# Valuation of a Financial Claim Contingent on the
Outcome of a Quantum Measurement

Lane P. Hughston1 and Leandro Sánchez-Betancourt2 1Department of Computing,
Goldsmiths University of London,
New Cross, London SE14 6NW, United Kingdom
2Department of Mathematics, King’s College London, Strand, London WC2R 2LS,
United Kingdom

###### Abstract

We consider a rational agent who at time $0$ enters into a financial contract
for which the payout is determined by a quantum measurement at some time
$T>0$. The state of the quantum system is given by a known density matrix
$\hat{p}$. How much will the agent be willing to pay at time $0$ to enter into
such a contract? In the case of a finite dimensional Hilbert space, each such
claim is represented by an observable $\hat{X}_{T}$ where the eigenvalues of
$\hat{X}_{T}$ determine the amount paid if the corresponding outcome is
obtained in the measurement. We prove, under reasonable axioms, that there
exists a pricing state $\hat{q}$ which is equivalent to the physical state
$\hat{p}$ on null spaces such that the pricing function $\Pi_{0T}$ takes the
form $\Pi_{0T}(\hat{X}_{T})=P_{0T}\,{\rm tr}(\hat{q}\hat{X}_{T})$ for any
claim $\hat{X}_{T}$, where $P_{0T}$ is the one-period discount factor. By
“equivalent” we mean that $\hat{p}$ and $\hat{q}$ share the same null space:
thus, for any $|\xi\rangle\in\mathcal{H}$ one has
$\langle\bar{\xi}|\hat{p}|\xi\rangle=0$ if and only if
$\langle\bar{\xi}|\hat{q}|\xi\rangle=0$. We introduce a class of optimization
problems and solve for the optimal contract payout structure for a claim based
on a given measurement. Then we consider the implications of the Kochen-
Specker theorem in such a setting and we look at the problem of forming
portfolios of such contracts.

Key words: Quantum mechanics, quantum measurement, contingent claims, discount
bonds,
absence of arbitrage, rate of return, density matrices, Gleason’s theorem,
Kochen-Specker theorem.

##  I Introduction

By “quantum finance” we mean the valuation, optimization and risk management
of financial contracts for which the outcomes (in the form of one or more
payments made between the various parties involved) are contingent on the
results of one or more quantum measurements. The financial contracts that we
consider can be easily implemented, at least in principle, in a suitable
laboratory. Our investigations are entirely within the scope of standard
quantum mechanics and we are not concerned here with modifications of the
standard framework or with interpretive issues. The resulting theory of
quantum finance is distinctly non-Kolmogorovian, inheriting as it does the
full generality of quantum probability.

Let time $0$ be the present and $T$ a fixed time in the future. We consider
the situation where an agent $A$ enters into a contract with another agent $B$
in accordance with which $A$ pays $B$ an amount $H_{0}$ (“the price”) at time
$0$ and then $B$ pays $A$ an amount $H_{T}$ (the “payout”) at time $T$, where
$H_{T}$ is contingent in some specified way on the outcome of a quantum
measurement. More elaborate setups can be considered, with multiple payments
as time progresses and multiple agents; but for simplicity we look at a one-
period market involving two agents.

As an example, suppose the payout is determined by a measurement of the spin
of a spin one-half particle along the $z$-axis. The outcome of such a
measurement either gives $+\mbox{$\textstyle\frac{1}{2}$}\hbar$, corresponding
to spin up along that axis, or $-\mbox{$\textstyle\frac{1}{2}$}\hbar$,
corresponding to spin down. Henceforth, we work with physical units such that
$\hbar=1$. For the basics of quantum theory, see, for instance, reference
isham . We fix a two-dimensional Hilbert space $\mathcal{H}^{2}$ and on it we
introduce the usual observable for the spin along the $z$-axis, given by the
Hermitian operator

$\displaystyle\hat{S}_{z}=\mbox{$\textstyle\frac{1}{2}$}|z_{1}\rangle\langle\bar{z}_{1}|-\mbox{$\textstyle\frac{1}{2}$}|z_{2}\rangle\langle\bar{z}_{2}|,$
(1)

where $|z_{1}\rangle$ is a unit Hilbert space vector corresponding to the
upward direction along the $z$-axis and $|z_{2}\rangle$ denotes an orthogonal
unit Hilbert space vector corresponding to the downward direction along the
$z$-axis. Thus, $\langle\bar{z}_{1}|z_{1}\rangle=1$,
$\langle\bar{z}_{2}|z_{2}\rangle=1$, $\langle\bar{z}_{1}|z_{2}\rangle=0$,
$\langle\bar{z}_{2}|z_{1}\rangle=0$, and the possible outcomes of the
measurement are the eigenvalues of $\hat{S}_{z}$, which are
$+\mbox{$\textstyle\frac{1}{2}$}$ and $-\mbox{$\textstyle\frac{1}{2}$}$.

The probabilities of these outcomes are determined by the state of the system,
which is represented by a density matrix $\hat{p}$. The density matrix in
quantum theory has a status that is analogous in certain respects to that of
the probability measure in classical probability theory. The density matrix is
assumed to be a positive Hermitian operator with trace unity, which in the
case of a two-dimensional Hilbert space takes the form

$\displaystyle\hat{p}=p_{1}|\psi_{1}\rangle\langle\bar{\psi}_{1}|+p_{2}|\psi_{2}\rangle\langle\bar{\psi}_{2}|,$
(2)

for some orthonormal basis $\\{|\psi_{1}\rangle,|\psi_{2}\rangle\\!\\}$ in
$\mathcal{H}^{2}$, where $p_{1}\geq 0,\,p_{2}\geq 0$, and $p_{1}+p_{2}=1$. In
general, such a matrix will have rank two, but if $p_{1}=0$ or $p_{2}=0$ then
it will have rank one. A state with rank one is called a “pure” state. The
probability for a given outcome is the trace of the product of the state and
the projection operator onto the Hilbert subspace associated to the eigenvalue
corresponding to that outcome (the “Born rule”). Thus we have

$\displaystyle{\rm
Prob}\,\left(S_{z}=\mbox{$\textstyle\frac{1}{2}$}\right)=\langle\bar{z}_{1}|\hat{p}|z_{1}\rangle,\quad{\rm
Prob}\,\left(S_{z}=-\mbox{$\textstyle\frac{1}{2}$}\right)=\langle\bar{z}_{2}|\hat{p}|z_{2}\rangle.$
(3)

In the case of a contingent claim where the payout is determined by the result
of such a spin measurement, it should be clear that the claim itself can also
be represented by a Hermitian operator on $\mathcal{H}^{2}$, in this case, an
operator of the form

$\displaystyle\hat{Z}_{T}=z_{1}|z_{1}\rangle\langle\bar{z}_{1}|+z_{2}|z_{2}\rangle\langle\bar{z}_{2}|,$
(4)

where $z_{1}$ denotes the payment made to agent $A$ in the case the
measurement outcome is spin $+\mbox{$\textstyle\frac{1}{2}$}$ and $z_{2}$ is
the payment made to $A$ when the measurement outcome is spin
$-\mbox{$\textstyle\frac{1}{2}$}$. One can think of such a contract as being
an example of a so-called real option dixit pindyck ; hughston zervos ;
trigeorgis . Payments are understood to be made in some fixed numeraire or
unit of account. Thus, we conclude that a contingent claim for which the
payouts are determined by the result of a quantum measurement can be
represented by an observable, in the usual quantum mechanical sense, whose
eigenvalues correspond to the possible cash flows at time $T$.

Among the various observables that can be represented in the form (4) there is
a special observable that takes the form

$\displaystyle\hat{P}_{0T}=1|z_{1}\rangle\langle\bar{z}_{1}|+1|z_{2}\rangle\langle\bar{z}_{2}|,$
(5)

which pays one unit of account at time $T$, regardless of the outcome of the
spin measurement. This is evidently a “risk-free” asset, since the payout is
fixed and guaranteed, and we write

$\displaystyle\hat{P}_{0T}=\hat{\bf 1}.$ (6)

Here $\hat{\bf 1}$ denotes the identity operator on $\mathcal{H}^{2}$. The
risk-free asset $\hat{P}_{0T}$ can be thought of as a discount bond that pays
one unit of account (e.g., one “dollar”) at maturity $T$. It has the
interesting property that it does not depend on the specific choice of axis
along which the spin measurement is taken.

In addition to contracts of the form (4), we can more generally consider
contracts of the same type, but where the measurement of the spin is taken
along some other axis. Each such contract is characterized by (a) the choice
of a basis in Hilbert space along which the spin measurement is made, together
with (b) the payouts that take place as a consequence of the results of the
measurement. Indeed, it is a theorem that any positive Hermitian operator
$\hat{Z}_{T}$ on $\mathcal{H}^{2}$ other than multiples of the identity can
expressed uniquely in the form (4) for some choice of the orthonormal basis
$\\{|z_{1}\rangle,|z_{2}\rangle\\!\\}$ in $\mathcal{H}^{2}$, modulo
multiplicative phase factors.

To complete the discussion we need to determine the price paid by agent $A$ to
agent $B$ in exchange for the payout corresponding to $\hat{X}_{T}$. In short,
we need a pricing function that maps each observable $\hat{X}_{T}$ to a
corresponding price $X_{0}$.

##  II Existence of Pricing Operator

It will be useful going forward to generalize our considerations to the case
of a Hilbert space $\mathcal{H}$ of arbitrary finite dimension $n$. As usual,
we can write $|\xi\rangle$ for a typical element of $\mathcal{H}$ and
$\langle\bar{\xi}|$ for its complex conjugate. The observable that determines
the payout will in the generic situation be a non-degenerate Hermitian
operator $\hat{X}_{T}$ on this space and hence admit $n$ distinct real
eigenvalues, each corresponding to a distinct cash flow.

For example, if the quantum system admits $n$ different energy levels, and the
underlying physical observable being measured is the energy of the system,
then the contract will in general result in a different cash flow
$\\{x_{j}\\}_{j=1,2,.\,.\,.\,,n}$ for each of the possible energy outcomes.
For the financial observable representing such a contract we can write

$\displaystyle\hat{X}_{T}=\sum_{j=1}^{n}x_{j}|x_{j}\rangle\langle\bar{x}_{j}|,\quad\langle\bar{x}_{j}|x_{k}\rangle=\delta_{jk}$
(7)

for some orthonormal basis $\\{|x_{j}\rangle\\!\\}_{j=1,2,.\,.\,.\,,n}$ in
Hilbert space. More generally, the set of all financial observables associated
with a given Hilbert space will include some that are degenerate in the sense
that the same payout will result for two or more distinct values of the
outcome $j$. Such a degeneracy can result either because there is a degeneracy
in the spectrum of the underlying physical observable, or because two or more
distinct eigenvalues of the physical observable are assigned the same cash
flow. An example of the latter is a unit discount bond, for which $x_{j}=1$
for all $j=1,2,.\,.\,.\,,n$ even though the underlying energy levels may be
distinct. Then the identity operator on $\mathcal{H}$ represents such a bond
and again we write (6) in that case.

