambda_{i}^{2}$ are also the eigenvalues of each
of the reduced density operators $\rho_{A}$ and $\rho_{B}$ representing each
entangled subystem. Since these eigenvalues fully determine the entanglement
entropy, the Schmidt decomposition reveals that one can calculate the
entanglement entropy of the bipartite system in two equivalent ways: by
computing the Shannon entropy of the probability distribution generated by the
square of the coefficients of $\sigma$ for the full bipartite system or, as is
more common, computing the von Neumann entropy of the reduced density
operators $\rho_{A}$ or $\rho_{B}$. Why not use the apparently more fine-
grained measure of entanglement?

My reasons are partially pragmatic. The first such reason is that such a
measure is more fine-grained than I need for present purposes since the
topological, measure-theoretic, and dynamical facts about quantum states that
I invoke in the remainder of this section can be proved using only information
about the Schmidt rank. The second is that there is virtue in adopting a
measure of entanglement can be extended to multipartite systems and to mixed
states. As a measure of entanglement, the Schmidt rank generalizes naturally
to such cases: the Schmidt measure (Eisert and Briegel, 2001), (Hein et al.,
2004, section II.c) or the Schmidt number (Terhal and Horodecki, 2000; Bruß,
2002; Sperling and Vogel, 2011) are generalizations of the Schmidt rank to
multipartite pure and mixed states. In fact, the availability of the Schmidt
measure will be important in section 5 when I consider a multipartite system
of N qubits. A third reason, closely related to the second, is that the
Schmidt measure has been used to give a definition of the algorithmic
information content (also called the algorithmic complexity or the Kolmogorov
complexity) of quantum states (Mora and Briegel, 2005, 2006). I will make use
of this connection when I turn to causal inference in section 5.

My reasons are not entirely pragmatic, however, and there are two conceptual
issues worth highlighting before moving on. There is a sense in which the
Schmidt rank and its generalizations are not measures of precisely the same
property as the entanglement entropy. Consider the following remarks from
(Preskill, 1998) and (Bruß, 2002):

> So a number used to quantify entanglement ought to have the property that
> local operations do not increase it. An obvious candidate is the Schmidt
> [rank], but on reflection it does not seem very satisfactory. Consider
>
>
> $\ket{\Psi_{\varepsilon}}=\sqrt{1-2\left|\varepsilon^{2}\right|}\ket{00}+\varepsilon\ket{11}+\varepsilon\ket{22}$
>
> which has Schmidt [rank] 3 for any $\left|\varepsilon\right|>0$. Should we
> really say that $\ket{\Psi_{\varepsilon}}$ is “more entangled” than
> $\ket{\phi^{+}}=\frac{1}{\sqrt{2}}(\ket{11}+\ket{00})$? Entanglement, after
> all, can be regarded as a resource – we might plan to use it for
> teleportation, for example. It seems clear that $\ket{\Psi_{\varepsilon}}$
> (for $\left|\varepsilon\right|\ll 1$) is a less valuable resource than
> $\ket{\phi^{+}}$ (Preskill, 1998, chapter 5.5).161616The Schmidt rank of a
> bipartite pure state is sometimes referred to interchangeably as its Schmidt
> number. This is unfortunate since I will occasionally mention a quantity
> introduced by (Terhal and Horodecki, 2000) that they call the Schmidt
> number, which is an extension of the Schmidt rank to mixed states. I’ve
> altered Preskill’s terminology to cohere with mine in this paper.

Bruß draws a similar distinction as Preskill while introducing a method for
determining the Schmidt number of an arbitrary mixed state:

> A slightly different question from “how much entangled is a state $\rho$?”
> can be addressed via the generalization of entanglement witnesses to so-
> called Schmidt witnesses. They give an answer to the question “how many
> degrees of freedom are entangled in $\rho$?” This corresponds to a finer
> classification of entangled states (Bruß, 2002, section V.C).

It is not hard to find similar remarks in the literature on entanglement which
suggest that rather than the entanglement entropy being a finer measure than
the Schmidt rank or vice versa, the two instead capture subtly different
aspects of entanglement.171717For example, see the remarks in (Sanpera et al.,
2001) or, more substantively, the proof in (Van den Nest, 2013) mentioned
below. This is not to say that there is never any relationship between the
entanglement entropy and the Schmidt rank; for example, see (Sperling and
Vogel, 2011, section III). For philosophical discussion of some of the
multiple notions of entanglement see (Earman, 2015). Heuristically, the
entanglement entropy quantifies how strongly subsystems are entangled while
the Schmidt rank captures how broadly the subsystems are entangled. This
distinction can track important practical differences in certain contexts: for
example, it was proved in (Van den Nest, 2013) that the Schmidt rank is an
informative measure of entanglement for determining the potential speedup of a
quantum computation over a classical one, but any entanglement measure that is
continuous and vanishes on product states – like the entanglement entropy – is
not.

The second conceptual issue concerns a connection between measures of
entanglement and measures of information that I will exploit when discussing
causal inference.181818See (Timpson, 2013, chapters 2.2 & 3.6) for a clear
presentation and philosophical discussion of the interpretation of the Shannon
and von Neumann entropies as measures of information. The von Neumann entropy
– what I have thus far been calling the entanglement entropy – can also be
given an information-theoretic interpretation as a quantum analogue of Shannon
information. The Shannon information does not quantify the information content
of an individual message, but rather characterizes an ensemble of messages
produced by a specified source that produces each bit $x_{i}$ with some
probability $p_{i}$. The Shannon information quantifies the minimum number of
bits required to encode an arbitrary message selected from an ensemble of
typical messages produced by the source. This is also true of the von Neumann
entropy; a quantum source prepares qubits in pure states $\ket{x_{i}}$ with
probabilities $p_{i}$ and the entanglement entropy quantifies the minimum
number of qubits required to transmit a typical message selected from the
ensemble of messages produced by the specified quantum source.

The causal inference methods I will use in the subsequent sections rely on the
notion of algorithmic information.191919See (Grunwald and Vitányi, 2010) for a
helpful comparison of Shannon information and algorithmic information. This is
not a measure of the amount of information required to describe a typical
object in a specified ensemble but of the “intrinsic” amount of information
contained in the object considered on its own. I will have more to say about
algorithmic information in section 4 but what matters presently is that the
algorithmic information content of an object is not a statistical property of
an ensemble containing that object. This suggests that the entanglement
entropy cannot provide a satisfactory measure of the algorithmic information
content of an individual pure quantum state. The Schmidt rank of a quantum
state is a property of that state alone and is thus a more appropriate measure
for drawing a connection between entanglement and the algorithmic information
content of a quantum state. (They do not motivate their choice in this way but
this does make the use of the Schmidt measure by (Mora and Briegel, 2005,
2006) a conceptually natural choice for explicitly connecting entanglement and
the algorithmic information content of a quantum state.)

With those justifications provided, I will now make use of the Schmidt rank to
introduce some topological, measure-theoretic, and dynamical facts about the
ubiquity of entangled pure states in the state space $\mathcal{H}_{AB}$ of a
bipartite quantum system. All of these facts are well-known; my aim in
rehearsing them is to lay out the physical grounds of the causal asymmetry for
which I argue at the end of this section, as well as to make maximally
plausible the similar conclusions I draw using causal inference methods in
section 5. If this discussion results in those conclusions seeming inevitable
then it will have been successful.

First consider the case where $\mathcal{H}_{AB}$ is finite-dimensional. Recall
that the rank of a matrix is the number of linearly independent rows (or
columns) it contains and that a matrix M of less-than-full rank has
$\det(M)=0$. This means that the density matrix corresponding to any bipartite
pure state of less-than-full Schmidt rank has determinant equal to zero. One
can use this fact to show that the set of pure states in $\mathcal{H}_{AB}$
with less-than-full Schmidt rank is nowhere-dense in the set of all pure
states in the topology induced by any norm on the space.202020See (Brock,
2005) for an elementary proof. The fact that all norms on a finite-dimensional
vector space are equivalent justifies the statement that this is true for “any
norm” on $\mathcal{H}_{AB}$. This means that each pure state in
$\mathcal{H}_{AB}$ of less-than-full Schmidt rank is enveloped by a ball of
fully entangled pure states; equivalently, each pure state of less-than-full
Schmidt rank is a limit point of a sequence of fully entangled pure states.

The set of pure states of less-than-full Schmidt rank is also sparse measure-
theoretically: it has measure zero in the set of all pure states in
$\mathcal{H}_{AB}$.212121In the finite-dimensional case, the restriction to
pure states is important: for mixed states of a bipartite system, the
separable states are no longer as sparse, and the set of separable mixed
states always contains an open ball around the maximally mixed state
(Życzkowski et al., 1998). For an infinite-dimensional Hilbert space, this is
no longer true: as (Clifton and Halvorson, 1999) showed, the set of mixed
states is nowhere dense in the set of all states, in the topology induced by
the trace-norm. This is well-known but I couldn’t find a proof to cite, so
here is a proof sketch. Since
$\mathcal{H}_{AB}\cong\mathbb{C}^{\text{N}\times\text{N}}$, the determinant
$\det\colon\mathbb{C}^{\text{N}\times\text{N}}\to\mathbb{C}$ is a polynomial
in the complex entries of an $\text{N}\times\text{N}$ matrix and any complex
polynomial is a holomorphic function. The zero set Z(f) of any holomorphic
function – the set of points in the domain of f on which that function equals
zero – is at most countable (Rudin, 1987, theorem 10.18) and so the
determinant equals zero on at most countably many points in
$\mathbb{C}^{\text{N}\times\text{N}}$. The determinant of a density matrix
vanishes if and only if it has less-than-full Schmidt rank so there can be at
most countably many such density matrices. Using
$\mathbb{C}^{\text{N}\times\text{N}}\cong\mathbb{R}^{2\text{N}\times
2\text{N}}$ we can conclude that the set of density matrices with less-than-
full Schmidt rank has Borel measure zero.