Another example of a degenerate observable is the analogue of a so-called
Arrow-Debrue (A-D) security arrow debrue , which for each value of $j$ has the
payout $x_{j}=\mathds{1}\\{j=k\\}$ for some fixed value of $k$. Here
$\mathds{1}\\{E\\}$ denotes the indicator function for the event $E$. Thus
$x_{j}=1$ if $j=k$, and $x_{j}=0$ if $j\neq k$. The A-D securities are
represented by pure projection operators, each with payout unity or zero,
depending on the result of the underlying quantum measurement, whose outcome
is also unity or zero. Thus the set of all Arrow-Debrue type contracts is
precisely the set of all pure projection operators on $\mathcal{H}$.

The state of a quantum system in $n$ dimensions is represented by a positive
semidefinite Hermitian matrix with trace unity. Such a matrix can be put in
the form

$\displaystyle\hat{p}=\sum_{j=1}^{n}p_{j}|\psi_{j}\rangle\langle\bar{\psi}_{j}|,$
(8)

for some orthonormal basis $\\{|\psi_{j}\rangle\\!\\}_{j=1,2,.\,.\,.\,,n}$,
with $p_{j}\geq 0$ for $j=1,2,.\,.\,.\,,n$ and $\sum_{j=1}^{n}p_{j}=1$.

In the case of a density matrix of maximal rank with distinct eigenvalues,
this basis is uniquely determined up to phase factors. If the density matrix
is of maximal rank but with a degenerate spectrum, the basis is determined
modulo unitary transformations on the degenerate subspaces. In particular, in
the case of a density matrix of lower rank, the basis is determined at best
only up to an arbitrary unitary transformation of the basis vectors that span
the null space of the density matrix.

Given two density matrices $\hat{p}$ and $\hat{q}$, we say that $\hat{q}$ is
absolutely continuous with respect to $\hat{p}$ if the null space of $\hat{p}$
is a subspace of the null space of $\hat{q}$. Thus, $\hat{q}$ is absolutely
continuous with respect to $\hat{p}$ if and only if for all
$|\psi\rangle\in\mathcal{H}$ such that $\hat{p}|\psi\rangle=0$ it holds that
$\hat{q}|\psi\rangle=0$. We say that $\hat{p}$ and $\hat{q}$ are equivalent if
each is absolutely continuous with respect to the other, that is to say, if
they share the same null space. For example, in four dimensions the operators

$\displaystyle\hat{p}=p_{1}|\psi_{1}\rangle\langle\bar{\psi}_{1}|+p_{2}|\psi_{2}\rangle\langle\bar{\psi}_{2}|+p_{3}|\psi_{3}\rangle\langle\bar{\psi}_{3}|+p_{4}|\psi_{4}\rangle\langle\bar{\psi}_{4}|$
(9)

and

$\displaystyle\hat{q}=q_{1}|\psi^{\prime}_{1}\rangle\langle\bar{\psi}^{\prime}_{1}|+q_{2}|\psi^{\prime}_{2}\rangle\langle\bar{\psi}^{\prime}_{2}|+q_{3}|\psi_{3}\rangle\langle\bar{\psi}_{3}|+q_{4}|\psi_{4}\rangle\langle\bar{\psi}_{4}|$
(10)

are equivalent when $p_{1}>0,p_{2}>0,p_{3}=0,p_{4}=0$,
$q_{1}>0,q_{2}>0,q_{3}=0,q_{4}=0$, if
$\\{|\psi_{1}\rangle,|\psi_{2}\rangle\\}$ and
$\\{|\psi^{\prime}_{1}\rangle,|\psi^{\prime}_{2}\rangle\\}$ are related by a
unitary transformation in the Hilbert subspace orthogonal to the subspace
spanned by $|\psi_{3}\rangle$ and $|\psi_{4}\rangle$. This is because the two
density matrices share a common null space, spanned by the orthogonal vectors
$|\psi_{3}\rangle$ and $|\psi_{4}\rangle$. It is easy to see that
“equivalence” in this sense is an equivalence relation in the usual
mathematical sense, and it follows that all density matrices of maximal rank
are equivalent.

We shall say that a claim $\hat{X}_{T}$ is positive if
$\langle\bar{\psi}|\hat{X}_{T}|\psi\rangle\geq 0$ for all
$|\psi\rangle\in\mathcal{H}$ and strictly positive if
$\langle\bar{\psi}|\hat{X}_{T}|\psi\rangle>0$ for all
$|\psi\rangle\in\mathcal{H}$. By (7) one sees that $\hat{X}_{T}$ is positive
if and only if $x_{j}\geq 0$ for all $j$ and strictly positive if and only if
$x_{j}>0$ for all $j$. It is legitimate to consider financial contracts
resulting in negative cash flows as well, but it will suffice for our purpose
to look at financial contracts with positive cash flows.

Let us now consider a one-period market represented by the set of all positive
claims on an $n$-dimensional Hilbert space. If $\hat{X}_{T}$ and $\hat{Y}_{T}$
are two claims, then so is the linear combination

$\displaystyle\hat{Z}_{T}=a\hat{X}_{T}+b\hat{Y}_{T}$ (11)

for $a,b\geq 0$. Hence the space of positive claims has a natural convex
structure. It should be evident that the experiments underlying the claims
$\hat{X}_{T}$ and $\hat{Y}_{T}$ are in general different and that the
experiment underlying $\hat{Z}_{T}$ is different yet again. If we write these
claims in their diagonalized forms

$\displaystyle\hat{X}_{T}=\sum_{j=1}^{n}x_{j}|x_{j}\rangle\langle\bar{x}_{j}|,\quad\hat{Y}_{T}=\sum_{j=1}^{n}y_{j}|y_{j}\rangle\langle\bar{y}_{j}|,$
(12)

with respect to the relevant basis vectors, one sees that the payouts and
basis vectors associated with these claims are uniquely determined, up to the
usual ambiguities associated with degeneracies and null spaces, and at the
same time the payouts and basis vectors of (11) are represented by the
decomposition

$\displaystyle\hat{Z}_{T}=\sum_{j=1}^{n}z_{j}|z_{j}\rangle\langle\bar{z}_{j}|.$
(13)

Here we see a curious feature arising in the analysis of such financial
instruments: if we have two contracts, each with positive payouts, depending
on separate measurements, then any linear combination of the operators
corresponding to the two contracts, with positive coefficients, will give rise
to the operator corresponding to yet another contracts, with a different set
of payouts, depending on still another measurement. Thus, a linear combination
(11) is not, generally, to be understood as representing a “portfolio” of its
constituents (we consider how to model portfolios in Section VI). One is
tempted, nonetheless, to conjecture that the price of the contract represented
by a linear combination of two contracts should equal the corresponding linear
combination of the prices of the constituents. But it is not obvious that this
will be the case, since the new contract involves a different payout structure
and a different experiment – so we do not simply wish to assume linearity in
general.

We can, however, quite reasonably assume that such a linear relationship holds
in certain special situations. In particular, if two contracts $\hat{U}_{T}$
and $\hat{V}_{T}$ depend on the outcome of the same experiment, and differ
from one another only in the amounts paid for the various outcomes of the
experiment, then the price of the contract

$\displaystyle\hat{W}_{T}=a\hat{U}_{T}+b\hat{V}_{T}$ (14)

should indeed be equal to the corresponding linear combination of the prices
of $\hat{U}_{T}$ and $\hat{V}_{T}$. More precisely, if the operators
$\hat{U}_{T}$ and $\hat{V}_{T}$ commute, then the prices should be additive.
The reasoning is as follows. If $\hat{U}_{T}$ and $\hat{V}_{T}$ commute, we
can find a common orthogonal basis $\\{|w_{j}\rangle\\}_{j=1,2,\,.\,.\,.\,,n}$
in which both are diagonalized:

$\displaystyle\hat{U}_{T}=\sum_{j=1}^{n}u_{j}|w_{j}\rangle\langle\bar{w}_{j}|,\quad\hat{V}_{T}=\sum_{j=1}^{n}v_{j}|w_{j}\rangle\langle\bar{w}_{j}|.$
(15)

Then if we form the linear combination (14) we obtain

$\displaystyle\hat{W}_{T}=\sum_{j=1}^{n}(au_{j}+bv_{j})|w_{j}\rangle\langle\bar{w}_{j}|,$
(16)

showing that the payouts for $\hat{W}_{T}$ are given by linear combinations of
the payouts of the constituents. Thus, for commuting observables it is obvious
that the price of a linear combination of contracts should be the
corresponding linear combination of the prices of the individual contracts;
but it is not obvious that linearity extends to non-commuting contracts.

At this point, it may be helpful if we codify our assumptions somewhat more
explicitly. As usual, we write $\mathbb{R}^{+}=\\{x\in\mathbb{R}:x\geq 0\\}$.
We fix a quantum system with state $\hat{p}$ on an $n$-dimensional Hilbert
space $\mathcal{H}$ and write $\mathcal{V}^{+}$ for the cone for positive
contracts on $\mathcal{H}$. Thus our market is characterized by the triple
$\\{\mathcal{H},\hat{p},\mathcal{V}^{+}\\}$. Let us write $P_{0T}$ for the
price of a unit discount bond. Our goal is to assign a price to each contract
$\hat{X}_{T}\in\mathcal{V}^{+}$.

By a pricing function on the market
$\\{\mathcal{H},\hat{p},\mathcal{V}^{+}\\}$ in a one-period setting we mean a
mapping $\Pi_{0T}:\mathcal{V}^{+}\to\mathbb{R}^{+}$ satisfying the following:

Axiom (1). For all $\hat{X}_{T}\in\mathcal{V}^{+}$ it holds that
$\Pi_{0T}[\hat{X}_{T}]=0$ if and only if ${\rm tr}(\hat{p}\hat{X}_{T})$ = 0.

Axiom (2). If the $m$ contracts represented by the Hermitian matrices
$\\{\hat{X}_{T}^{k}\\}_{k=1,2,\,.\,.\,.\,,m}$
commute, then for all $\\{a_{k}\geq 0\\}_{k=1,2,\,.\,.\,.\,,m}$ one has

$\displaystyle\Pi_{0T}\\!\left[\sum_{k=1}^{m}a_{k}\,\hat{X}_{T}^{k}\right]=\sum_{k=1}^{m}a_{k}\,\Pi_{0T}\\!\left[\hat{X}_{T}^{k}\right]\\!.$
(17)

Axiom (3). $\Pi_{0T}[\,\hat{\bf 1}\,]=P_{0T}$.