In light of the sparseness of separable pure states it may seem obvious that
non-trivial time evolution on $\mathcal{H}_{AB}$ – any time evolution that
includes interactions between Alice and Bob’s subsystems – has to take any
pure state $\ket{\psi}_{AB}$ that is not fully entangled into a fully
entangled state, and that the curve traced out by U(t)$\ket{\psi}_{AB}$ will
consist almost entirely of fully entangled states.222222I will drop the “non-
trivial” qualifier for the remainder of the paper. Considering time evolutions
that are non-trivial in this sense is equivalent to requiring that the time
evolution operator U(t) does not factorize into a product
$\text{U(t)}_{A}\otimes\text{U(t)}_{B}$ of operators evolving Alice and Bob’s
subsystems independently. Any unitary operator that factorizes in this way
cannot change the Schmidt coefficients of a quantum state, as I mentioned
previously, and so cannot create entanglement between subsystems. This is
true.232323A general argument is sketched in (Binney and Skinner, 2013,
section 6.1). There are an immense number of concrete examples of dynamically
generated entanglement; see, for instance, the discussions in many-body
physics (Calabrese and Cardy, 2006; Eisert and Osborne, 2006; Amico et al.,
2008), quantum field theory (Peschanski and Seki, 2016; Cervera-Lierta et al.,
2017; Kharzeev and Levin, 2017), and non-relativistic quantum mechanics
(Mishima et al., 2004; Schroeder, 2017). In fact, the previous discussion
ensures that any initial bipartite state with less-than-full Schmidt rank
will, after an infinitesimal time interval, develop into a state with full
Schmidt rank and will then remain in a state of full Schmidt rank for all
subsequent times except for a set of measure zero.

This should not be surprising at this point. The topological sparseness of
pure states of less-than-full Schmidt rank means that there is nowhere else
for those states to evolve except into states with full Schmidt rank, at least
for infinitesimal time evolution, and their measure-theoretic sparseness means
that any curve through $\mathcal{H}_{AB}$ generated by time evolution for any
finite time interval will pass through a state of less-than-full Schmidt rank
in at most countably many instants. The upshot of this is that if one chooses
an arbitrary state $\ket{k}$ of less-than-full Schmidt rank k and evolves it
for any finite time t, except for set of isntants of measure zero the state
U(t)$\ket{k}$ will be a state of full Schmidt rank.

Now suppose instead that one is interested in the time evolution of an
arbitrary state $\ket{N}$ of full Schmidt rank N. Are there any time evolution
operators U(t) such that U(t)$\ket{N}$ has less-than-full Schmidt rank? Since
specifying a Hilbert space representation for a quantum system requires fixing
a Hamiltonian $H$, the only freedom in choosing U(t) comes from fixing
t.242424This is clearest in the algebraic approach to quantum theories, where
a quantum system is defined by a C∗ algebra. The GNS reconstruction theorem
enables one to move from the abstract C∗ algebra to a Hilbert space
representation for the system, but this requires data about the full C∗
algebra, including the Hamiltonian.. Thus this question is equivalent to
asking how many instants $t_{i}$ there are such that U($t_{i}$)$\ket{N}$ has
less-than-full Schmidt rank.

One such operator can obviously be reverse-engineered. Find a state of less-
than-full Schmidt rank $\ket{k}$ such that U(t)$\ket{k}=\ket{N}$. Applying
$\text{U(t)}^{-1}$ to $\ket{N}$ will then produce $\ket{k}$, a state of less-
than-full Schmidt rank. Are there more such operators?

There can be at most countably many. Let $\tau_{R}$ be the recurrence time for
the system and consider the operator U($\tau_{R}$).252525See (Wallace, 2015)
for proofs that in any finite-dimensional quantum system, and any infinite-
dimensional quantum system that satisfies modest constraints on the
Hamiltonian, there exists a single time $\tau_{R}$ such that after $\tau_{R}$
every state in $\mathcal{H}_{AB}$ will have returned to a state arbitrarily
close to itself. Suppose that one starts with the state $\ket{N}$ and that the
dynamics are such that $\ket{N}$ visits every state of less-than-full Schmidt
rank in $\mathcal{H}_{AB}$ between $t=0$ and $t=\tau_{R}$. Then the curve
$\text{U}(\tau_{R})\ket{N}$ will visit countably many states of less-than-full
Schmidt rank on its tour through $\mathcal{H}_{AB}$. Any finite time interval
t can be split up into at most n intervals of length $\tau_{R}$ (plus some
remainder q) so we can write $\text{U(t)}=\text{U}(n\tau_{R}+q)$ and, since a
finite union of countable sets is itself countable, the state $\ket{N}$
evolves into a state of less-than-full Schmidt rank for at most countably many
instants. This means that the set of times $t_{i}$ for which
U($t_{i}$)$\ket{N}$ has less-than-full Schmidt rank has measure zero.

Recall that I began with a simple problem: if one is given two bipartite pure
states $\ket{\alpha}$ and $\ket{\beta}$ and knows that they are related by
unitary time evolution, can they determine whether $\ket{\alpha}$ was the
cause of $\ket{\beta}$ or vice versa? My discussion of the sparseness of pure
states of less-than-full Schmidt rank in $\mathcal{H}_{AB}$ lays the
groundwork for an answer. Suppose that either of the following is
true:262626Note that this excludes the case where both $\ket{\alpha}$ and
$\ket{\beta}$ are entangled but neither has full Schmidt rank. I’m not aware
of any demonstration that within the set of bipartite pure states of less-
than-full Schmidt rank, the states of lower rank are less prevalent than those
of higher rank. The causal inference methods used in section 5 will do better
for this case.

  1. 1.

The state $\ket{\alpha}$ is separable (has Schmidt rank 1) and $\ket{\beta}$
is entangled (has Schmidt rank $>$1);

or

  2. 2.

Both of the states $\ket{\alpha}$ and $\ket{\beta}$ are entangled but
$\ket{\alpha}$ has less-than-full Schmidt rank and $\ket{\beta}$ has full
Schmidt rank.

It is then overwhelmingly likely that $\ket{\beta}$ was caused by time-
evolving $\ket{\alpha}$ rather than vice versa.

The intuitive reason for this is that it is overwhelmingly likely that a state
of low Schmidt rank will evolve into a state of higher Schmidt rank but the
converse is extremely unlikely. If the state $\ket{\alpha}$ is the cause then
it is almost certain that the state $\text{U(t)}\ket{\alpha}$ would have
higher Schmidt rank, for an arbitrarily chosen t. One does not have to know
anything about the dynamics – about how $\ket{\alpha}$ would evolve over
different intervals t – to be essentially certain that the resultant state
will have full Schmidt rank. If one evolved the state $\ket{\alpha}$ for
$t+dt$ or $t+10^{23}$, or $t+\pi t$, or $\ldots$ the resultant state would not
be $\ket{\beta}$, but one can be essentially certain that it would be a fully
entangled state. That it happens to be $\ket{\beta}$ in particular reflects
the unmysterious fact that one evolved the state $\ket{\alpha}$ for t rather
than some other interval of time. In short, identifying $\ket{\alpha}$ as the
cause renders the effect $\ket{\beta}$ utterly unsurprising.

If instead one adopts $\ket{\beta}$ as the cause, then the effect
$\ket{\alpha}$ is rendered fantastically surprising. It is impossibly unlikely
that $\ket{\beta}$ just happened to evolve for precisely an interval t lying
in the measure-zero set of times that would result in $\text{U(t)}\ket{\beta}$
having less-than-full Schmidt rank. It suggests an incredible degree of fine-
tuning; indeed, it is nearly impossible to imagine a human experimenter who
was trying to accomplish such a precise fine-tuning doing so successfully.
Even with complete knowledge of both the state $\ket{\beta}$ and its behavior
under the dynamics – how it would evolve under U(t) for different t – and the
explicit goal of producing a state of lower-Schmidt rank, they would have to
control the experiment with a level of precision typically reserved for
deities.272727An observation about the impracticality of arranging a similar
evolution was made in a different context by (Englert et al., 1988). They
consider sending a spin-1/2 particle in a $\sigma_{x}$ eigenstate through a
Stern-Gerlach device set to measure $\sigma_{z}$, which will split the
incoming beam into a superposition of the two eigenstates of $\sigma_{z}$.
They investigate the degree of precision with which an experimenter would need
to control the magnetic field in the Stern-Gerlach device to ensure that the
spin-1/2 particle returns to its original $\sigma_{x}$ eigenstate after
passing through the Stern-Gerlach device. They show that an exact return to
the original $\sigma_{x}$ eigenstate is unattainable in practice and that even
to reproduce the original state with 99% accuracy would require the ability to
control the gradient of the macroscopic magnetic field in the Stern-Gerlach
device to at least 5 decimal places. In fact, insofar as one thinks that a
cause ought to explain its effects then identifying $\ket{\beta}$ as the cause
is unacceptable: it not only fails to explain $\ket{\alpha}$ in any meaningful
sense but also raises additional explanatory concerns more urgent than those
with which one began.

Some readers may have noticed a parallel with classical statistical mechanics:
precisely this form of reasoning has been used in that context to argue that
there is an identifiable explanatory asymmetry between two microstates despite
the time-symmetry of the dynamical laws describing their time evolution. For
example, (Maudlin, 2007, p. 132) introduces the problem as follows:

> $\ldots$ postulate a macroscopically atypical [low entropy] but
> microscopically typical [chosen at random from microstates compatible with
> the atypical macrostate] state, plus the laws, and one can explain the
> macroscopically typical [high entropy] but microscopically atypical [evolves
> to a lower entropy macrostate in one temporal direction] state from them:
> the latter was generated from the former by means of the operation of the
> laws. But equally: postulate a macroscopically typical but microscopically
> atypical state at one end, plus the laws, and one can ‘generate’ a
> macroscopically atypical but microscopically typical state from them. Pick
> one end, add the laws, and you can explain the other end: which end you pick
> as explanans and which as explanandum is up to you.

Maudlin argues that this apparent explanatory symmetry is illusory, invoking a
principle that lies at the foundation of the causal inference methods I
discuss in section 4. He points out that the asymmetry is hiding in the way
that one specifies the explanans microstate and the explanandum microstate in
the two candidate explanations. Consider explaining an atypical microstate in
a high-entropy macrostate by demonstrating that it resulted from the time
evolution of a typical microstate in a low-entropy macrostate. The meaning of
“typical” used to describe the explanans is measure-theoretic: the low-entropy
macrostate is atypical because it occupies a small volume of the system’s
phase space and the microstate is typical because it was chosen at random from
that low-entropy macrostate. Importantly, one can entirely specify an
explanans that will produce the desired explanandum while knowing basically
nothing about the dynamics governing the system.

This is no longer true if one tries to invert the order of explanation. If one
chooses as the explanans an atypical microstate in a high-entropy macrostate,
the meaning of “atypical” becomes crucially different. “Atypical” now means
dynamically atypical: the microstate is atypical in the new sense if its time
evolution takes it into a low-entropy macrostate. Such atypical microstates
are scattered throughout the volume of the high-entropy macrostate and share
no common property that could be used to identify them except their dynamical
behavior. Specifying an atypical microstate in this sense requires thorough
knowledge of the dynamics governing its evolution. Not only can such
microstates only be specified by describing their dynamical behavior, but
their identity as atypical microstates depends very sensitively on the
specific form of the dynamics governing the system. As Maudlin points out,

> $\ldots$ a slight modification of the dynamical laws would lead to
> essentially no change in which initial states are macroscopically atypical,
> in that they have low entropy, but would completely alter the set of
> atypical high-entropy states whose time evolution in either direction leads
> to low entropy (Maudlin, 2007, p. 133) [my emphasis].

The explanatory asymmetry comes from this asymmetry in how we have to specify
the explanans and the explanandum in the two cases. In one direction, one can
randomly pick a microstate from a low-entropy macrostate, knowing essentially
nothing about the specific dynamics governing the system, and then show that
those dynamics will evolve that explanans microstate into a microstate lying
in a high-entropy macrostate. In the other direction, one needs detailed
knowledge about the specific form of the dynamics to even know what counts as
an atypical microstate, let alone to select as their explanans an atypical
microstate that the dynamics will evolve into a low-entropy macrostate.
Maudlin argues that this is no explanation at all: it amounts to explaining
the system’s evolution into a low-entropy macrostate by saying that the
initial conditions were such that they would evolve into a low-entropy
macrostate, given the dynamics governing the system. One might as well ascribe
such microstates a virtus entropia.282828Sklar expresses a similar
dissatisfaction with such candidate explanations of why subsystems of a larger
system obey the second law: “$\ldots$ we would simply posit an initial state
that gives rise to parallel entropic increase in branch systems with each
other and with the main system. But to characterise the state in that way
would, of course, not be offering us an explanation of the sort we expected.
It would be one thing to be able to characterize the initial state in some
simple way$\ldots$ and be able to derive the Second Law from that. But to
derive the Second Law from a bald assertion that “initial conditions were such
that they would lead to Second Law behavior” hardly seems of much interest”
(Sklar, 1993, p. 330).

The parallel strategy for using entanglement to identify explanatory
asymmetries between quantum states is clear. The set of all states in
$\mathcal{H}_{AB}$ can be partitioned into disjoint subsets containing states
with Schmidt rank 1, 2, $\ldots$, $\dim(\mathcal{H}_{AB})$. We know from the
above discussion that almost all states in $\mathcal{H}_{AB}$ will be in the
subset of states with full Schmidt rank; call a quantum state in
$\mathcal{H}_{AB}$ “macroscopically atypical” if it lies in any set of states
with less-than-full Schmidt rank. A state within that set is “microscopically
typical” if it is chosen from that set at random. A state is “macroscopically
typical” in $\mathcal{H}_{AB}$ if it is in the set of states with full Schmidt
rank; a state within that set is “microscopically atypical”, given a specific
time-evolution operator U($t_{i}$), if it evolves into a state of less-than-
full Schmidt rank.

The discussion above establishes that time evolution from full Schmidt rank to
less-than-full Schmidt rank is highly non-generic; the only way to specify an
explanans state $\ket{\phi}$ that would do so under U($t_{i}$) requires
extremely detailed knowledge of both the details of the Hamiltonian $H$
appearing in U($t_{i}$) and the duration of time evolution $t_{i}$. Just like
in the classical statistical mechanical case, “explaining” a state
$\ket{\phi_{1}}$ of less-than-full Schmidt rank as resulting from the time
evolution of a state $\ket{N}$ of full Schmidt rank amounts to saying that
$\ket{\phi_{1}}$ occurred because the initial condition $\ket{N}$ was such
that it would evolve to a state of lower Schmidt rank, given the dynamics
U($t_{i}$).

Maudlin extracts from his discussion a principle for evaluating explanations,
though he doesn’t dwell on it:292929Although Maudlin doesn’t dwell on it,
(Woodward, 2020) has given an extended and enlightening examination of the
very similar principle that forms the foundation of the causal inference
methods I describe in section 4.

> The problem is this: in order to account for the universe as we see it, we
> need more than the laws: we need a constraint on one of the boundaries. That
> constraint, together with the operations of the laws, then suffices to
> account for the nature of the other boundary. But in order for this to work
> [be explanatory] the constraint must itself be specifiable independently of
> what will result from the operation of the laws (Maudlin, 2007, p. 132).

I think there is an argument to be made at this point that as long as two
quantum states satisfy one of the two conditions outlined above, my initial
goal of identifying a causal asymmetry between quantum states related by time-
symmetric dynamics has been accomplished. Recall that I have adopted a minimal
interventionist notion of causation in this paper: X is a cause of Y iff there
is an intervention that can be performed on the value of X, while holding all
other variables fixed, that produces a change in the value of Y. Two quantum
states $\ket{\alpha}$ and $\ket{\beta}$ related by a unitary time evolution
U(t) satisfy this condition, but they satisfy it for both candidate causal
orderings: intervening to set $\ket{\alpha}$ to $\ket{\alpha^{\prime}}$ will
produce a change in $\ket{\beta}$, since
U(t)$\ket{\alpha^{\prime}}\neq\ket{\beta}$. But this is also true if the
candidate causal order is inverted: intervening to set $\ket{\beta}$ to
$\ket{\beta^{\prime}}$ will produce a change in $\ket{\alpha}$ since
U(t)-1$\ket{\beta^{\prime}}\neq\ket{\alpha}$.

Some additional information is required to break the symmetry. The above
discussion suggests that, under conditions widely satisfied in quantum
theories, one has such information: one should identify the “cause” state as
the state that can produce the effect state and can be specified without any
reference to its behavior under the dynamics U(t). In the case under
consideration, this amounts to identifying the “cause” state as either (1) the
state with Schmidt rank 1, if the other state has Schmidt rank $>$ 1 or (2)
the state with less-than-full Schmidt rank, if the other state has full
Schmidt rank. Many people have claimed that the time-symmetry of the dynamics
governing microscopic systems makes any such asymmetry impossible; I have
argued that this is not correct.

That said, I want to emphasize some limitations of my discussion thus far.
First, I have discussed only microscopic systems par excellence: pure states
of bipartite systems related by unitary evolution. Second, conditions (1) or
(2) are satisfied for a pair of states only when one of those states comes
from a set of measure zero: the set of states of less-than-full Schmidt rank.
It is true that such states, particularly product states, play a more
significant role in foundational and mathematical discussions of quantum
mechanics than their sparseness in $\mathcal{H}_{AB}$ alone might suggest, but
they are sparse nonetheless. I invoked an analogy with the case of classical
statistical mechanics above, but this is an important disanalogy: the measure
of the set of low-entropy microstates is exponentially small and vanishes as
the size of the system $N\to 0$, but it is non-zero for any finite system; for
estimates and discussion of classical and quantum multipartite systems
(including exceptions to this estimate), see (Goldstein et al., 2015b) and
(Goldstein et al., 2017, section 7). I will return to this disanalogy in
section 5.

The final limitation is that while conditions (1) and (2) show that one can,
in principle, identify a causal asymmetry in quantum theories, they are of
little practical value for the real-world epistemology of causation. The
argument presented in this section does identify an asymmetry present in the
mathematical foundations of quantum theories, but it relies on a standard of
precision that unsatisfiable in practice. I have been considering states that
are exactly product states or, more generally, states whose Schmidt
decomposition has exactly k non-zero coefficients. This presumption of
precision is often made without comment in foundational discussions but is
never achievable – or at least never verifiably achievable – in any real world
situation. At best, one can verify that a state is approximately a product
state or that a state has k Schmidt coefficients larger than some
$\varepsilon>0$. However, the topological sparseness of the states of less-
than-full Schmidt rank means that if $\ket{\psi}$ has less-than-full Schmidt
rank, any state $\ket{\phi}$ that is within $\varepsilon$ of $\ket{\psi}$ in
the Hilbert space norm will have full Schmidt rank. Conditions (1) and (2) no
longer identify a causal asymmetry if modified to

  1. $1_{\varepsilon}$: The state $\ket{\alpha}$ is within $\varepsilon$ of a state that is separable (has Schmidt rank 1) and $\ket{\beta}$ is within $\varepsilon$ of a state that is entangled (has Schmidt rank $>$1), or;

  2. $2_{\varepsilon}$: Both of the states $\ket{\alpha}$ and $\ket{\beta}$ are entangled but $\ket{\alpha}$ is within $\varepsilon$ of a state that has less-than-full Schmidt rank and $\ket{\beta}$ is within $\varepsilon$ of a state that has full Schmidt rank.

In section 5 I will show that one can do better than this while still making
use of a principle much like the one Maudlin employed to identify an
explanatory asymmetry in classical statistical mechanics and which I have used
in quantum theories. Such a principle provides part of the foundation of a set
of mathematical and conceptual methods for causal inference that have proven
empirically reliable in a number of disparate domains. My aim is to make use
of those methods to give a formal demonstration that the presence of
entanglement can be used to infer causal direction in quantum theories under
conditions that are less restrictive than those invoked in this section.

##  4 Causal Inference Methods

I began with the problem of determining whether, and under what conditions,
one could identify a causally relevant asymmetry between quantum states
related by unitary evolution. Faced with such a problem, it would be natural
to turn to the tools of causal inference. Unfortunately, if one takes up
commonly used tools to infer causal relationships from observational data that
rely solely on conditional statistical independence, such as those in (Pearl,
2009, chapter 2), they will find those tools inadequate for the problem at
hand. The reason for this is fairly straightfoward.

Any causal graph relating classical statistical variables V1, V2, $\ldots$,
V${}_{\text{N}}$ defines a joint probability distribution P(V1, V2, $\ldots$,
V${}_{\text{N}}$) over those variables.303030The restriction to classical
statistical variables is important because a joint probability distribution
for non-commuting variables, like conjugate observables in quantum mechanics,
is generally not well-defined. If one restricts to quantum observables that
commute then one can define a joint probability distribution over their
possible values. See (Fine, 1982) for a review and connection to hidden
variable theories. Consider a simple graph relating V1, V2, and V3:

$V_{3}$$V_{2}$$V_{1}$

This graph defines a set of possible joint probability distributions P(V1, V2,
V3). Suppose that all causal relations are deterministic; then possible joint
distributions could be P(V1=1, V2=0, V3=1)=1, P(V1=0, V2=0, V3=1)=0, and so
on. Causal inference problems begin with a joint probability distribution over
variables V1, V2, $\ldots$, V${}_{\text{N}}$ and attempt to reconstruct the
causal graph representing the true causal structure that generates the given
probability distribution.