The axioms can be interpreted as follows. Axiom (1) ensures the absence of
arbitrage: the price of a positive contract vanishes if and only if the
expected payout vanishes. Axiom (2) ensures that the pricing function is
linear when it acts on a collection of contracts represented by commuting
observables. Axiom (3) fixes the price of the risk-free asset. Then we obtain
the following general characterization of the price of a contract:

######  Proposition 1.

If $n\geq 3$ then there exists a state $\hat{q}$ on
$\\{\mathcal{H},\hat{p},\mathcal{V}^{+}\\}$ that is equivalent to $\hat{p}$
such that for any contract $\hat{X}_{T}\in\mathcal{V}^{+}$ the price of
$\hat{X}_{T}$ is given by

$\displaystyle\Pi_{0T}\large[\hat{X}_{T}\large]=P_{0T}\,{\rm
tr}(\hat{q}\hat{X}_{T}).$ (18)

Proof. It suffices to consider the pricing of A-D securities. For each such
contract, the underlying measurement takes the form of a projection operator
$\hat{\Lambda}=|\lambda\rangle\langle\bar{\lambda}|$ for some normalized
vector $|\lambda\rangle\in\mathcal{H}$. The pricing function gives a map from
the space of pure projection operators on $\mathcal{H}$ to $\mathbb{R}^{+}$.
Now, it is well known that the space of pure projection operators on a Hilbert
space of dimension $n$ is isomorphic to the complex projective space
$\mathbb{CP}^{n-1}$. Thus we obtain a real function
$\Pi_{0T}:\mathbb{CP}^{n-1}\to\mathbb{R}^{+}$ with the property that for any
set of $n$ points $\\{\lambda_{j}\in\mathbb{CP}^{n-1}\\}_{j=1,2,.\,.\,.\,,n}$
corresponding to an orthogonal basis in $\mathcal{H}$ one has

$\displaystyle\sum_{j=1}^{n}\Pi_{0T}(\lambda_{j})=P_{0T}.$ (19)

This is because the projection operators associated with an orthonormal basis
commute and hence by Axiom (2) the sum of the prices of the projection
operators must equal the price of the sum of the projection operators. But the
latter sum gives the identity operator, which offers a risk-free payout of
unity. Thus we obtain a unit discount bond, for which the price is $P_{0T}$ by
Axiom (3). Gleason’s theorem gleason  can now be applied to the problem and it
follows that there exists a state $\hat{q}$ such that the price of any claim
of the form $\hat{\Lambda}$ is given by

$\displaystyle\Pi_{0T}\large[\hat{\Lambda}\large]=P_{0T}\,{\rm
tr}(\hat{q}\hat{\Lambda}).$ (20)

Now, any contract $\hat{X}_{T}$ can be constructed as a linear combination of
orthogonal pure projection operators with positive coefficients. Since these
operators commute, Axiom (2) implies that the price of such a contract will be
given by the sum of the prices of its elements, and this gives us (18). The
fact that the “pricing” operator $\hat{q}$ must be equivalent to the
“physical” state $\hat{p}$ follows as a consequence of Axiom (1), which
ensures that the price of a positive contract $\hat{X}_{T}$ vanishes if and
only if $\hat{X}_{T}$ is concentrated on the null space of $\hat{p}$. $\Box$

The key point here is that we do not assume a priori the existence of a
pricing state. The idea rather is to prove the existence of such a state under
the prima facie much weaker assumptions implicit in our axioms. The
requirement that the pricing function is linear when it is applied to any
commuting family of A-D securities coupled with the assumption that the price
of a one-period discount bond is known allows us to deduce that the pricing
function takes the form (18). In the case of a finite-dimension Hilbert space,
the associated projective Hilbert space takes the form of a complex projective
space $\mathbb{CP}^{n-1}$ equipped with the Fubini-Study metric BH2001 .
Gleason’s theorem shows that any map $f:\mathbb{CP}^{n-1}\to[0,1]$ with the
property that $\sum_{j=1}^{n}f(\lambda_{j})=1$ for any set of $n$ points
$\\{\lambda_{j}\\}_{j=1,2,.\,.\,.\,n}\in\mathbb{CP}^{n-1}$ that are maximally
distant from each other under the Fubini-Study metric necessarily takes the
form
$f(\lambda)=\langle\bar{\lambda}|\hat{q}|\lambda\rangle/\langle\bar{\lambda}|\lambda\rangle$
for some positive operator $\hat{q}$ with trace unity. The principle of no
arbitrage (“no free lunch”) then implies that $\hat{q}$ is equivalent to
$\hat{p}$.

It should be noted that the physical state $\hat{p}$ refers to the state of
the quantum system upon which measurement of a given physical observable
determines the payment made under the terms of the financial contract. Thus
$\hat{p}$ can be used to calculate the probability distribution of the payout,
but gives no information about the price, except that minimal statement which
is mandated by the absence of arbitrage – namely, that the price should be
zero if and only if the probability of a payout greater than zero is zero.

The operator $P_{0T}\,\hat{q}$ plays the role of a pricing kernel in our
theory. In the case of an $n$-dimensional Hilbert space the prices of any
$n^{2}-1$ linearly independent financial contracts, alongside the price of the
unit discount bond, will be sufficient to completely calibrate the pricing
kernel, which can then be used to price other contracts. It may seem
surprising that the knowledge of such a system of prices gives no information
about the physical state $\hat{p}$, except to determine its null space, but
the analogue of this phenomenon is well known in the classical theory of
finance bingham kiesel ; dothan ; etheridge ; duffie ; follmer schied . At
first glance, one might think that Proposition 1 is rather weak, since the
pricing operator $\hat{q}$ is completely arbitrary apart from its having the
same null space as $\hat{p}$; but such a conclusion would be incorrect – the
point is that the existence of a pricing operator is not assumed but rather is
deduced from the minimal axioms we have chosen to characterize a pricing
function. Thus, beginning with only the assumed existence of a pricing
function, which might in principle be nonlinear, one can whittle the
candidates for such a map down to a linear function of the specific form (18).

##  III Optimal Investment

A typical problem in classical finance theory is to determine, given a budget
$X_{0}$, the investment that maximizes the expectation of the utility gained
by the investor at time $T$ when the proceeds of the investment are
liquidated. It is reasonable then to pose a similar problem in quantum
finance. Let us assume (a) that agent $A$’s attitudes towards risk and return
can be expressed by a standard utility function $\\{U(x)\\}_{x>0}$, (b) that
the physical state $\hat{p}$ of the quantum system is known, (c) that the
basis under which the physical measurement is being made is known, and (d)
that the pricing operator is known. Thus the investment will be characterized
by an observable of the form (7), where the basis
$\\{|x_{j}\rangle\\}_{j=1,.\,.\,.\,,n}$ is fixed, and the cash flows
$\\{x_{j}\\}_{j=1,.\,.\,.\,,n}$ must be determined in such a way that the
budget is saturated and the expected utility is maximized. What makes the
problem interesting in the present setting is that the expected utility of the
payout is calculated by use of the physical state $\hat{p}$ whereas the budget
constraint involves the pricing state $\hat{q}$, and that neither $\hat{p}$
nor $\hat{q}$ necessarily has any special relation to the measurement basis.

######  Definition 1.

By a standard utility function we mean a map
$U:\mathbb{R}^{+}\backslash\\{0\\}\to\mathbb{R}$ that satisfies the following
conditions: (i) $U\in{\rm C}^{2}(\mathbb{R}^{+}\backslash\\{0\\})$, (ii)
$U^{\prime}(x)>0$ for all $x>0$, (iii) $U^{\prime\prime}(x)<0$ for all $x>0$,
(iv) $\lim_{x\to\infty}U^{\prime}(x)=0$, and (v) $\lim_{x\to
0}U^{\prime}(x)=\infty$.

These requirements can be relaxed somewhat, but the “standard” conditions
typically lead to well-posed problems for which solutions can be shown to
exist and hence prove to be natural as a basis for modelling. Note that a
standard utility function is a strictly convex, strictly increasing function
defined for all strictly positive values of its argument. We refer to the
function
$U^{\prime}:\mathbb{R}^{+}\backslash\\{0\\}\to\mathbb{R}^{+}\backslash\\{0\\}$
as the marginal utility. The final two conditions of the definition ensure
that a standard utility function has the property that there exists an inverse
marginal utility function $\\{I(y)\\}_{y>0}$ such that $I(U^{\prime}(x))=x$
for all $x>0$. Examples of standard utility functions are (a) logarithmic
utility, for which $U(x)=\log(x)$ for $x>0$, and (b) power utility with index
$p\in(-\infty,1)\backslash\\{0\\}$, for which $U(x)=p^{-1}x^{p}$ for $x>0$.
For logarithmic utility one finds that $I(y)=1/y$ and for power utility
$I(y)=y^{1/(p-1)}$.

The goal of agent $A$’s optimization problem is to determine the cash flows
$\\{x_{j}\\}_{j=1,2,.\,.\,.\,,n}$ that maximize the expected value of the
utility, providing that these cash flows can be realized with the specified
budget. Thus, given a standard utility function $\\{U(x)\\}_{x>0}$ we set

$\displaystyle\\{x^{*}_{j}\\}_{j=1,2,\,.\,.\,.\,,n}=\operatorname*{argmax}_{\\{x_{j}\\}}\,{\rm
tr}\left[\hat{p}\,\hat{U}(\\{x_{j}\\})\right]$ (21)

where
$\hat{U}(\\{x_{j}\\})=\sum_{j=1}^{n}U(x_{j})|x_{j}\rangle\langle\bar{x}_{j}|$
and the argmax is subject to the budget constraint

$\displaystyle X_{0}=P_{0T}\,{\rm
tr}(\hat{q}\,\hat{X}_{T}),\quad\hat{X}_{T}=\sum_{j=1}^{n}x_{j}|x_{j}\rangle\langle\bar{x}_{j}|.$
(22)

######  Proposition 2.

Let the physical state of a quantum system on an $n$-dimensional Hilbert space
be $\hat{p}$. Let the pricing state for a financial market based on
measurements of the system be $\hat{q}$, with one-period discount factor
$P_{0T}$. Let the risk preferences of the investor be represented by a
standard utility function $U:\mathbb{R}^{+}\backslash\\{0\\}\to\mathbb{R}$ and
write $I$ for the associated inverse marginal utility function. Then the
optimal cash flow structure $\\{x^{*}_{j}\\}$ for an investment with budget
$X_{0}$ paying out according to the measurement of a financial observable of
the form

$\displaystyle\hat{X}=\sum_{j=1}^{n}x_{j}|x_{j}\rangle\langle\bar{x}_{j}|,$
(23)

for some fixed orthonormal basis $\\{|x_{j}\rangle\\!\\}_{j=1,.\,.\,.\,,n}$,
is given by

$\displaystyle
x^{*}_{j}=I\\!\left[\lambda\frac{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}\right]\\!,$
(24)

where for any choice of $X_{0}>0$ the parameter $\lambda$ is uniquely
determined by the relation

$\displaystyle
P_{0T}\sum_{j=1}^{n}I\\!\left[\lambda\frac{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}\right]\langle
x_{j}|\hat{q}|\bar{x}_{j}\rangle=X_{0}.$ (25)

Proof. The method of Lagrange multipliers can be used to obtain a candidate
for the argmax. We introduce a Lagrange multiplier $\lambda$ and seek a
solution to the unconstrained problem