For each candidate causal graph, one typically requires that a probability
distribution satisfies two conditions relative to that graph: the Causal
Markov Condition and a faithfulness condition.313131See, for example, (Peters
et al., 2017, chapter 6.5), (Spirtes et al., 2000, chapter 3), or (Pearl,
2009, chapter 2.4), where the faithfulness condition is called “stability”.
The Causal Markov Condition is a constraint on how the joint probability
distribution P(V1, V2, $\ldots$, V${}_{\text{N}}$) factorizes into a product
of conditional dependence relations according to a candidate causal graph.
Speaking loosely, one requires that if all of the direct causes of a variable
O in the causal graph are specified, then one cannot learn any additional
information about the value of O from any variable that is not itself a
descendent of O. Formally, one requires that any candidate causal graph
satisfy:

$P(\text{V}_{1},\,\ldots,\,\text{V}_{\text{N}})=\prod_{k=1}^{n}P(\text{V}_{\text{k}}\mid
PA(\text{V}_{\text{k}}))$

where $PA(\text{V}_{\text{k}})$ represents the direct causes (or “parents”) of
the variable Vk. This captures the requirement that all of the conditional
dependence relations in the joint distribution P(V1, V2, $\ldots$,
V${}_{\text{N}}$) be accounted for by causal relations between V1, V2,
$\ldots$, V${}_{\text{N}}$ in the causal graph.

Faithfulness requires that only the conditional dependence relations in the
joint probability distribution are reflected in the causal relations between
V1, V2, $\ldots$, V${}_{\text{N}}$ in the causal graph. The idea is easiest to
illustrate with an example. Suppose one knows the joint distribution P(V1, V2,
V3) and suppose that V${}_{1}\perp\\!\\!\\!\perp\text{V}_{2}$, but
V${}_{1}\not{\perp\\!\\!\\!\perp}\text{V}_{2}\,\mid\text{V}_{3}$. (Notation:
$\perp\\!\\!\\!\perp$ indicates statistical independence and
$\not{\perp\\!\\!\\!\perp}$ indicates statistical dependence.) This is
consistent with both of the following causal graphs:

$V_{3}$$V_{2}$$V_{1}$

$V_{3}$$V_{2}$$V_{1}$

Further suppose that, in the graph on the right, the causal influence of V1 on
V2 is precisely canceled by the causal influence of V3 on V2. Then both of
these graphs entail V${}_{1}\perp\\!\\!\\!\perp\text{V}_{2}$ and
V${}_{1}\not{\perp\\!\\!\\!\perp}\text{V}_{2}\,\mid\text{V}_{3}$, but there is
something unsatisfying about the graph on the right: the required conditional
dependence relations have been recovered only by fine-tuning causal influences
to cancel precisely. This is why (Pearl, 2009, chapter 2.4) calls this a
stability condition: holding fixed the strength of the causal influence V1 on
V2 while perturbing the strength of the causal influence of V3 on V2 by any
$\delta>0$ will destroy the conditional independence relations. It is these
kind of finely-tuned graphs that are ruled out by
faithfulness.323232Faithfulness is often motivated by the fact that the set of
parameter values quantifying causal influence that produce this type of
cancellation are measure zero in the set of all parameter values Meek (1995),
(Spirtes et al., 2000, theorem 3.2). For a clarifying discussion of
alternative justifications for imposing faithfulness, see (Weinberger, 2018).

In short, if a probability distribution satisfies the Causal Markov and
Faithfulness conditions relative to a candidate causal graph that entails all,
and only, the conditional dependence relations present in the joint
distribution P(V1, V2, $\ldots$, V${}_{\text{N}}$) are reflected in the causal
structure of the graph. If a joint probability distribution satisfies these
two properties relative to multiple graphs, those graphs are said to form a
Markov equivalent set.

One can now see why these conditions are insufficient for causal inference
with two variables V1 and V2. Suppose one already knows that any statistical
dependence between the two variables is due to a direct causal relationship
between them, allowing them to rule out confounders. Even then, the above
conditions are insufficient to identify whether V1 causes V2 or vice versa.
One is stuck with the following Markov equivalence class of graphs:

$V_{1}$$V_{2}$

$V_{2}$$V_{1}$

The reason is simple: any causal graph relating the two variables V1 and V2
will be fully connected, and fully connected graphs do not predict any
conditional independence relations. This is why causal inference from
conditional independences in observational statistical data, using only the
Causal Markov Property and Faithfulness, can successfully identify the causal
relationship between two variables only if they are embedded in a larger set
of at least three variables.

Happily, it turns out that one can do better than this if they are willing to
introduce additional conditions on a satisfactory causal graph. A set of
causal inference strategies has recently been developed that aims to address
the problem of causal inference from observational data for only two variables
(Janzing and Schölkopf, 2010; Daniušis et al., 2010; Janzing et al., 2012;
Peters et al., 2017). The central condition they introduce is simple: given
two candidate causal graphs that generate the joint distribution P(V1, V2),
the true causal graph is the one that entails that the distribution for the
“cause” variable P(V1) does not contain any information about the conditional
distribution P(V${}_{2}\mid\text{V}_{1}$) for the “effect” variable, and vice
versa. A perhaps more intuitive way to describe this condition is as follows:
the mechanism that determines the probability distribution over the cause
variable operates independently of the mechanism that determines the
conditional distribution over the effect variable, given the distribution over
the cause; (Peters et al., 2017, chapter 2) label this “The Principle of
Independent Mechanisms”.

Figure 1: A jagged potential.

The range of contexts in which such a principle is justified has been given an
enlightening examination in (Woodward, 2020), but is quite reasonable in the
particular context of physical theory: one can specify an initial “cause”
state – or a probability distribution over initial “cause” states – without
any detailed knowledge of the dynamical mechanism that will take that initial
state as input and output a conditional distribution over final, or “effect”,
states. If that is possible according to the way P(V1, V2) factorizes
according to one candidate causal graph but impossible according to the
second, then it seems quite reasonable that the first causal graph is the
correct one.

One can illustrate the principle with a simple example. Suppose one prepares a
beam of N classical particles and scatters them off a jagged potential like
the one in Figure 1.333333This is roughly a combination of two different
examples from (Peters et al., 2017, chapter 2) and (Janzing et al., 2016).
Draw a line that all N particles will cross, both before they encounter the
potential and after they are scattered back off it, some distance away from
the region where the potential is non-zero. Repeat the experiment many times
and record the position of the particles each time they cross the line on
their way in and again on their way out. This will produce a joint probability
distribution over the positions of the N particles P(X${}_{\text{in}}$,
X${}_{\text{out}}$), and the scattering potential is the mechanism that
determines the conditional distributions
P(X${}_{\text{in}}\mid\text{X}_{\text{out}})$ and
P(X${}_{\text{out}}\mid\text{X}_{\text{in}})$. The causal inference problem is
whether one could infer that one distribution is the cause and the other the
effect in the absence of any temporal information.

The Causal Markov and Faithfulness conditions are insufficient, as discussed
above, but it is easy to see how the Principle of Independent Mechanisms can
make this possible. The distribution P(X${}_{\text{out}}$) will be highly
disordered, with the positions of the N particles distributed roughly randomly
across the line. However P(X${}_{\text{in}}$) will be quite uniform: the beam
of N particles will cross the line at roughly the same positions on each run
of the experiment. For P(X${}_{\text{out}}$) to be the cause distribution, it
would have to be such that the potential would funnel the random distribution
into a highly ordered one. This would be impossibly unlikely given the
geometry of the scattering potential unless each state of the incoming beams
of N particles had been extremely carefully fine-tuned to be funnelled into a
more ordered state by that particular scattering potential. Any such
preparation procedure would require extremely detailed information about the
geometry of the mechanism determining P(X${}_{\text{in}}\mid X_{\text{out}})$
– the scattering potential – and so identifying P(X${}_{\text{out}}$) as the
cause would violate the Principle of Independent Mechanisms.

A natural precisification of this principle uses algorithmic information
theory: the true causal graph should render the mutual information
$\text{I}(P_{\text{cause}}\,:\,P_{\text{effect}\mid\text{cause}})$ between the
probability distribution over the cause and the conditional distribution over
the effect equal to zero. This can be stated in an equivalent form that is
more transparent in the context of physical theory: the true causal graph
should render equal to zero the mutual information $\text{I}(s\,:\,M)$ between
the “cause” state s and the dynamical mechanism M that determines the
conditional distribution over the effect, given s. In fact, Janzing,
Schölkopf, and collaborators have constructed a framework for causal inference
that is founded on the use of the tools of algorithmic information theory in
which The Principle of Indepenent Mechanisms plays a foundational role.

A valuable feature of this framework is that it is non-probabilistic: its
foundational concepts are those of algorithmic information theory, not
statistics (Janzing and Schölkopf, 2010; Peters et al., 2017). This is not to
say that it somehow doesn’t work for the more familiar cases of causal
inference from statistical data; in fact, the examination of algorithmic
informational dependencies can reveal novel statistical dependencies entailed
by a causal graph (Janzing and Schölkopf, 2010, section 3). The point is that
it also applies more generally, enabling causal inference between objects even
when one does not have statistical data about those objects, like a joint
probability distribution over their values. Although I do not rely on it here,
this may prove valuable for doing causal inference in quantum theory where
joint probability distributions are not guaranteed to be well-defined (see fn.
30).

I will begin with a brief introduction to some relevant concepts of
algorithmic information theory.343434I very loosely follow (Janzing and
Schölkopf, 2010, section 2.1) here. See (Li and Vitányi, 2019) for a textbook
introduction. Suppose one has a countable set of objects
$\left\\{\mathcal{O}_{1},\,\mathcal{O}_{2},\,\ldots,\,\mathcal{O}_{n},\,\ldots\right\\}$
and a system for identifying each object $\mathcal{O}_{i}$ by a binary
description $s_{i}$. Note that “objects” here is quite broad: it can include
things like probability distributions, states of a physical system, PDF files,
Blood Meridian, Gila monsters, surfboards, etc. The algorithmic information
content of an object $\mathcal{O}_{i}$ is meant to quantify the amount of
information required by the shortest complete description of the object
$\mathcal{O}_{i}$.

There will always be some shortest binary string that uniquely identifies an
object $\mathcal{O}$, denoted $s^{*}$. The algorithmic information of
$\mathcal{O}$ (also called the algorithmic complexity or Kolmogorov
complexity) is the length $\ell(p)$ of the shortest program p that, if run on
a (prefix-free) universal Turing machine U, would output the string $s^{*}$
and halt. Formally:353535In algorithmic information theory, equalities are
generally only equalities up to a constant that is independent of the object
itself, but may depend on the alphabet or programming language chosen for the
encoding or the particular universal Turing machine being considered. For
example, a program to output Hamlet may be shorter when written in Python than
in FORTRAN. This tells us something about Python and FORTRAN, but nothing
about the algorithmic information content of Hamlet itself. This “equality up
to a constant” is denoted by $\stackrel{{\scriptstyle+}}{{=}}$

$K(s)=\min_{p}\left\\{\ell(p)\mid U(p)=s\right\\}$

Note that the upper bound on $K(s)$ for an n-bit string is n since one can
always write a program of the form “Print $s_{1},\,\ldots,\,s_{n}$” that
reproduces the string $K(s)$ bit-by-bit. This illustrates one sense in which
$K(s)$ is a measure of the information contained in the object $\mathcal{O}$:
it captures how much information is required to algorithmically reconstruct
its complete binary description.363636Note that this has the somewhat
counterintuitive consequence that a binary sequence that is completely random
has maximal algorithmic information.

One can similarly define the conditional algorithmic information of one
object, given a second. Let $t$ and $s$ be the shortest binary descriptions of
objects $\mathcal{O}_{t}$ and $\mathcal{O}_{s}$. Then the conditional
algorithmic information of $\mathcal{O}_{s}$ given $\mathcal{O}_{t}$, denoted
$K(s\mid t)$, is defined as as the length of the shortest program that takes
as input t then generates s as output and halts. This measures how much
information about $\mathcal{O}_{s}$ one obtains if given $\mathcal{O}_{t}$,
and thus how much computational work is saved by knowing t when computing s.
If t contains no information about s then $K(s\mid t)=K(s)$.

The conditional algorithmic information allows a precise definition of the
natural intuition that a cause does not contain information about the
mechanism that maps it to the effect. The mutual algorithmic information
between two objects $\mathcal{O}_{t}$ and $\mathcal{O}_{s}$ is:

$I(s\,:\,t)=K(t)-K(t\mid s^{*})$

One can also use the joint algorithmic information
$K(s,t)\stackrel{{\scriptstyle+}}{{=}}K(s)+K(t\mid s^{*})$ to give a symmetric
formulation of the mutual algorithmic information:

$I(s\,:\,t)\stackrel{{\scriptstyle+}}{{=}}K(s)+K(t)-K(s,t)$

This gives the intuitive result that if knowing $s^{*}$ does not enable a
shorter computation of t (or vice-versa) then s and t do not share any mutual
information.

In (Janzing and Schölkopf, 2010, section 2) they used a related concept, the
conditional mutual information

$I(s\,:\,t\mid r)\stackrel{{\scriptstyle+}}{{=}}K(s\mid r)+K(t\mid
r)-K(s,t\mid r)$

to define an algorithmic information theoretic formulation of the Causal
Markov Property. If $x_{1},\,x_{2},\,\ldots,\,x_{n}$ are binary strings
describing observations related by a graph, then each string $x_{j}$ shares no
mutual information with the strings $nd_{j}$ associated with its non-
descendents in the graph, conditional on the shortest binary string associated
with the parents $pa_{j}^{*}$ of $x_{j}$ in the graph:

$I(x_{j}\,:\,nd_{j}\mid pa_{j}^{*})\stackrel{{\scriptstyle+}}{{=}}0$

In general, Janzing, Schölkopf, and collaborators have made use of algorithmic
information theory for a multitude of causal inference tasks; see (Mooij et
al., 2016) for a review of how some of these methods perform on a variety of
empirical data sets.

My interest is in the application of some of these tools to inferring causal
direction in physical theory. In particular, I will focus on a connection
drawn in (Janzing et al., 2016) between dynamical evolution and the increase
of algorithmic complexity in a toy model of statistical mechanics, with the
aim of extending it to quantum theories.

They begin by assuming the Principle of Independent Mechanisms: if s is the
initial state of an N-particle system and the operator $D_{t}$ represents
applying the dynamics governing the system for some time interval t, then
$I(s\,:\,D_{t})\stackrel{{\scriptstyle+}}{{=}}0$. (More precisely, s and
$D_{t}$ are the shortest binary encodings of the initial state and the
dynamics.) This asserts that not only does s contain no information about the
dynamics D (the geometry and strength of a potential, for example), s also
contains no information about how much time it will be subjected to those
dynamics (i.e. it contains no information about the interval t).

Although it seems so weak as to be nearly tautological, Janzing, Chaves, and
Schölkopf immediately derive from it that the algorithmic entropy $K(s)$ of a
toy N-particle system must increase under dynamical evolution.373737The
equation of algorithmic entropy with $K(s)$ by (Janzing et al., 2016) assumes
that the microstate s is perfectly known to the observer. More generally, one
defines the algorithmic entropy of a microstate s as the sum of the
algorithmic information and the thermodynamic entropy
$\mathcal{S}(s)=K(s)+H(s)$; see (Zurek, 1989) or (Li and Vitányi, 2019,
chapter 8). The proof is sufficiently simple that I will include it here
before flagging its main limitation:

No entropy decrease: If the dynamics of a system is an invertible mapping
$D_{t}$ of a finite set $\mathcal{S}$ of states then
$I(s\,:\,D_{t})\stackrel{{\scriptstyle+}}{{=}}0$ implies that the algorithmic
information can never decrease when applying $D_{t}$ to the initial state,
i.e.

$K(D(s))\stackrel{{\scriptstyle+}}{{\geq}}K(s)$

for all $s\in\mathcal{S}$.

Proof: Imposing $I(s\,:\,D_{t})\stackrel{{\scriptstyle+}}{{=}}0$ entails that
$K(s)\stackrel{{\scriptstyle+}}{{=}}K(s\mid D_{t})$. Since $D_{t}$ is
invertible, s can be computed from $D_{t}(s)$ and vice versa, which implies
$K(s\mid D_{t})\stackrel{{\scriptstyle+}}{{=}}K(D_{t}(s)\mid D_{t})$. From
this one has $K(s)\stackrel{{\scriptstyle+}}{{=}}K(s\mid
D_{t})\stackrel{{\scriptstyle+}}{{=}}K(D_{t}(s)\mid
D_{t})\stackrel{{\scriptstyle+}}{{\leq}}K(D_{t}(s))$. $\square$

The intuitive idea is simple: if $D_{t}(s)$ had a shorter description than s
one could obtain a shorter binary description of the initial state by encoding
$D_{t}(s)$ and adding “then apply $D_{t}^{-1}$”. But that is impossible since
we’ve assumed s is the shortest binary description of the initial state. Upon
establishing this theorem they illustrate that it holds for a toy system of N
particles modeled as a cellular automaton (Janzing et al., 2016, section 2).

The theorem is suggestive but, as it stands, ultimately insufficient for
inferring causal direction for time evolution in quantum systems. The reason
for this is simple: the set of possible states of any quantum system is
infinite. That said, I showed by construction at the end of section 3 that
something much like No entropy decrease should be true for quantum systems:
the Schmidt rank of a bipartite quantum system in a pure state is guaranteed
to not decrease under time evolution, except for a set of time intervals
$t_{i}$ of measure zero, because time evolution generically creates
entanglement.

To connect my discussion at the end of section 3 to the methods of causal
inference using algorithmic information theory, I need a measure of the
algorithmic information of a quantum state. In the next section, I will show
that one can use such a measure and the causal inference methods described
above to derive a more general, and more useful, version of the conclusion I
reached at the end of section 3.

##  5 Entanglement, Algorithmic Information, and Causal Direction

To embed the inference of causal direction made on physical grounds in section
3 into the framework of causal inference using algorithmic information theory,
one needs a measure of the algorithmic information of a quantum state. It
would be preferable for the measure to be natural, either in the sense that it
shares many or all of the conceptual virtues of classical algorithmic
complexity or because it seems to appropriately generalize those properties to
the novel physical and mathematical setting of quantum theories. Ideally,
there would be a unique such measure.

There are multiple proposals for quantitative meaures of quantum algorithmic
information, each with some legitimate claim to being a natural generalization
of classical algorithmic information to quantum states (see (Vitányi, 2001)
for an early overview and (Mora et al., 2007) for a more recent one). We do
not live in the best of all possible worlds, however: the measures are
demonstrably inequivalent and one thus has to make a choice. I will focus on a
measure of algorithmic information that tracks the Schmidt rank of a quantum
state (or the Schmidt measure, for multipartite systems). Ultimately I will
show that one can reproduce a less restrictive version of the criteria for
identifying causal direction offered at the end of section 3 as a theorem that
follows from imposing the Principle of Independent Mechanisms. I will first
say a bit about the problems faced by any extension of algorithmic information
to quantum states.

The question of how much a measure of quantum algorithmic information needs to
have in common with its classical precursor to deserve the name is somewhat
subjective, but one can identify at least three properties that seem non-
negotiable. First, it should be definable from the quantum state alone rather
than, say, only for a member of a specified ensemble of quantum states. This
was why I deemed the von Neumann entropy unsatisfactory in section 3. Second,
there should be some recognizable sense in which it measures the amount of
information required to compute, or reconstruct, the state in question using
some algorithmic procedure. Finally, just as there is an upper bound on the
algorithmic information required to specify any classical bit string of
specified length n, there should be some analogous upper bound on the
algorithmic information of a quantum state that scales with the size of the
system.

A number of other seemingly essential properties of classical algorithmic
information are up for grabs, however. Must quantum algorithmic information be
defined in terms of a computation carried out by a quantum generalization of a
Turing machine or is a different notion of “algorithmic procedure”
appropriate? Should the algorithmic information be measured in classical bits
or qubits? Does quantum algorithmic information need to reduce smoothly to
classical information in some context? How should the upper bound on the
algorithmic information of a quantum system scale with the size of the system?
Almost all classical bit strings of length n maximize the classical
algorithmic information; should the same be true of the quantum algorithmic
information content of quantum states in, say, an N-dimensional Hilbert space?
Different proposals for measures of quantum algorithmic complexity have
adopted different answers to these questions (Berthiaume et al., 2001;
Vitányi, 2001; Mueller, 2007).