$\displaystyle\\{x^{*}_{j}\\}=\operatorname*{argmax}_{\\{x_{j}\\}}\,\left({\rm
tr}\left[\hat{p}\,\hat{U}(\\{x_{j}\\})\right]-\lambda P_{0T}\,{\rm
tr}(\hat{q}\,\hat{X}_{T})\right)\\!,$ (26)

or equivalently

$\displaystyle\\{x^{*}_{j}\\}=\operatorname*{argmax}_{\\{x_{j}\\}}\,\left(\sum_{j=1}^{n}U(x_{j})\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle-\lambda
P_{0T}\sum_{j=1}^{n}x_{j}\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle\right)\\!.$
(27)

Differentiating with respect to $x_{j}$ and setting the results to zero, we
find that

$\displaystyle
U^{\prime}(x_{j})=\lambda\frac{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}$
(28)

for each value of $j$. Applying the inverse marginal utility function to each
side of this equation, we are then led to (24) and the budget constraint (22)
gives (25). That (25) admits a unique solution for $\lambda$ for any $X_{0}>0$
follows from the fact that the monotonic decreasing map
$I:\mathbb{R}^{+}\backslash\\{0\\}\to\mathbb{R}^{+}\backslash\\{0\\}$ is
surjective, which is a consequence of the conditions (iv) and (v) satisfied by
a standard utility function. That the candidate solution is indeed a true
solution can be checked by use of the fundamental inequality
$U(I(y))-I(y)y\geq U(x)-xy$, which holds for all $x>0$ and $y>0$ in the case
of a standard utility function. It follows then from (24) that for any
alternative choice of payout structure $\\{x_{j}\\}$ we have

$\displaystyle
U(x^{*}_{j})-x^{*}_{j}\lambda\frac{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}\geq
U(x_{j})-x_{j}\lambda\frac{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle},$
(29)

for each $j=1,2,.\,.\,.\,,n.$ Multiplying by $p_{j}$ and summing we obtain

$\displaystyle\sum_{j=1}^{n}p_{j}U(x^{*}_{j})-\sum_{j=1}^{n}p_{j}U(x_{j})\geq\sum_{j=1}^{n}p_{j}x^{*}_{j}\lambda\frac{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}-\sum_{j=1}^{n}p_{j}x_{j}\lambda\frac{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}.$
(30)

Then since $p_{j}=\langle x_{j}|\hat{p}|\bar{x}_{j}\rangle$ we have

$\displaystyle\sum_{j=1}^{n}p_{j}U(x^{*}_{j})-\sum_{j=1}^{n}p_{j}U(x_{j})\geq\lambda\left[\sum_{j=1}^{n}x^{*}_{j}\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle-\sum_{j=1}^{n}x_{j}\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle\right]\\!.$
(31)

Now, we know by (25) that $\lambda$ has been chosen to ensure that the
candidate solution $\\{x^{*}_{j}\\}$ satisfies the budget constraint

$\displaystyle
P_{0T}\sum_{j=1}^{n}x^{*}_{j}\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle=X_{0}.$
(32)

If we require that the alternative choice of payout structure should also
satisfy the budget constraint, or perhaps operate under budget, so

$\displaystyle
P_{0T}\sum_{j=1}^{n}x_{j}\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle\leq X_{0},$
(33)

then the two terms on the right-hand side of (31) cancel, or else leave a
difference that is positive (if the alternative choice is under budget), which
gives

$\displaystyle\sum_{j=1}^{n}p_{j}U(x^{*}_{j})\geq\sum_{j=1}^{n}p_{j}U(x_{j}),$
(34)

showing that the candidate solution for the optimal payout gives an expected
utility that is no less than that of any alternative choice of payout
structure with a budget no greater than that of the candidate solution. $\Box$

##  IV Rate of return

As an example, we can look in detail at the case of logarithmic utility.
Suppose we set $U(x)=\log x$ for $x>0$. Then the inverse marginal utility
function is given by $I(y)=1/y$ for $y>0$. It follows that for log utility the
optimal payout structure takes the form

$\displaystyle
x^{*}_{j}=\lambda^{-1}\frac{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}.$
(35)

Inserting this expression into the budget constraint (32) we obtain

$\displaystyle
P_{0T}\lambda^{-1}\sum_{j=1}^{n}\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle=X_{0}.$
(36)

But the sum appearing in the expression above is unity since
$\sum_{j=1}^{n}|x_{j}\rangle\langle\bar{x}_{j}|=\hat{\bf 1}$ and the trace of
$\hat{p}$ is one. Thus for log utility we deduce that
$P_{0T}\lambda^{-1}=X_{0}$ and hence

$\displaystyle
x^{*}_{j}=(P_{0T})^{-1}X_{0}\frac{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}.$
(37)

We observe that when the physical state and the pricing state are one and the
same, the payouts of the optimal investment are identical for each outcome of
chance, each giving $(P_{0T})^{-1}X_{0}$, the usual “future value” of the
initial investment. In that case, the optimal investment is to put the initial
endowment into unit discount bonds, totalling $X_{0}$ in value. Then we have
$\hat{X}_{T}=(P_{0T})^{-1}X_{0}\,\hat{\bf 1}$ and one sees that if the pricing
state is the physical state, the market assigns no premium to the return on a
risky investment, ensuring that the optimal investment is in a discount bond
and the rate of return is the interest rate.

The same conclusion applies, more generally, for any choice of the utility.
This follows from (24) and (25), from which one concludes that if
$\hat{p}=\hat{q}$ then $x^{*}_{j}=(P_{0T})^{-1}X_{0}$ for all $j$. It is
interesting therefore to enquire what happens when the pricing state is
different from the physical state. The return $R_{0T}$ on an investment
$\hat{X}_{T}$ is given by the ratio of the expectation of $\hat{X}_{T}$ under
$\hat{p}$ to the amount initially invested, namely $X_{0}$. Thus, quite
generally, we have

$\displaystyle R_{0T}=(X_{0})^{-1}{\rm tr}(\hat{p}\hat{X}_{T}).$ (38)

But $X_{0}=P_{0T}\,{\rm tr}(\hat{q}\hat{X}_{T})$ by (18), so we deduce that

$\displaystyle R_{0T}=(P_{0T})^{-1}\frac{{\rm tr}(\hat{p}\hat{X}_{T})}{{\rm
tr}(\hat{q}\hat{X}_{T})},$ (39)

and it should be clear that if $\hat{p}=\hat{q}$, except possibly on the null
space of $\hat{X}_{T}$, then the rate of return on the investment is the one-
period interest rate.

Specializing now to the case of an optimal investment for an agent with
logarithmic utility, let us calculate the rate of return. We have

$\displaystyle\hat{X}_{T}=\sum_{j=1}^{n}x^{*}_{j}\,|x_{j}\rangle\langle\bar{x}_{j}|,$
(40)

where the optimal payout structure $\\{x^{*}_{j}\\}$ is given by (37). It
follows then that

$\displaystyle R_{0T}$ $\displaystyle=$ $\displaystyle(X_{0})^{-1}{\rm
tr}(\hat{p}\hat{X}_{T})$ (41) $\displaystyle=$ $\displaystyle(X_{0})^{-1}{\rm
tr}\left(\hat{p}\,\sum_{j=1}^{n}x^{*}_{j}\,|x_{j}\rangle\langle\bar{x}_{j}|\right)$
$\displaystyle=$
$\displaystyle(X_{0})^{-1}\sum_{j=1}^{n}x^{*}_{j}\,\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle$
$\displaystyle=$
$\displaystyle(P_{0T})^{-1}\sum_{j=1}^{n}\frac{\,\,\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle^{2}}{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}.$

If we set $R_{0T}=\mbox{$\rm e$}^{\mu T}$ then the rate of return $\mu$ can be
split into two parts, namely a risk-free one-period interest rate and a so-
called excess rate of return, which is the part of the rate of return that
exceeds the interest rate. We can represent this in a standard way by writing
$R_{0T}=\mbox{$\rm e$}^{(r+\beta)T}$ where $r$ is the interest rate and
$\beta$ denotes the excess rate of return. The interest rate is determined by
the relation $\mbox{$\rm e$}^{rT}=(P_{0T})^{-1}$ and the excess rate of return
is determined by the relation

$\displaystyle\mbox{$\rm e$}^{\beta
T}=\sum_{j=1}^{n}\frac{\,\,\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle^{2}}{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}.$
(42)

######  Proposition 3.

The optimal investment in the case of an investor with logarithmic utility has
a positive excess rate of return. The utility gained from such an investment
in a market where the physical state and pricing state differ is greater than
or equal to the utility gained from an investment in a risk-free bond.

Proof. For general utility, the expected utility gained from the payout of an
optimal investment is given by

$\displaystyle{\rm
tr}\left[\hat{p}\,\hat{U}\right]=\sum_{j=1}^{n}U(x^{*}_{j})\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle.$
(43)

Let us set $U(x^{*}_{j})=\log x^{*}_{j}$ for logarithmic utility and insert
(37). The result is

$\displaystyle\sum_{j=1}^{n}U(x^{*}_{j})\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle=\log\left[(P_{0T})^{-1}X_{0}\right]+\sum_{j=1}^{n}\left[\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle\log\frac{\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle}{\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle}\right]\\!.$
(44)

The first term on the right-hand side of this equation isolates the part of
the utility gain due to the interest rate. The second term can be interpreted
as a relative entropy. In particular, if we set
$p_{j}=\langle\bar{x}_{j}|\hat{p}|x_{j}\rangle$ and
$q_{j}=\langle\bar{x}_{j}|\hat{q}|x_{j}\rangle$ then it is evident that
$\\{p_{j}\\}_{j=1,2,.\,.\,.\,,n}$ and $\\{q_{j}\\}_{j=1,2,.\,.\,.\,,n}$
constitute a pair of absolutely continuous probability distributions. The
second term on the right then takes the form of a Kullback-Liebler divergence
Kullback Leibler 1951 :

$\displaystyle
D_{KL}(p,q)=\sum_{j=1}^{n}p_{j}\log\left(\frac{p_{j}}{q_{j}}\right)\\!.$ (45)

Thus, the utility thus gained gives a measure of the divergence between the
physical density matrix and the pricing density matrix. Now, it is well known
that the Kullback-Liebler divergence is non-negative. It follows, then, that
the utility gained from an optimal risky investment in a market where
$\hat{p}$ and $\hat{q}$ are distinct will be greater than or equal to the
utility gained from a risk-free bond investment, as claimed. Moreover, we have
the following. The standard logarithmic inequality $\log z\leq z-1$, which
holds for $z>0$, implies that

$\displaystyle\log\left(\frac{p_{j}}{q_{j}}\right)\leq\left(\frac{p_{j}}{q_{j}}\right)-1$
(46)

for each $j$. Hence, multiplying by $p_{j}$ and summing we obtain

$\displaystyle\sum_{j=1}^{n}p_{j}\log\left(\frac{p_{j}}{q_{j}}\right)\leq\sum_{j=1}^{n}\left(\frac{p_{j}^{2}}{q_{j}}\right)-1.$
(47)

Thus, we have

$\displaystyle\mbox{$\rm e$}^{\beta
T}=\sum_{j=1}^{n}\left(\frac{p_{j}^{2}}{q_{j}}\right)\geq 1+D_{KL}(p,q),$ (48)

and by the positivity of the Kullback-Liebler divergence we deduce that the
excess rate of return $\beta$ is positive for an optimal investment under
logarithmic utility, as claimed. $\Box$

##  V Kolmogorov vs Bell-Kochen-Specker

It is often maintained by physicists that quantum probability is more general
than Kolmogorov’s well-established “classical” theory of probability
kolmogorov1933  and that the latter is contained as a special case of the
former. There is no doubt that quantum probability, when laid out as a
mathematical theory, does have a different look and feel when it is compared
to Kolmogorov’s theory; but despite the fact that numerous well-argued
accounts of quantum probability can be found in the literature davies ;
deutsch ; gudder ; holevo ; kraus ; mackey ; segal1947 ; streater2000  (see
also BCFFS ; deutsch ; wallace 2010 ), some even taking an axiomatic approach,
it is not that easy to pinpoint the exact sense in which quantum theory is
essentially non-Kolmogorovian – rather than, say, merely an intricate
reworking of Kolmogorov’s theory in a radically different form. This issue is
compounded by the fact that, except in the most loose terms, it is difficult
to say what one means by “probability” without embedding the concept in a
rigorous framework; and it seems that there are at least two such frameworks
available to us at the moment – namely, classical probability and quantum
probability, each with a host of applications.