The measure I will focus on defines the amount of algorithmic information in a
quantum state $\ket{\phi}$ as the classical algorithmic information required
by the shortest description of an algorithmic procedure for preparing it from
some reference state $\ket{0}$ (Mora and Briegel, 2005, 2006). One models the
algorithmic preparation procedure as a quantum circut $\mathcal{C}_{\phi}$: a
finite sequence of unitary operations – quantum gates – chosen from a finite
set $\mathcal{G}=\left\\{G_{1},\,\ldots,\,G_{n}\right\\}$ that, when performed
on the reference state $\ket{0}$, produce the desired state
$\mathcal{C}_{\phi}\ket{0}=\ket{\phi}$ up to some fidelity
$\varepsilon$.383838This means that
$\absolutevalue{\matrixelement{\phi}{\mathcal{C}}{0}}^{2}\geq 1-\varepsilon$.
The set $\mathcal{G}$ is finite, so each unitary operation – and thus each
circuit $\mathcal{C}_{\phi}$ – can be given a finite binary encoding which has
a finite quantity of classical algorithmic information. The algorithmic
information of a quantum state $\ket{\phi}$ is identified with the classical
algorithmic information of the binary encoding of the simplest circuit
$\mathcal{C}_{\phi}$:

$K^{\varepsilon}_{Q}(\ket{\phi})=\min_{\mathcal{C}_{\phi}}K(\mathcal{C}_{\phi})$

where the notation $\mathcal{C}_{\phi}$ is doing double-duty as the quantum
circuit or the binary encoding of that circuit, depending on context.

The value of $K^{\varepsilon}_{Q}$ appears to depend on four quantities not
obviously related to the state itself: (i) the choice of the set $\mathcal{G}$
of quantum gates, (ii) the alphabet $\Omega$ used to encode the circuit, (iii)
the degree of fidelity $\varepsilon$, and (iv) the particular circuit
$\mathcal{C}_{\phi}$. As (Mora and Briegel, 2005, 2006) show, (ii) and (iv)
are unproblematic: the values of $K^{\varepsilon}_{Q}(\ket{\phi})$ for two
different alphabets $\Omega_{1}$ and $\Omega_{2}$ differ by at most a
constant, and minimizing over all circuits $\mathcal{C}_{\phi}$ that prepare
the state $\ket{\phi}$ removes any dependence on an arbitrary choice of
circuit. On reflection, it is physically quite reasonable that
$K^{\varepsilon}_{Q}$ depend on the degree of fidelity $\varepsilon$: changing
the precision with which an algorithm has to prepare a state will, and should,
change the amount of information the algorithm requires. The arbitrariness in
$K^{\varepsilon}_{Q}$ associated with (i), the chosen set of quantum gates
$\mathcal{G}$, is genuine, although (Mora and Briegel, 2005, 2006) offer
several palliative remarks on that front.

To see the connection between this notion of algorithmic information and
entanglement, consider a quantum system consisting of N qubits with Hilbert
space $\mathcal{H}_{N}$. In this case, one can derive the specific dependence
of $K^{\varepsilon}_{Q}$ on the fidelity $\varepsilon$ by invoking the fact
that beginning with the reference state
$\ket{0}=\ket{0}_{1}\,\cdots\,\ket{0}_{N}\in\mathcal{H}_{N}$, one can prepare
any state $\ket{\phi}\in\mathcal{H}_{N}$ with at most $\order{2^{N}}$ quantum
gates $G_{i}$. Taking into account the dependence on the fidelity
$\varepsilon$, (Mora and Briegel, 2006, section 5) show that the number of
quantum gates required to compose a circuit $\mathcal{C}_{\phi}$ that prepares
an arbitrary state $\ket{\phi}$ from the reference state $\ket{0}$ is
$\mathcal{O}(2^{N}\log\frac{1}{\varepsilon})$. This is an intuitive result:
the bigger the system or the greater fidelity demanded, the more unitary
operations required.

Figure 2: An $\varepsilon$-radius patch around $\ket{\psi}$.

Note that by specifying a degree of fidelity $\varepsilon$ one identifies a
patch around $\ket{\phi}$. More precisely, it specifies a patch of radius
$\varepsilon$ on the unit sphere corresponding to all pure states in
$\mathcal{H}_{N}$; see Figure 2. The quantity
$K^{\varepsilon}_{Q}(\ket{\phi})$ captures the amount of information required
to specify a circuit $\mathcal{C}_{\phi}$ that will put $\ket{0}$ in that
patch.393939The two limiting cases may be clarifying. Let $\varepsilon\to 1$;
then the “patch” is the entire sphere of pure states and
$K^{\varepsilon=1}_{Q}(\ket{\phi})$ quantifies the information required to
construct a circuit $\mathcal{C}$ that will act on $\ket{0}$ to put it into
any pure state inside $\mathcal{H}_{N}$. In this limit $K^{\varepsilon=1}_{Q}$
is zero for all states, as it obviously should be: $\ket{0}$ is already such a
state, so just leave it alone. Now let $\varepsilon\to 0$; then the “patch”
approaches the single state $\ket{\phi}$ and
$K^{\varepsilon=0}_{Q}(\ket{\phi})$ quantifies the information required to
prepare exactly $\ket{\phi}$ from $\ket{0}$. This obviously diverges at the
limit; for example, it would require preparing an amplitude to be exactly
$\frac{1}{\sqrt{2}}$ rather than the closest rational approximation to
$\frac{1}{\sqrt{2}}$. Choosing a degree of fidelity $\varepsilon$ thus defines
an equivalence class of states close to $\ket{\phi}$ by assigning each of them
the same algorithmic information content $K^{\varepsilon}_{Q}(\ket{\phi})$.
For $\varepsilon>0$ such “patches” of states have small but finite measure in
the set of pure states in $\mathcal{H}_{N}$; indeed, one can cover the unit
sphere of pure states with finitely many of them404040See (Nielsen and Chuang,
2010, section 4.5.4) for how the number of patches required scales with the
fidelity $\varepsilon$ and system size N. Topologically, this means that every
state $\ket{\phi}$ is surrounded by a ball of radius $\varepsilon$ of states
of equivalent algorithmic information.

The information required to describe the circuit $\mathcal{C}_{\phi}$ is just
the information required to describe the quantum gates that compose it. From
the fact that $\mathcal{C}_{\phi}$ consists of, at most, a sequence of $\sim
2^{N}\log\frac{1}{\varepsilon}$ unitary operations, (Mora and Briegel, 2006)
conclude that its binary encoding – and thus the quantum algorithmic
information of the state $\ket{\phi}$ – has the upper bound

$K^{\varepsilon}_{Q}(\ket{\phi})=\min_{\mathcal{C}_{\phi}}K(\mathcal{C}_{\phi})\lesssim
2^{N}\log\frac{1}{\varepsilon}$

It is significant that while the classical algorithmic information of an n-bit
string grows at most linearly with n, the algorithmic information of a quantum
state can grow exponentially with the size of the system N. Given that the
dimension of the space of states of a composite classical system in physics
generally grows linearly in the dimension of the subsystems, while the
dimension of the space of states of a composite quantum system grows
exponentially due to the possibility of entanglement, one might think that the
upper bound of $K^{\varepsilon}_{Q}(\ket{\phi})$ reflects the presence of
entangled states in $\mathcal{H}_{N}$. As (Mora and Briegel, 2005, 2006) show,
that is correct.

At this point the Schmidt rank of a quantum state once again becomes relevant.
They adopt as their measure of entanglement the Schmidt measure (Eisert and
Briegel, 2001), a generalization of the Schmidt rank to multipartite quantum
systems. For an N-particle system with Hilbert space
$\mathcal{H}=\mathcal{H}_{1}\otimes\cdots\otimes\mathcal{H}_{N}$, any state
$\ket{\Psi}$ in $\mathcal{H}$ can be written as414141I am presuming a
preferred factorization of the big Hilbert space $\mathcal{H}$ into tensor
factors. In general, entanglement measures for multipartite systems are
sensitive to different choices of partition; this is part of the difficulty of
extending such measures beyond bipartite systems. Where a preferred partition
isn’t available, one could more generally define the Schmidt measure as the
minimum value of r over all possible partitions. See (Zanardi et al., 2004),
(Cotler et al., 2019), or (Carroll and Singh, 2020) for different proposals
for identifying preferred factorizations in certain contexts.

$\ket{\Psi}=\sum^{R}_{i}a_{i}\ket{\psi_{i}^{(1)}}\otimes\cdots\otimes\ket{\psi_{i}^{(N)}}$

Let r be the minimum number of product states
$\ket{\psi^{(1)}}\otimes\cdots\otimes\ket{\psi^{(N)}}$ needed to write
$\ket{\Psi}$ in the above form. The Schmidt measure of the state $\ket{\Psi}$
is defined as $S(\ket{\Psi})=\log r$. A separable state will have Schmidt
measure $\log 1=0$ and a fully entangled state will have Schmidt measure
$\log\dim(\mathcal{H})$, which will be $\log 2^{N}=N\log 2=N$ for the system
of N qubits.

The connection between algorithmic information and entanglement follows from
the fact one can use the Schmidt measure to bound the algorithmic information
$K^{\varepsilon}_{Q}$ of a quantum state. As (Mora and Briegel, 2006) show,
knowing the Schmidt measure of a quantum state makes possible a more
informative bound on its complexity $K^{\varepsilon}_{Q}$:

$K^{\varepsilon}_{Q}(\ket{\phi})=\min_{\mathcal{C}_{\phi}}K(\mathcal{C}_{\phi})\lesssim
3N2^{S(\ket{\phi})}\log\frac{1}{\varepsilon}$

The algorithmic complexity of a separable state $\ket{1}$ can grow at most
linearly with $N$, while the upper bound for all states in $\mathcal{H}_{N}$
is saturated by a state $\ket{N}$ only if it has full Schmidt measure. Given a
state $\ket{k}$ that is neither separable nor fully entangled – i.e. a state
which has Schmidt measure $0<k<N$ – the upper bound on the algorithmic
complexity of $\ket{k}$ will thus satisfy
$K^{\varepsilon}_{Q}(\ket{\text{SEP}})<K^{\varepsilon}_{Q}(\ket{k})<K^{\varepsilon}_{Q}(\ket{N})$.