It is fortunate then that we have the results of Gleason gleason , Bell
bell1964 ; bell1966 ; bell1987 , Kochen & Specker kochen-specker ; held , and
others following in their footsteps, which add clarity to the matter. The
point is that one has to work rather hard to come up with examples of
situations in quantum probability that cannot be reduced to a classical
probability model. But a number of such examples have been worked out
involving finite-dimensional Hilbert spaces, so this creates the prospect of
constructing financial models for claims based on the results of quantum
measurements, in settings for which quantum probability is required in their
analysis.

Since most of what we know of modern finance theory is based quite explicitly
on Kolmogorov’s framework, both for the derivation of rigorous theorems
concerning the mathematics of financial markets as well as for the
implementation of the large-scale computer models used for trading and risk
management at financial institutions, it may be worthwhile to take note of a
few examples of situations where quantum probability comes into play.

Among the numerous attempts that have been made to generalize or extend the
Kochen-Specker construction mermin ; peres ; kernaghan ; kernaghan-peres ;
penrose ; cabello et al ; cabello 1997 . Perhaps the simplest of these yet put
forward is that of Cabello et al, which entails the specification of nine
different non-commuting observables on a four-dimensional Hilbert space, each
admitting four distinct eigenvalues. In a financial context, one can think of
the setup as involving a single quantum system being prepared in a state
$\hat{p}$ with nine different “draft financial contracts” drawn up, each
requiring measurement of one of the nine observables. The contracts specify
the payments that will be made in the event that one of the four possible
results occurs for the measurement associated with a specific contract. It is
of the nature of quantum probability that only one of the nine contracts can
be implemented, so we can envisage a rational investor being presented with
the alternatives and choosing one optimally in accordance with their needs.

0 0 0 1 0 1 0 0 1 0 1 0 1 0 $\bar{1}$ 00 0 1 0 0 1 0 0 1 0 0 1 1 0 0
$\bar{1}$1 $\bar{1}$ 1 $\bar{1}$ 1 $\bar{1}$ $\bar{1}$ 1 1 1 0 0 0 0 1 11
$\bar{1}$ 1 $\bar{1}$ 1 1 1 1 1 0 $\bar{1}$ 0 0 1 0 $\bar{1}$1 $\bar{1}$
$\bar{1}$ 1 1 1 1 1 1 0 0 $\bar{1}$ 0 1 $\bar{1}$ 01 1 $\bar{1}$ 1 1 1 1
$\bar{1}$ 1 $\bar{1}$ 0 0 0 0 1 11 1 $\bar{1}$ 1 $\bar{1}$ 1 1 1 1 0 1 0 0 1 0
$\bar{1}$1 1 1 $\bar{1}$ $\bar{1}$ 1 1 1 1 0 0 1 0 1 $\bar{1}$ 00 0 0 1 0 0 1
0 1 1 0 0 1 $\bar{1}$ 0 0 Figure 1: Diagram illustrating a result of the
Kochen-Specker type in a 4-dimensional Hilbert space. Each of the 9 vertices
are met by 4 lines and each of the 18 lines join 2 vertices. The 18 lines
represent a set of normalized projection operators with the property that the
4 projection operators meeting a given vertex are mutually orthogonal and sum
to the identity operator. It is easy to see that it is impossible to “colour”
the lines so that one blue line meets each vertex and 3 red lines meet each
vertex. This illustrates the fact that in the standard Kolmogorov setup one
cannot find a set of 18 random variables on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$, each taking values in the set $\\{0,1\\}$,
such that when the 18 random variables are assigned to the 18 lines, the sum
of the 4 random variables meeting any given vertex will be one for all
$\omega\in\Omega$.

The setup is an elaborate although feasible one, and we can use the methods
discussed to calculate the probabilities of the results for the nine different
measurements and hence the expected utility gained from each choice. Each
observable has four possible outcomes, thus determining an orthonormal tetrad
in Hilbert space. These are the four eigenvectors of the Hermitian matrix
corresponding to a given observable. The result of the measurement is to
select one of these eigenvectors. Equivalently, each measurement measures four
commuting projection operators, namely the projection operators associated
with the four legs of the tetrad. The outcome of one of these four
measurements will be unity and the rest nil.

The clever idea behind results of the Kochen-Specker type is to choose the
observables so that some of the tetrads legs overlap when one moves from one
observable to another. In the present situation, involving nine observables,
the overlap structure is shown in Figure 1. Alongside each vertex of the
enneagon one sees the corresponding tetrad, where to ease the typography we
write $\bar{1}$ for $-1$. When two vertices are connected by a dotted line,
this means that the associated tetrads share a vector in common. The analysis
is simplified somewhat by the fact that the tetrads in this example can all be
taken to be real.

If we label the nine observables $\\{\hat{X}_{r}\\}_{r=1,2,\,.\,.\,.,9}$ and
if for each value of $r$ the four projection operators associated with
$\hat{X}_{r}$ are denoted $\\{\hat{\pi}_{rj}\\}_{j=1,2,3,4}$ , then the
probability that outcome $j$ will result, if contract $r$ is chosen, is given
by ${\rm tr}(\hat{\pi}_{rj}\,\hat{p})$. The construction of an analogous setup
within Kolmogorov’s system turns out to be impossible. Since this is a rather
sweeping statement, let us be a little more precise about what is being
claimed. The point is that in Kolmogorov’s theory, one would have to model the
setup with 36 random variables on a single probability space. The 36 random
variables are grouped into nine sets of four. Let’s call these hypothetical
random variables $\\{X_{rj}\\}_{r=1,2,\,.\,.\,.,9,\,j=1,2,3,4}$ (with no
hats). Each random variable can take the value zero or one. Thus we have a
total of 36 maps of the form

$\displaystyle X_{rj}:\Omega\to\\{0,1\\}.$ (49)

There are two requirements that have to be satisfied to match the layout of
the quantum setup. First, the sum of the four random variables for a given
value of $r$ must be unity. This means that one of them must be equal to one
and the other three must be equal to zero for any given outcome of chance
$\omega\in\Omega$. Secondly (this is where the rabbit goes into the hat) the
36 random variables have to be equal in pairs, in conformation with the
structure of the diagram in Figure 1. Thus, the 36 random variables are cut
down in effect to 18 by the requirement that they must match in pairs.

Can one find such a set of 18 random variables? The answer, perhaps
surprisingly, is no. This can be checked by a colouring argument. Given Figure
1, can one colour each line red or blue in such a way that exactly one blue
line meets each vertex? Suppose one finds a way of colouring four of the lines
blue, no vertex being hit by more than one blue line. That would leave one
vertex unmet by a blue line. Suppose then one tried to colour five lines blue.
Well, that would mean at least one vertex was hit by more than one blue line.
This shows that it is impossible to construct a set of 18 random variables on
a probability space in such a way that the required properties are satisfied.

In financial terms, this means that we cannot model the payouts of the nine
contracts as random variables on a probability space in such a way that the
outcome of chance determines the payouts of all nine. A sceptic might ask,
“Isn’t it unlikely in practice that one will come up against such a
configuration of contracts?” Well, that may be so, but the point is that
quantum finance can handle such configurations whereas classical finance
can’t.

##  VI Portfolios as multi-particle systems

Let’s return now to the matter of portfolios. There are two rather distinct
notions of portfolio that arise in quantum finance. The first notion involves
a portfolio of contracts all depending in their payouts on the same
experiment. In that case, we can fix the $n$ axes of the $n$-dimensional
Hilbert space determining the frame of the measurement and write
$\\{\hat{\pi}_{j}\\}_{j=1,2,\,.\,.\,.\,n}$ for the associated projection
operators. Then for a given outcome of the experiment one of these projection
operators will give the result unity and the rest zero. The projection
operators can be regarded as the A-D securities for that experiment and it
should be evident that any contingent claim based on the outcome of the given
experiment can be written as a portfolio of $n$ such A-D securities. Thus, for
such claims we can write

$\displaystyle\hat{X}=\sum_{j=1}^{n}\theta_{j}\,\hat{\pi}_{j},$ (50)

where the $\\{\theta_{j}\\}_{j=1,2,\,.\,.\,.\,n}$ represent the holdings in
the various A-D securities. More generally, if we allow short positions in the
A-D securities, then the resulting overall position can be expressed uniquely
as the difference between two positive claims, with the understanding that we
net claims involving long and short positions in the same A-D security.

Clearly, a linear combination of two portfolios in this setting gives another
portfolio. Furthermore, it should be evident that the operator corresponding
to the portfolio can be represented as the sum of a trace part, proportional
to the identity operator, and a trace-free part. The trace part represents a
position (long or short) in the risk-free asset, and the remainder consists of
investments in risky assets. For example, in two dimensions, a portfolio of
the form $2|z_{1}\rangle\langle\bar{z}_{1}|+|z_{2}\rangle\langle\bar{z}_{2}|$
consists of a long position of three-halves of one unit of the risk free
asset, a long position of one-half of a unit in the A-D security
$\hat{\pi}_{1}$ and a short position of one-half of a unit in the A-D security
$\hat{\pi}_{2}$, since we have
$2\hat{\pi}_{1}+\hat{\pi}_{2}=\frac{3}{2}(\hat{\pi}_{1}+\hat{\pi}_{2})+\mbox{$\textstyle\frac{1}{2}$}(\hat{\pi}_{1}-\hat{\pi}_{2})$.
In this way we can isolate the risk-free part of a portfolio. This first
notion of a portfolio corresponds rather closely to the notion of a portfolio
in a one-period market that arises in classical finance theory arrow debrue ;
bingham kiesel ; duffie ; dothan ; etheridge ; follmer schied  and can be
pursued further in that spirit. The point is that once the measurement basis
for the underlying experiment has been fixed, the various associated operators
arising for positions with different portfolio weightings commute.