There are several comforting features of this definition of quantum
algorithmic information. The first is that it lays the groundwork for
addressing two major limitations of the causal inference criteria of section
3: that they applied only if one state from the pair comes from a set of
states that has measure zero in the set of all pure states in
$\mathcal{H}_{AB}$, and that they no longer held if one replaced states
$\ket{\alpha}$ that were exactly of less-than-full Schmidt rank with states
that were within $\varepsilon$ of a state of less-than-full Schmidt rank. In
algorithmic information theoretic terms, states $\ket{k}$ of less-than-full
Schmidt rank now correspond to $\varepsilon$-patches of non-maximal
algorithmic information $K^{\varepsilon}_{Q}(\ket{k})$. As mentioned above,
those patches have non-zero measure on the sphere of pure states in
$\mathcal{H}_{N}$. Furthermore, these patches include all states that
approximate $\ket{k}$ with fidelity $\varepsilon$. According to Schmidt rank
alone, the two states

$\ket{\Psi}=\ket{00}\hskip
24.0pt\ket{\Psi_{\varepsilon}}=\sqrt{1-\varepsilon}\ket{00}+\varepsilon\ket{11}$

are no less different than are

$\ket{\Psi}=\ket{00}\hskip
24.0pt\ket{\Psi_{\text{EPR}}}=\frac{1}{\sqrt{2}}\ket{00}+\frac{1}{\sqrt{2}}\ket{11}$

The algorithmic information $K^{\varepsilon}_{Q}$ is less sensitive: it treats
$\ket{\Psi}$ and $\ket{\Psi_{\varepsilon}}$ as equivalent, but distinguishes
$\ket{\Psi}$ from $\ket{\Psi_{\text{EPR}}}$. It represents an improvement over
the Schmidt rank in this regard.

Nevertheless it can still reproduce an algorithmic information-theoretic
version of the topological and measure-theoretic facts relied upon in section
3. In (Mora and Briegel, 2006) they show that for a system of N qubits, the
relative frequency of pure states in $\mathcal{H}_{N}$ with non-maximal
algorithmic information is exponentially small: it scales as $\sim
2^{2^{N}\log\varepsilon}$.424242Since the algorithmic information of a quantum
state is the classical algorithmic information of the circuit $\mathcal{C}$
that prepares it, this is a straightforward adaptation of the standard proof
that the relative frequency of compressible strings in the set of all n-bit
strings goes to zero as $n\to\infty$. Recall that (i) a state in
$\mathcal{H}_{N}$ is maximally complex only if it has full Schmidt measure and
(ii) the Schmidt measure reduces to the Schmidt rank for pure states of a
bipartite system. With that in mind, consideration of the (divergent) limit
$\varepsilon\to 0$ shows that the relative frequency of pure states of non-
maximal algorithmic information goes to zero in the set of all pure states as
one demands perfect fidelity. One can thus reproduce in algorithmic
information-theoretic terms the measure-theoretic sparseness of states that
are not fully entangled, not only for bipartite pure states but also for pure
states of multipartite systems of N qubits.

With all this machinery in place, an analogue of the result of (Janzing et
al., 2016) for quantum algorithmic information now follows by essentially
identical reasoning.

No entanglement decrease: If the dynamics of a quantum system are given by a
unitary operator $U_{t}$ on a finite-dimensional Hilbert space $\mathcal{H}$
of states $\ket{\phi}$, then
$I(\ket{\phi}\,:\,U_{t})\stackrel{{\scriptstyle+}}{{=}}0$ implies that the
quantum algorithmic information can never decrease when applying $U_{t}$ to
the initial state, i.e.

$K^{\varepsilon}_{Q}(U_{t}\ket{\phi})\stackrel{{\scriptstyle+}}{{\geq}}K^{\varepsilon}_{Q}(\ket{\phi})$

for all $\ket{\phi}\in\mathcal{H}$.

Proof: The assumption that
$I(\ket{\phi}\,:\,U_{t})\stackrel{{\scriptstyle+}}{{=}}0$ entails that
$K^{\varepsilon}_{Q}(\ket{\phi})\stackrel{{\scriptstyle+}}{{=}}K^{\varepsilon}_{Q}(\ket{\phi}\mid
U_{t})$. Unitary time evolution is invertible, so $\ket{\phi}$ can be computed
from $U_{t}\ket{\phi}$ and vice versa. This implies that
$K^{\varepsilon}_{Q}(\ket{\phi}\mid
U_{t})\stackrel{{\scriptstyle+}}{{=}}K^{\varepsilon}_{Q}(U_{t}\ket{\phi}\mid
U_{t})$. From this, one obtains
$K^{\varepsilon}_{Q}(\ket{\phi})\stackrel{{\scriptstyle+}}{{=}}K^{\varepsilon}_{Q}(\ket{\phi}\mid
U_{t})\stackrel{{\scriptstyle+}}{{=}}K^{\varepsilon}_{Q}(U_{t}\ket{\phi}\mid
U_{t})\stackrel{{\scriptstyle+}}{{\leq}}K^{\varepsilon}_{Q}(U_{t}\ket{\phi})$.
$\square$

This is unsurprising in light of the facts reviewed above concerning the
relative (in)frequency of states of non-maximal complexity in the set of all
pure states.

One can also recognize the same intuitive reasoning at work as in the result
of (Janzing et al., 2016): Suppose that $U_{t}\ket{\phi}$ had less quantum
algorithmic information than $\ket{\phi}$. By definition there would be a
circuit $\mathcal{C}_{U_{t}\phi}$, the concatenation of $\mathcal{C}_{U_{t}}$
and $\mathcal{C}_{\phi}$, with a simpler binary encoding than
$\mathcal{C}_{\phi}$.434343Time evolution for a large class of Hamiltonians
can be well-approximated as a quantum circuit $\mathcal{C}_{U_{t}}$; see
(Nielsen and Chuang, 2010, section 4.7). That would entail one could obtain a
shorter description of the initial circuit $\mathcal{C}_{\phi}$ by applying
$\mathcal{C}_{U_{t}\phi}$ to the reference state $\ket{0}$ and appending
$\mathcal{C}_{U^{-1}_{t}}$ to prepare $\ket{\phi}$:
$\mathcal{C}_{U^{-1}_{t}}\mathcal{C}_{U_{t}\phi}\ket{0}=\ket{\phi}$. But by
definition $\mathcal{C}_{\phi}$ is the shortest binary encoding of any circuit
that will prepare $\ket{\phi}$ from $\ket{0}$, so this is impossible.

One may wonder how this is consistent with the measure-zero set of pure states
whose Schmidt rank decreases for a given dynamical evolution U($t_{i}$). The
answer is simple: they are excluded by imposing the Principle of Independent
Mechanisms $I(\ket{\phi}\,:\,U_{t})=0$. Specifying a particular time interval
$t_{i}$ of dynamical evolution tells one quite a lot about the candidate
“cause” states for an entanglement-decreasing evolution. Indeed, it is only
conditional on knowing U($t_{i}$) that one can even specify such states. Even
then they can only be picked out from the set of all pure states in
$\mathcal{H}_{N}$ by including their dynamical behavior in their description.
They are thus ruled out as candidate “cause” states because they fail to
satisfy $I(\ket{\phi}\,:\,U_{t})=0$.

From a physical perspective No entanglement decrease is quite reasonable. It
would be baffling if a quantum system contained information about the
dynamics, like the geometry and strength of the potential, prior to having
experienced them. It is obvious that the system will have acquired some
information about the dynamics after it is subjected to them. This
informational asymmetry will be encoded in the quantum state – in multiple
ways, surely, but I have focused on the extent to which this information about
the dynamics is captured by the degree of entanglement in the quantum state.
This gives us a new criterion for causal inference:

  1. Given two states $\ket{\alpha}$ and $\ket{\beta}$ related by unitary evolution, if $K^{\varepsilon}_{Q}(\ket{\alpha})<K^{\varepsilon}_{Q}(\ket{\beta})$ then $\ket{\beta}$ was caused by time-evolving $\ket{\alpha}$.

This criterion represents an improvement in several respects on the criteria
identified at the end of section 3. First, the introduction of a fidelity
$\varepsilon$ means that it no longer depends on an unsatisfiable standard of
precision, as noted above.

Second, the set of states for which it holds has finite measure in the set of
all pure states in $\mathcal{H}_{N}$, as discussed above. The algorithmic
information $K^{\varepsilon}_{Q}$ is defined on patches of pure states of
small, but finite, volume and so states of non-maximal algorithmic information
will occupy finite volume in the set of all pure states in $\mathcal{H}_{N}$.
A similar conclusion follows from recalling that in the set of all pure
states, the relative frequency of pure states of maximal algorithmic
complexity is $\approx 1-2^{2^{N}\log\varepsilon}$.

It is illuminating to contrast this with the case of classical statistical
mechanics; for example, Goldstein et al. (2017) estimate that for a system of
$N\sim 10^{20}$ particles, the relative volume of phase space occupied by
microstates in the equilibrium macrostate is $\approx
1-e^{-10^{-15}N}$.444444More precisely, this the volume occupied by the
equilibrium macrostate for any energy shell in phase space containing
microstates with energies in the range $E+\Delta E$. The relative frequency of
pure states of maximal algorithmic complexity thus represents a difference in
degree from the classical statistical mechanical case – possibly a dramatic
degree, depending on the system size N and chosen fidelity $\varepsilon$ – but
not a difference in kind.

Third, the criteria proposed in section 3 were too weak to identify a causal
direction between two pure states each of less-than-full Schmidt rank, as
noted in fn. 26. The algorithmic information-theoretic criterion can do
better: given two pure states of any Schmidt measure, No entanglement decrease
entails that the state with strictly lower algorithmic information is the
cause state. Finally, the criteria in section 3 were restricted to bipartite
systems, while this algorithmic information-theoretic criterion extends also
to multipartite systems of qubits. Note, however, that it does not
straightforwardly extend to infinite-dimensional systems, while the criteria
in section 3 applied to both finite and infinite-dimensional systems.

One may naturally wonder if No entanglement decrease is an artifact of the
chosen definition of quantum algorithmic information. There is good reason to
be optimistic on this score. Many proposals for extending algorithmic
information to quantum states assign higher algorithmic information to
entangled states (see (Mora et al., 2007) for a review) so one can reasonably
expect that some version of the conclusion here concerning
$K^{\varepsilon}_{Q}$ could be reproduced for other proposed measures of
quantum algorithmic information. Additionally, the measure here is defined
using quantum gates – the unitary operations composing the circuit
$\mathcal{C}_{\phi}$ – and one may be concerned that this limits its
application to systems of qubits. That concern can be alleviated: there is no
principled obstacle to extending the use of quantum gates beyond two-level
systems to more interesting multi-level systems, or even to quantum systems
with continuous degrees of freedom Braunstein and Van Loock (2005), and thus
to extending the notion of algorithmic information employed in this paper to
those systems.