As we pointed out in Section II, it does not make sense to form a portfolio of
several contracts each based on the same quantum system but with different
measurement frames, since such measurements will in general be incompatible
and cannot be simultaneously realized. In this respect, our approach differs
completely from that of reference bao rebentrost  who attempt to set up a
portfolio theory on exactly that basis, where all of the assets are associated
with the same Hilbert space. This leads us to conclude their main result (an
analogue of the fundamental theorem of asset pricing) is in some respects ill-
posed. The authors of bao rebentrost  give no clear statement (let alone a
definition) of what is meant in real terms by a “quantum asset” and merely
assume the existence of a pricing state rather than deducing its existence.
Nonetheless, that said, there is some interesting overlap with the present
work.

In our approach to the problem we consider portfolios of assets for which the
payouts are based on separate measurements being made on two or more distinct
quantum systems. Imagine, for example, a financial institution where in one
room an experiment is carried out on Quantum System I, with certain results
obtained, and another experiment is carried out in another room on Quantum
System II, with certain results obtained. In each case, there are contracts
leading to payouts depending on the results obtained.

Since the measurements do not interfere with one another (after all, they are
carried out in different rooms) they can be carried out simultaneously, each
delivering a certain number of units of account, so it makes sense to speak of
holding a portfolio in the two assets, for which the payout is simply the
totality of the payouts of the constituents of the portfolio, with appropriate
weightings.

Let’s see how we model such a situation. To simplify the discussion, we stick
with the case where there are just two quantum systems involved, with
measurements made on each of them. The setup can be easily generalized to the
case where there are $N$ such systems. The key idea is that to model a
portfolio of two such quantum assets, we need to consider the tensor product
of the Hilbert spaces of the individual systems. In fact, the two Hilbert
spaces might even be of different dimensions.

The usual Dirac notation does not hold up so well in such a setting, so we use
an index notation instead, which works quite smoothly geroch ; BH2001 . Thus,
let $\mathcal{H}_{1}$ be a Hilbert space of dimension $n$ and let
$\mathcal{H}_{2}$ be a Hilbert space of dimension $n^{\prime}$, where $n$ and
$n^{\prime}$ are not necessarily the same. We write $\xi^{a}$ and
$\xi^{a^{\prime}}$ for typical elements of $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$ respectively, where $a=1,2,\,.\,.\,.\,n$ and
$a^{\prime}=1,2,\,.\,.\,.\,n^{\prime}$. Thus indices without dashes refer to
the first Hilbert space and indices with dashes refer to the second Hilbert
space. We write $\eta_{a}$ and $\eta_{a^{\prime}}$ for typical elements of the
corresponding dual spaces $\mathcal{H}_{1}^{*}$ and $\mathcal{H}_{2}^{*}$. The
complex conjugates of $\xi^{a}$ and $\xi^{a^{\prime}}$ are denoted
$\bar{\xi}_{a}$ and $\bar{\xi}_{a^{\prime}}$ respectively. Then for the inner
product between $\xi^{a}$ and $\eta_{a}$ we write $\xi^{a}\,\eta_{a}$ and for
the inner product between $\xi^{a^{\prime}}$ and $\eta_{a^{\prime}}$ we write
$\xi^{a^{\prime}}\,\eta_{a^{\prime}}$, with the usual summation convention.

We are interested in the product Hilbert space
$\mathcal{H}_{12}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, and we write
$\xi^{aa^{\prime}}\in\mathcal{H}_{12}$ for a typical element of this space.
Then we write $\eta_{aa^{\prime}}$ for a typical element of
$\mathcal{H}_{12}^{*}$ and $\bar{\xi}_{aa^{\prime}}$ for the complex conjugate
of $\xi^{aa^{\prime}}$, and for the inner product of $\xi^{aa^{\prime}}$ and
$\eta_{aa^{\prime}}$ we write $\xi^{aa^{\prime}}\,\eta_{aa^{\prime}}$. The
state of a two-particle system takes the form of a density matrix
$p^{aa^{\prime}}_{bb^{\prime}}$. Thus we require that it should be Hermitian,
of unit trace, and positive, so

$\displaystyle
p^{aa^{\prime}}_{bb^{\prime}}=\bar{p}^{aa^{\prime}}_{bb^{\prime}},\quad
p^{cc^{\prime}}_{cc^{\prime}}=1,\quad
p^{aa^{\prime}}_{bb^{\prime}}\alpha^{b}\bar{\alpha}_{a}\beta^{b^{\prime}}\bar{\beta}_{a^{\prime}}\geq
0$ (51)

for all $\alpha^{a},\beta^{a^{\prime}}$. A two-particle density matrix is
pure if
$p^{aa^{\prime}}_{bb^{\prime}}=\xi^{aa^{\prime}}\,\bar{\xi}_{bb^{\prime}}$ for
some state vector $\xi^{aa^{\prime}}$. We say that the particles are
independent if

$\displaystyle
p^{aa^{\prime}}_{bb^{\prime}}=p^{a}_{b}\,p^{a^{\prime}}_{b^{\prime}}$ (52)

for some pair of one-particle states $p^{a}_{b}$ and
$p^{a^{\prime}}_{b^{\prime}}$. The state is said to be separable if it can be
written in the form

$\displaystyle
p^{aa^{\prime}}_{bb^{\prime}}=\sum_{r=1}^{k}p^{a}_{b}(r)\,p^{a^{\prime}}_{b^{\prime}}(r),$
(53)

for some collection of $2k$ one-particle states
$\\{p^{a}_{b}(r)\\}_{r=1,2,\,.\,.\,.\,k}$ and
$\\{p^{a^{\prime}}_{b^{\prime}}(r)\\}_{r=1,2,\,.\,.\,.\,k}$. But if the two-
particle state is not separable then we say that the particles are entangled.

Now we are in a position to discuss the idea of measurements on a two-particle
system and the contracts one can associate with such measurements. A generic
contract based on the outcome of a measurement made on a two-particle system
is described by a Hermitian operator $X^{aa^{\prime}}_{bb^{\prime}}$. We are
interested in the case when the measurement splits into a measurement on
System I and a measurement on System II and one adds the results. Such a
contract takes the form

$\displaystyle
X^{aa^{\prime}}_{bb^{\prime}}=U^{a}_{b}\delta^{a^{\prime}}_{b^{\prime}}+\delta^{a}_{b}V^{a^{\prime}}_{b^{\prime}}\\!,$
(54)

where $\delta^{a}_{b}$ and $\delta^{a^{\prime}}_{b^{\prime}}$ denote the
identity operators on $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ respectively.
The eigenstates of such an operator are of the form

$\displaystyle
p^{aa^{\prime}}_{bb^{\prime}}=\alpha^{a}\bar{\alpha}_{b}\,\beta^{a^{\prime}}\bar{\beta}_{b^{\prime}},$
(55)

where $\alpha^{a}$ is an eigenvector of $U^{a}_{b}$ and $\beta^{a^{\prime}}$
is an eigenvector of $V^{a^{\prime}}_{b^{\prime}}$. Thus
$U^{a}_{b}\alpha^{b}=u\alpha^{a}$ and
$V^{a^{\prime}}_{b^{\prime}}\beta^{b^{\prime}}=v\beta^{a^{\prime}}$ for some
$u,v\in\mathbb{R}^{+}\backslash\\{0\\}$ and the sum $u+v$ gives the overall
outcome of the measurement. Such a contract can be thought of as representing
a portfolio consisting of one unit of a contract based on System I and one
unit of a contract based on System II. More generally, for a portfolio
consisting of $\theta_{1}$ units of the first contract and $\theta_{2}$ units
of the second contract we have

$\displaystyle
X^{aa^{\prime}}_{bb^{\prime}}(\theta_{1},\theta_{2})=\theta_{1}\,U^{a}_{b}\delta^{a^{\prime}}_{b^{\prime}}+\theta_{2}\,\delta^{a}_{b}V^{a^{\prime}}_{b^{\prime}}$
(56)

and the payout will be of the form $\theta_{1}u+\theta_{2}v$. The setup for a
portfolio of arbitrary size can be constructed analogously. In particular, one
can check that the expected payout of a portfolio is equal to the sum of the
expectations of the constituents. This is because whenever the density matrix
of the two-particle state hits one of the identity operators in the portfolio
operator, all but one of the systems gets traced out and one is left with the
trace of the product of a single particle density operator and the observable
associated with that system. For example, in the case of a two-particle system
one finds that

$\displaystyle
p_{aa^{\prime}}^{bb^{\prime}}X^{aa^{\prime}}_{bb^{\prime}}(\theta_{1},\theta_{2})$
$\displaystyle=$ $\displaystyle
p_{aa^{\prime}}^{bb^{\prime}}(\theta_{1}\,U^{a}_{b}\delta^{a^{\prime}}_{b^{\prime}}+\theta_{2}\,\delta^{a}_{b}V^{a^{\prime}}_{b^{\prime}})$
(57) $\displaystyle=$
$\displaystyle\theta_{1}\,p_{aa^{\prime}}^{bb^{\prime}}\,U^{a}_{b}\delta^{a^{\prime}}_{b^{\prime}}+\theta_{2}\,p_{aa^{\prime}}^{bb^{\prime}}\delta^{a}_{b}V^{a^{\prime}}_{b^{\prime}}$
$\displaystyle=$
$\displaystyle\theta_{1}\,p_{a}^{b}\,U^{a}_{b}+\theta_{2}\,p_{a^{\prime}}^{b^{\prime}}V^{a^{\prime}}_{b^{\prime}},$

where $p_{a}^{b}=p_{ac^{\prime}}^{bc^{\prime}}$ and
$p_{a^{\prime}}^{b^{\prime}}=p_{ca^{\prime}}^{cb^{\prime}}$. Likewise one can
check that the price of a portfolio is equal to the weighted sum of the prices
of its constituents. This may not be obvious, but the point is that the two-
particle system is itself a quantum system with a financial observable based
on it, of the form (54), so by Proposition 1 there exists a pricing operator
$q_{aa^{\prime}}^{bb^{\prime}}$ such that

$\displaystyle
q_{aa^{\prime}}^{bb^{\prime}}X^{aa^{\prime}}_{bb^{\prime}}(\theta_{1},\theta_{2})=\theta_{1}\,q_{a}^{b}\,U^{a}_{b}+\theta_{2}\,q_{a^{\prime}}^{b^{\prime}}V^{a^{\prime}}_{b^{\prime}},$
(58)

where the traced-out operators $q_{a}^{b}=q_{ac^{\prime}}^{bc^{\prime}}$ and
$q_{a^{\prime}}^{b^{\prime}}=q_{ca^{\prime}}^{cb^{\prime}}$ are the pricing
operators associated with the respective individual systems.