There is a significant limitation to this result, however, which is a well-
known drawback of any use of algorithmic information theory: the algorithmic
complexity of an object is not computable. Making use of this causal inference
criterion in any practical setting would thus require identifying contextually
appropriate, computable substitutes for the algorithmic complexity. Certain
computable substitutes have already been proposed in different contexts; for
example, the “information-geometric approach” of (Daniušis et al., 2010;
Janzing et al., 2012). (The practical applicability of this method is itself
limited by the fact that it requires deterministic relations between cause and
effect variables.) It is also the case that the particular scenario here –
inferring the causal ordering of two known pure states – is itself of limited
practical application in the laboratory; after all, one doesn’t typically find
pure states lying around in the world (to borrow a phrase from a referee). As
a philosophical result, it succeed in demonstrating that contrary to what
Russell and modern-day Russellians say about causation in “fundamental”
physics, the time-symmetric nature of dynamical evolution does not make it
impossible for that dynamical evolution to ground relations of cause and
effect. However, as a procedure intended for practical use in the laboratory,
this is only a small first step in applying the methods of Janzing, Schölkopf,
and collaborators to a quantum context.

That said, focusing solely on the strength of the causal inference criterion
underwritten by No entanglement decrease may be too parochial. Arguably the
primary interest of the criterion proposed here lies as much in the method
used to reach it as its content. By attributing to quantum states a measure of
algorithmic information and incorporating them into an empirically successful
framework for causal inference, one can identify causal asymmetries between
quantum states related by unitary time evolution. All that is required is a
very weak assumption: the Principle of Independent Mechanisms. The extension
of the methods of (Janzing and Schölkopf, 2010; Peters et al., 2017) to
quantum theories introduces a novel set of mathematical and conceptual tools
into the landscape for quantum causal inference. The fact that those methods
confirm a conclusion that can also be reached on physical grounds provides a
comforting verification of their reliability in this context, while their
ability to extend that conclusion in select ways encourages their broader
exploration.

##  6 Conclusion

In this paper I have argued that one can give a general and principled
identification of a causal asymmetry between pure states of certain quantum
systems related by unitary time evolution. The physical fact underlying this
identification is simple: interaction between quantum systems entangles those
systems but almost never disentangles them. Although there are many ways to
make that physical fact precise, I have focused on one in particular for
identifying causal asymmetry: the Schmidt rank (more generally, the Schmidt
measure) of the two quantum states in question. I argued for the presence of
this asymmetry on physical grounds in section 3, while in section 5 I did so
by applying algorithmic information-theoretic concepts and methods from causal
inference. In both cases, the Principle of Independent Mechanisms played a
central role: the requirement that one ought to be able to specify the “cause”
state without any information about the dynamics identified the causal
asymmetry. Also in both cases, the asymmetry arose not in the laws governing
dynamical evolution themselves, but between the states related by that
dynamical evolution. In this respect, the asymmetry was identified by
abandoning what (Woodward, 2020, section 11) calls the “cause-in-laws”
attitude: the belief that if the laws of dynamical evolution themselves do not
reflect an asymmetry, then there is no asymmetry to be found. Instead, my
discussion here is a reflection of the more general fact that “the directional
features of causation are closely bound up with facts about the initial and
boundary conditions of the systems we are analyzing and the way in which these
are related to or interact with the [laws] governing those systems” (Woodward,
2020, section 11). That one can identify causal asymmetries in the dynamical
evolution of quantum theories is indication that, pace Russell and others, one
can locate a meaningful notion of causation in microscopic physics after all.

I want to conclude by highlighting a couple of philosophical questions that
the previous sections raise. The first is the relationship between causal
asymmetry and temporal asymmetry. Couldn’t the conclusions here about causal
asymmetry be simply recast as conclusions about inferring temporal asymmetry?
That is, couldn’t this have been described as an exercies in determining which
of $\ket{\alpha}$ or $\ket{\beta}$ was the initial or earlier state, rather
than whether $\ket{\beta}$ was caused by time-evolving $\ket{\alpha}$ or vice
versa? This is certainly how Janzing, Schölkopf, and collaborators interpret
much of their work; when applied to a time series or within the context of
physical theory, for instance, they frequently describe their aim
interchangeably as distinguishing between cause and effect or between past and
future.

I think that it is quite reasonable to accept that in the setup I have
considered, a causal asymmetry entails a temporal asymmetry. Indeed, one might
think that when applied to states and dynamics in physical theories, some pre-
theoretic understanding that causal asymmetries are almost always asymmetries
in time is what lends the Principle of Independent Mechanisms its
plausibility. However, I ultimately do think that what is being inferred from
these methods is, first and foremost, a causal relationship. One reason for
thinking this is that the methods of Janzing et al. that I use in this paper
have been shown to perform well at distinguishing cause from effect even for
data sets that are not time series data (e.g. the relationship between
altitude and temperature, or longitude and precipitation in (Mooij et al.,
2016).). That suggests that the structure being identified by these methods is
causal structure; it just so happens that in the restricted setup being
considered in this paper, that causal asymmetry coincides with the temporal
asymmetry.

Second, one might wonder how this relates to other recognized temporal
asymmetries in quantum theories. The clearest relationship is with
decoherence.454545See (Wallace, 2012, chapter 9) for a discussion of
decoherence and temporal asymmetry. As is well known, in quantum systems of
many particles the entanglement between any two subsystems rapidly decoheres
as each subsystem interacts, and becomes increasingly entangled, with other
particles in the system.464646In language that may be more familiar: typically
one splits the multiparticle system into two and calls the subsystem(s) of
interest “the system” and the rest of the particles “the environment”. The
entanglement between the subsystems of interest is then decohered by their
respective entanglements with the environment. It essentially never happens
that a multiparticle system that is initially, or evolves into, a highly
entangled state is later disentangled through some series of interactions
between the particles. This is a manifestation of the general physical fact at
the foundation of this paper: interactions between quantum systems generically
entangle those systems but almost never disentangle them. The non-unitarity of
time evolution associated with decoherence that arises when one traces out all
but the subystem(s) of interest is the sharpest formal sense in which
decoherence is asymmetric in time. There is an asymmetry that arises prior to
the tracing out, however: the increasing entanglement between
subsystems.474747As I mentioned in section 1, the relationship between
entanglement and the direction of time has received extensive exploration.

With this in mind, one could focus on the physical fact underlying my formal
discussion of causal inference: the generic creation of entanglement between
quantum systems by interactions. One could then view my discussion as merely
one way to formalize that fact in a restricted context, in a way chosen
specificially to be amenable to a particular set of formal methods of causal
inference. In that case, one could see that physical fact as identifying an
asymmetry that is conceptually prior both to the asymmetry I identifed using
causal inference methods and to the asymmetry introduced when, after two
entangled subsystems are decohered through interactions with some
“environment”, that environment is traced out to produce a non-unitary
evolution for the subsystems of interest. With that attitude, the causal
asymmetry I have identified for bipartite (and multipartite) pure states is
just the maximally microscopic manifestation of the very same physical
conditions that produce the asymmetry manifest in decoherence.

Finally, I have spoken consistently of inferring or identifying causal
asymmetry. As this epistemic language suggests, I have not primarily been
concerned with providing an analysis of what causation or causal asymmetry is,
in any strong metaphysical sense, but rather with certain circumstances in
quantum theories under which one can identify it. That said, empirically
accessible manifestations of an object in, or property of, the natural world –
and thus successful methods for detecting that object or property – are not
independent of the way that object, or that property, actually is. By focusing
on how one can learn about causation under particular circumstances, one can
hope to learn something about the worldly properties that provide the physical
basis for, or ground, our reasoning about causal relations in various
contexts. I am particularly influenced on this score by (Woodward,
2020):484848See also (Eberhardt, 2009, p. 914):  “$\ldots$ metaphysical
accounts have provided essentially no guidance for methods of discovery
because it remains unclear how they could be operationalized into discovery
procedures that do not depend on the availability of causal knowledge in the
first place. Epistemological headway was made by a completely different
strategy that largely ignored metaphysical considerations.” I would only
emphasize that I think that once one has made this epistemological headway, it
can be valuable to use it as a basis for circling back to address some of the
metaphysical considerations that one initially set aside to make epistemic
progress.

> This is the project, alluded to earlier, of elucidating the worldly
> infrastructure that underlies and grounds assessments of causal direction. I
> see this project as connecting epistemological concerns having to do with
> how we find out about causal direction with the “what is out there” concerns
> of metaphysicians, although my answer to the what is out there question does
> not involve any kind of elaborate metaphysics. My general picture is that
> causal thinking “works” to the extent that it does because it picks up on or
> is supported by certain generic features of our world $\ldots$ (Woodward,
> 2020, p. 5).

The causal inference methods of Janzing, Schölkopf, and collaborators, or of
(Pearl, 2009) or (Spirtes et al., 2000), represent relatively unified
methodological strategies for identifying causal relations in the world
without presuming any unified metaphysical ground of those relations. These
methodological frameworks allow that the physical basis, or metaphysical,
ground of any particular causal relationship identified by those methods will
likely vary with the worldly situations to which those methods are applied.
Aside from the satisfaction of certain statistical or algorithmic information-
theoretic properties, there may be little in their respective worldly grounds
to link them together as causal relationships; the relationships may be
metaphysically necessary or contingent, may be fundamental or emergent, and so
on. The ability of these methodological frameworks to act as reliable
diagnostics for such a diversely instantiated class of causal relations is
part of their immense epistemic value. In the case I have considered, the
worldly structure that grounds the identifiation of a causal asymmetry is a
differential degree of entanglement in two states related by unitary time
evolution. This asymmetry, in turn, is grounded in the physical fact that
interactions between subsystems in quantum theories generically produce
entanglement. The fact that one can identify this causal asymmetry using
methods of causal inference reflects not merely a fact about causal
epistemology, but a fact about the nature of the physical world.

## Acknowledgments

I would like to thank John Dougherty, Casey McCoy, Michael Miller, Siddarth
Muthukrishnan, and an audience at Oxford University for helpful feedback. I
would like to thank David Albert, Naftali Weinberg, Jim Woodward, and two very
helpful referees for valuable and constructive feedback on a previous draft. I
am especially indebted to Jim for conversations that sparked the idea for the
paper and for subsequent encouragement and helpful discussion.

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