There is one further aspect of the portfolio problem that can be analyzed and
this concerns the matter of correlations. If the state of the two-particle
system is of the form (52), so the two particles are independent, then the
outcomes of the experiments on the two systems will be uncorrelated. But if
the systems are entangled, then the correlation will in general be non-
vanishing, leading to relations such as

$\displaystyle
p_{aa^{\prime}}^{bb^{\prime}}\big{[}U^{a}_{b}-\delta^{a}_{b}\,p_{c}^{d}\,U^{c}_{d}\big{]}\big{[}V^{a^{\prime}}_{b^{\prime}}-\delta^{a^{\prime}}_{b^{\prime}}\,p_{c^{\prime}}^{d^{\prime}}\,V^{c^{\prime}}_{d^{\prime}}\big{]}\neq
0.$ (59)

The point about entanglement is that even if the two systems are in separate
rooms (or even different cities) the outcomes may be correlated, owing to the
original construction of the state of the two-particle system to which they
belong. The same is true of the prices: if $q^{aa^{\prime}}_{bb^{\prime}}$ is
entangled, then there will be correlations in the prices, as shown in
relations such as

$\displaystyle
q_{aa^{\prime}}^{bb^{\prime}}\big{[}U^{a}_{b}-\delta^{a}_{b}\,q_{c}^{d}\,U^{c}_{d}\big{]}\big{[}V^{a^{\prime}}_{b^{\prime}}-\delta^{a^{\prime}}_{b^{\prime}}\,q_{c^{\prime}}^{d^{\prime}}\,V^{c^{\prime}}_{d^{\prime}}\big{]}\neq
0.$ (60)

Thus, in the general situation we see that when there is a market based on
contracts associated with measurements being made on a number of different
quantum systems, there will be correlations between outcomes of measurements
and correlations between prices, where the former are determined by the
structure of physical density operator for the market as a whole and the
latter by the structure of the pricing operator for the market as a whole.

The physical density operator is objective in nature, the only limitations in
its determination being in the usual practicalities of the laboratory settings
where the states are manufactured. The pricing operator, on the other hand, if
classical finance theory is any guide in the matter arrow debrue ; bingham
kiesel ; dothan ; duffie ; etheridge ; follmer schied , will be determined by
the collective appetite for risk and reward among market participants and also
perhaps by supply and demand related considerations. Hence, as in all markets,
prices will be subject to fluctuation and change over time and may even be
amenable to a Bayesian treatment. In the one-period setting that we have
investigated here, the pricing operator simply is what it is, and all we can
say of a definite nature about it is that it exists and that the pricing
operator and the physical density operator are equivalent, as we have seen in
Proposition 1.

In the one-period version of the theory we have presented, one can be agnostic
on the matter of dynamics. This is because $\hat{p}$, $\hat{q}$ and $P_{0T}$
are specified at time $0$ and no further data are required. By the time one
reaches $T$, however, the physical state will have evolved from the initial
state $\hat{p}$ to a new state $\hat{p}_{T}$ given by

$\displaystyle\hat{p}_{T}=\mbox{$\rm e$}^{{\rm
i}\hat{H}T}\,\hat{p}\,\mbox{$\rm e$}^{{-\rm i}\hat{H}T},$ (61)

where $\hat{H}$ denotes the Hamiltonian, assuming that there is no physical
interaction with the environment. This new state can then be used for working
out the probabilities of outcomes for the next period in a multi-period
setting. One might conjecture that $\hat{q}$ will undergo a similar unitary
transformation, in the absence of any Bayesian updating. This will ensure that
the physical state and the pricing state will continue to share the same null
space. But one can also arrive at this conclusion if one works in the
Heisenberg representation from the outset, in which case the sole function of
the unitary operators is to rotate the measurement frames for each observable,
leaving $\hat{p}$ and $\hat{q}$ fixed, in the absence of external
interventions or state changes arising as a consequence of measurements having
been made.

That the non-Kolmogorovian character of quantum probability may have
implications for the development of quantum technologies is widely
appreciated. See, e.g., holik  and references cited therein. And indeed, if
quantum computers eventually replace the classical computers currently used
for algorithmic trading by financial institutions, as they no doubt will, then
the role of valuations of the type we have considered here may be important.
There is also a widely held view that quantum probability may play a part in
cognitive science and hence behavioural finance as well. See, e.g., busemeyer
bruza 2012 ; khrennikov 2010 ; haven khrennikov 2016 ; pothosbusemeyer2022 ;
pothos busemeyer 2013 ; yukalov sornette 2017  and references cited therein.

It would be out of the scope of the present discussion to look at either of
these proposals in any detail here, but on the latter point one could well
imagine that if judgements and decisions are made on the basis of quantum
probability then in some situations these assessments would involve
valuations, rather than probability estimates, and it would be the pricing
operator, rather than the physical density operator, that would come into play
in making these valuations. In such cases, external intervention in the form
of Bayesian updating could be modelled, for example, as in reference brody
2023 . This is consistent with the point we made earlier about the pricing
operator being specific to the risk and reward profiles of market operatives
and in a state of flux as new information arrives. These and other further
developments of the theory we hope to explore elsewhere.

###### Acknowledgements.

The authors wish to thank D C Brody and B K Meister for helpful comments.

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# Hoare-Like Triples and
Kleene Algebras with Top and Tests

Towards a Holistic Perspective on Hoare Logic, Incorrectness Logic, and Beyond

Lena Verscht  Saarland University<EMAIL_ADDRESS>and  Benjamin
Kaminski  Saarland University and University College London<EMAIL_ADDRESS>saarland.de

##  1\. Introduction

“Program correctness and incorrectness are
two sides of the same coin.”

— O’Hearn (2019)

We argue that the object of discourse is not a coin and that it has at least
three sides or rather _dimensions_ :

  1. (1)

_correctness_ vs.  _incorrectness_

  2. (2)

_totality_ vs.  _partiality_

  3. (3)

_reachability_ vs.  _unreachability_

We will explore how one can pigeonhole _total_ and _partial correctness_ , as
well as _incorrectness_ into this view and explore further program properties
that emerge from exhausting all combinations of the above dimensions. We will
furthermore explore how to express validity of those properties in the
language of Kleene algebras with top and tests (Zhang et al., 2022).

##  2\. Canonical Exegeses of Triples

The central notion of Hoare logic (Hoare, 1969) are triples of the form
$\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}$, where $p$ is a program, $b$
is a _precondition_ on initial states, and $c$ is a _postcondition_ on final
states. To give meaning to a triple, it must be _exegeted_ when a triple is
considered _valid_ and when it is not. For Hoare logic, there are two such
exegeses: validity for _total_ and for _partial_ correctness.

### Exegesis I: Partial Correctness.

$\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}$ is valid for _partial
correctness_ iff all executions of $p$ that start in $b$ can only ever
terminate in $c$. The execution _is_ allowed to diverge. For example, validity
for partial correctness of

$\displaystyle\left\\{\text{wrong password}\right\\}\mathrel{\quad
p_{\text{login}}\quad}\left\\{\text{login failed}\right\\}$

specifies that when the user enters a wrong password, $p_{\text{login}}$ will
either diverge or terminate with a failed login. In particular, successful
login is impossible with the wrong password.

### Exegesis II: Total Correctness.

$\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}$ is valid for _total
correctness_ iff all executions of $p$ that start in $b$ definitely
_terminate_ and they do so in $c$. Reusing the example above, validity for
total correctness of

$\displaystyle\left\\{\text{correct password}\right\\}\mathrel{\quad
p_{\text{login}}\quad}\left\\{\text{login successful}\right\\}~{},$

specifies that when the user enters the correct password, the login will
definitely be successful.

It is noteworthy that both partial and total correctness are about
_coreachability of all initial states_ :
$\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}$ is valid if _all_ initial
states in $b$ are (partially) mapped by $p$ _into_ (but not necessarily
_onto_) the final states in $c$.

### Weakest (Liberal) Preconditions.

Proposed by Dijkstra (Dijkstra et al., 1976), _weakest (liberal)
preconditions_ are a very versatile alternative to Hoare logic and led to
numerous extensions (Morgan et al., 1996; Kaminski et al., 2018; Kaminski,
2019; Zhang and Kaminski, 2022). Given only a program $p$ and a postcondition
$c$, the _weakest liberal precondition_ is the weakest predicate
$\mathsf{wlp}\llbracket p\rrbracket\left(c\right)$, such that all executions
of $p$ that start in $\mathsf{wlp}\llbracket p\rrbracket\left(c\right)$ either
diverge or terminate in $c$. One can (re)define partial correctness in terms
of weakest liberal preconditions:
$\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}$ is valid for partial
correctness iff $b\implies\mathsf{wlp}\llbracket p\rrbracket\left(c\right)$.

For _total_ correctness, the weakest precondition is the weakest predicate
$\mathsf{wp}\llbracket p\rrbracket\left(c\right)$, such that all executions of
$p$ that start in $\mathsf{wp}\llbracket p\rrbracket\left(c\right)$ definitely
terminate and they do so in $c$. One can define total correctness in terms of
weakest preconditions: $\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}$ is
valid for total correctness iff $b\implies\mathsf{wp}\llbracket
p\rrbracket\left(c\right)$.

### Exegesis III: Incorrectness.

Incorrectness triples were introduced by De Vries and Koutavas (2011) and
later independently reintroduced by O’Hearn (2019).
$\left[b\right]\mathrel{p}\left[c\right]$ is valid for incorrectness iff all
executions of $p$ that terminate in $c$ could have started in $b$. In other
words: iff $p$ maps (a subset of) $b$ _onto_ $c$. In that sense, incorrectness
is really about _reachability of all final states_. If $c$ is a set of error
states, then validity of $\left[b\right]\mathrel{p}\left[c\right]$ for
incorrectness indeed proves that executing $p$ on $b$ is not safe, because
doing so can reach an error in $c$. For example, validity for incorrectness of

$\displaystyle\left[\text{correct password}\right]\mathrel{\quad
p_{\text{login}}\quad}\left[\text{program crash}\right]$

specifies that $p_{\text{login}}$ can crash on entering a correct password.

### Strongest Postconditions.

$b\implies\mathsf{wp}\llbracket p\rrbracket\left(c\right)$total
correctnessg$\mathsf{wlp}\llbracket p\rrbracket\left(c\right)\implies
b$???g$b\implies\mathsf{wlp}\llbracket
p\rrbracket\left(c\right)$$\mathsf{wp}\llbracket
p\rrbracket\left(c\right)\implies b$$\mathsf{sp}\llbracket
p\rrbracket\left(b\right)\implies c$$\mathsf{slp}\llbracket
p\rrbracket\left(b\right)\implies c$¿¿¿gpartial
correctnessg$c\implies\mathsf{slp}\llbracket
p\rrbracket\left(b\right)$$c\implies\mathsf{sp}\llbracket
p\rrbracket\left(b\right)$incorrectnessgpartial incorrectnessg Figure 1.  An
overview of the exegeses of Hoare triples and how they are related. The dotted
green arrows in the middle symbolize contrapositives and the orange arrows
symbolize equivalences through Galois connections.

Given only a program $p$ and a _pre_ condition $b$, the _strongest
postcondition_ (Dijkstra and Scholten, 1990) is the strongest predicate
$\mathsf{sp}\llbracket p\rrbracket\left(b\right)$, such that any state in
$\mathsf{sp}\llbracket p\rrbracket\left(b\right)$ is reachable from $b$ by
executing $p$. We can redefine incorrectness in terms of strongest
postconditions (Zhang and Kaminski, 2022):
$\left[b\right]\mathrel{p}\left[c\right]$ is valid for incorrectness iff
$c\implies\mathsf{sp}\llbracket p\rrbracket\left(b\right)$.

$\mathsf{sp}$ has a Galois connection to $\mathsf{wlp}$, namely

($\dagger$) $\displaystyle b\implies\mathsf{wlp}\llbracket
p\rrbracket\left(c\right)\qquad{\textnormal{iff}}\qquad\mathsf{sp}\llbracket
p\rrbracket\left(b\right)\implies c~{}.$

Because of this, we can also define partial correctness in terms of strongest
postconditions: $\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}$ is valid for
partial correctness iff $\mathsf{sp}\llbracket
p\rrbracket\left(b\right)\implies c$. Notice that the difference between
incorrectness and partial correctness is merely the direction of the
implication between $c$ and $\mathsf{sp}\llbracket p\rrbracket\left(b\right)$.

##  3\. Completive Exegeses of Triples

Given a triple of precondition $b$, program $p$, and postcondition $c$, we
have so far seen three exegesis of such triples, all given in terms of
$\mathsf{wlp}$/$\mathsf{wp}$/$\mathsf{sp}$ and implications:

total correctness | $b\implies\mathsf{wp}\llbracket p\rrbracket\left(c\right)$
---|---
partial correctness | $b\implies\mathsf{wlp}\llbracket p\rrbracket\left(c\right)$
$\mathsf{sp}\llbracket p\rrbracket\left(b\right)\implies c$
incorrectness | $c\implies\mathsf{sp}\llbracket p\rrbracket\left(b\right)$

From inspecting this table, we can make out three degrees of freedom one has
in exegeting triples:

  1. (1)

The difference between total correctness and partial correctness is the usage
of $\mathsf{wp}$ vs. $\mathsf{wlp}$.

  2. (2)

The difference between partial correctness and incorrectness is the direction
of the implication.

  3. (3)

The difference between total correctness and incorrectness is $\mathsf{wp}$
vs. $\mathsf{sp}$.

This immediately gives rise to two questions: What about
$\mathsf{wp}\llbracket p\rrbracket\left(c\right){\implies}b$ and
$\mathsf{wlp}\llbracket p\rrbracket\left(c\right){\implies}b$? And can we
define strongest liberal postconditions?

### Strongest Liberal Postconditions.

Indeed, these can be sensibly defined (Zhang and Kaminski, 2022). The
difference between $\mathsf{wlp}$ and $\mathsf{wp}$ is that the liberal
variant additionally contains all states that diverge, i.e. those _initial
states for which there exists no final state_ in which the execution of $p$
could terminate. Dually, $\mathsf{slp}\llbracket p\rrbracket\left(b\right)$
contains also all states that _cannot be reached at all_ by executing $p$,
i.e. those _final states for which there exists no initial state_ in which the
execution of $p$ could have started.

$\mathsf{slp}$ also has a Galois connection (Zhang and Kaminski, 2022), but to
$\mathsf{wp}$, namely

($\ddagger$) $\displaystyle\mathsf{wp}\llbracket
p\rrbracket\left(c\right)\implies b\qquad{\textnormal{iff}}\qquad
c\implies\mathsf{slp}\llbracket p\rrbracket\left(b\right)~{}.$

Exhaustively combining $\mathsf{wp}$, $\mathsf{wlp}$, $\mathsf{sp}$, and
$\mathsf{slp}$ with all implication directions gives a completed canon of in
total 8 different exegeses of triples, four of which reduce to only two via
Galois connections, see Figure 1.

### Exegesis IV: Partial Incorrectness.

The difference between total and partial correctness is the use of
$\mathsf{wp}$ vs. $\mathsf{wlp}$. Consequently, when we replace $\mathsf{sp}$
by $\mathsf{slp}$ in the defining implication $c\implies\mathsf{sp}\llbracket
p\rrbracket\left(b\right)$ of incorrectness, we should arrive at the exegesis
of _partial incorrectness_ , as suggested (but not further explored) by Zhang
and Kaminski (2022): $[b]p[c]$ is valid for _partial incorrectness_ iff
$c\implies\mathsf{slp}\llbracket p\rrbracket\left(b\right)$ (and by
($\ddagger$) equivalently iff $\mathsf{wp}\llbracket
p\rrbracket\left(c\right)\implies b$), which means that all states in $c$ are
either unreachable (from anywhere) or reachable from $b$ by executing $p$. For
example, validity for partial incorrectness of

$\displaystyle\left[\text{wrong password}\right]\mathrel{\quad
p_{\text{login}}\quad}\left[\text{login successful}\right]$

specifies that any state where the login was successful is either entirely
unreachable, or reachable by entering a wrong password, thus constituting a
potential bug.

### Exegesis V and VI

There are two remaining triple exegeses, defined by $\mathsf{wlp}\llbracket
p\rrbracket\left(c\right)\implies b$ and $\mathsf{slp}\llbracket
p\rrbracket\left(b\right)\implies c$, respectively. These have also been
suggested by Zhang and Kaminski (2022), but not further explored.

### Contrapositives

Besides some actual _equivalences_ via Galois connections, some triples are
contrapositives of each other (Zhang and Kaminski, 2022). In fact, the entire
left-hand-side (in some sense the _correctness side_) is the contrapositive of
the right-hand-side (the _incorrectness side_) in Figure 1. For example,

$\displaystyle b\implies\mathsf{wlp}\llbracket
p\rrbracket\left(c\right)\quad\text{iff}\quad\mathsf{wp}\llbracket
p\rrbracket\left(\neg c\right)\implies\neg b$

and hence partial correctness is the contrapositive of partial incorrectness
and so both describe in some sense the same class of specifications. We would
claim, however, that it might be more intuitive for a working programmer to
specify either a partial incorrectness triple or a partial correctness triple,
depending on the property they have in mind.

##  4\. Kleene Algebra with Top and Tests

_Kleene algebra with tests_ (KAT), introduced by Kozen (1997), is another
alternative for specifying and reasoning about program properties. KAT terms
are generalized regular expressions over a _two-sorted alphabet_ comprising
(i) programs ($p,q,\ldots$) and (ii) tests ($b,c,\ldots$). We interpret these
symbols as relations: a program $p$ relates (maps) initial states to final
states by its execution. Initial states on which $p$ diverges are not related
with any final states. Similarly, unreachable final states are not related
with any initial states. A test $b$ maps initial states satisfying $b$ to
themselves, and those not satisfying $b$ to no state. In this sense, tests act
as filters.

Testing for false is denoted by $0$ and gives the _empty_ relation. In the
lattice of relations (ordered by set inclusion), $0$ is the least/bottom
element. In _Kleene algebra with top and tests_ (TopKAT), we now additionally
add a $\top$ element to KAT, which codes for the _universal_ relation relating
every initial state to every final state (Zhang et al., 2022).

Composing symbols means composing relations: For example, the term $bpqc$
intuitively means: first test for $b$, then execute $p$ (but only on states
that satisfy $b$), then execute $q$, then test for $c$. Executions that fail
to satisfy initially $b$ or finally $c$ are filtered out and not part of the
resulting relation.

### Partial Correctness in TopKAT

We can express partial correctness by the TopKAT equation $\top bpc=\top bp$.
Prepending the top element on both relations in the equation amounts to
comparing their _codomains_ , i.e. their _final states_. Hence, $\top bpc=\top
bp$ expresses that the set of final states in $c$ in which $p$ can terminate
starting from $b$ is exactly those final states in which $p$ can terminate
from $b$ at all. No statement is made about initial states in $b$ on which $p$
diverges.

### Incorrectness in TopKAT

It is not possible to express incorrectness in KAT, but this _is_ possible in
TopKAT (Zhang et al., 2022), namely by $\top bpc=\top c$. On the left-hand-
side of the equation, we select all final states in $c$ that were reachable by
executing $p$ on $b$. On the right-hand-side, we select _all_ final states in
$c$.

### Nondeterminism and Correctness in KAT

So far, we have kept quiet about nondeterminism, but in Kleene algebras it is
very natural to model nondeterministic programs. Considering total correctness
of $\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}$, one then has to decide:
(i) must _all_ execution paths emerging from any state in $b$ terminate in
$c$, or (ii) must there merely exist _some_ path that does so? The former is
called the _demonic_ exegesis of nondeterminism while the latter is called
_angelic_. On that note, Zhang and Kaminski (2022) always interpret
nondeterminism _angelically_ for their _non_ liberal pre- and postconditions,
and _demonically_ for their liberal ones. Only this way, they obtain the
Galois connections ($\dagger$) and ($\ddagger$). However, the standard notion
of total correctness is _demonic_ and not _angelic_.

Unfortunately, it is known that _demonic_ total correctness is inexpressible
in KAT (von Wright, 2002) because it lacks a way of reasoning about
nontermination: Nonterminating executions are just not part of the resulting
relation. For the angelic variant, however, $bpc\top=b\top$ indeed expresses
that from all intial states in $b$, it is _possible_ for $p$ to terminate in
$c$, thus:
$\displaystyle\begin{array}[]{c}\left\\{b\right\\}\mathrel{p}\left\\{c\right\\}\textnormal{
is valid for angelic total correctness}\\\\[5.0pt] \textnormal{iff}\\\\[5.0pt]
bpc\top=b\top\end{array}$ This is, to the best of our knowledge, a novel
result. Moreover, this result renders _all_ triples depicted in Figure 1
expressible in TopKAT in a syntactically very similar manner:

total correctness | $bpc\top=b\top$
---|---
partial correctness | $\top bpc=\top bp$
incorrectness | $\top bpc=\top c$

The opposite-hand-sides are expressible via contrapositives. For example,
validity of $\left[b\right]\mathrel{p}\left[c\right]$ for partial
incorrectness is expressible via $bpc\top=pc\top$. And more equations fit into
this pattern, for example $\top bpc=\top pc$ and $bpc\top=bp\top$.
Interestingly, these two express the angelic variants of partial correctness
and incorrectness, respectively, and thus lead to further exegeses of triples.

##  5\. Ongoing and Future Work

There are many open questions left to investigate. One direction is to
investigate deeper the different exegesis of nondeterminism, which would lead
to a total combination of 16 possible exegeses, of which we conjecture that
still only four of them reduce to two via Galois connections, thus ending up
with 14 possible exegeses. We would then like to investigate the impact of
different assumptions about the program on this picture: For example, what
happens if we assume that executions started in $b$ always terminate? What
happens if all states in $c$ are reachable? What if the program is
deterministic or reversible?

As for a different direction, none of the aforementioned triples expresses the
existence of (at least) one path from $b$ to $c$. However, being able to more
directly specify a bug like “it is possible to enter a wrong password and yet
obtain successful login” would be quite desirable. The following equations
would express a property of this form:

$\displaystyle bpc\neq 0\quad\text{and equivalently}\quad\mathsf{wp}\llbracket
p\rrbracket\left(c\right)\land b\neq\text{false}$

In contrast to the existing triples, this property speaks neither about _all_
initial states in $b$ nor _all_ final states in $c$.

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