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Quantum information theory has given rise to a renewed interest in, and a new perspective on, the old issue of understanding the ways in which quantum mechanics differs from classical mechanics. The task of distinguishing between quantum and

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From Physics to Information Theory and Back
Wayne C. MyrvoldDepartment of PhilosophyUniversity of Western OntarioTo appear in Alisa Bokulich and Gregg Jaeger, eds.,
Foundations of Quantum Information and Entanglement
Cambridge University Press
Abstract
Quantum information theory has given rise to a renewed interest in, and a newperspective on, the old issue of understanding the ways in which quantum mechan-ics diﬀers from classical mechanics. The task of distinguishing between quantum andclassical theory is facilitated by neutral frameworks that embrace both classical andquantum theory. In this paper, I discuss two approaches to this endeavour, the alge-braic approach, and the convex set approach, with an eye to the strengths of each,and the relations between the two. I end with a discussion of one particular model,the toy theory devised by Rob Spekkens, which, with minor modiﬁcations, ﬁts neatlywithin the convex sets framework, and which displays in an elegant manner some of the similarities and diﬀerences between classical and quantum theories. The conclusionsuggested by this investigation is that Schr¨odinger was right to ﬁnd the essential dif-ference between classical and quantum theory in their handling of composite systems,though Schr¨odinger’s contention that it is entanglement that is the distinctive featureof quantum mechanics needs to be modiﬁed.
1 Introduction
Quantum information theory is the study of how the peculiar features of quantum mechanicscan be exploited for the purposes of information processing and transmission. A centraltheme of such a study is the ways in which quantum mechanics opens up possibilities thatgo beyond what can be achieved classically. This has in turn led to a renewed interest in,and a new perspective on, the diﬀerences between the classical and the quantum. Althoughmuch of the work along these lines has been motivated by quantum information theory—and some of it has been motivated by the conviction that quantum theory is essentially1
about possibilities of information processing and transmission—the results obtained, andthe frameworks developed, have interest even for those of us who are not of that conviction.Indeed, much of the recent work echoes, and builds upon, work that predates the inceptionof quantum information theory. The signiﬁcance of such work extends beyond the setting of quantum information theory; the work done on distinguishing the quantum from the classicalin the context of frameworks that embrace both is something worthy of the attention of anyone interested in the foundational issues surrounding quantum theory.One of the striking features of quantum mechanics lies in its probabilistic character.A quantum state yields, not a deﬁnite prediction of the outcome of an experiment, buta probability measures on the space of possible outcomes. Of course, probabilities occuralso in a classical context. In this context they have to do with situations in which theexperimenter does not have complete control over the classical state to be prepared, and,as a consequence, we do not have complete knowledge of the classical state of the systemsubjected to the preparation procedure. The question arises, therefore, whether quantumprobabilities can be construed as being like this. One way of framing this question is interms of hidden variables: ought we to think of quantum-mechanical pure states as beingprobabilistic mixtures of states of a more encompassing theory, whose pure states wouldascribe deﬁnite values to all variables? It is interesting to ask which of the peculiar featuresof quantum mechanics are traceable to ineliminable statistical dispersion in its states. Suchfeatures will be reproducible in an essentially classical theory with suitable restrictions onpossibilities of state preparation.Some of the recent work in quantum information theory has shown that some of thefeatures of quantum mechanics that one might be inclined to think of as peculiarly quantumcan, indeed, be recovered from a theory in which the preparable states are probabilisticmixtures of such classical states. The no-cloning theorem is a case in point. Arbitrarypairs of quantum states cannot be cloned; those pairs that can be cloned are orthogonalpairs. Though srcinally formulated in the context of quantum mechanics, it admits of a formulation applicable to classical mixed states, which are probability measures over aclassical phase space. A pair of classical states is orthogonal iﬀ the probability measureshave disjoint support, and it can be shown that cloneable pairs are orthogonal in this sense.The similarity between these two theorems suggests that they are special cases of a moregeneral theorem, and indeed, this is the case. Implicit in the proof of Lemma 3 of Clifton,Bub, and Halvorson (2003) is a proof that a pair of pure states of a
C
∗
-algebra are cloneableiﬀ they are orthogonal. Barnum, Barrett, Leifer, and Wilce (2006) prove a theorem of muchgreater generality. Within the convex sets framework (see section 4, below), they prove thata ﬁnite set of states on a compact, ﬁnite-dimensional state space is cloneable if and only if it is a jointly distinguishable set of states.Some, notably Fuchs (2002) Spekkens (2001, 2007), have found in this fact—thatsome phenomena that might be thought to be distinctively quantum can be reproduced in aclassical theory by imposing restrictions on state preparation—encouragement for the viewthat quantum probabilities are just like classical probabilities, epistemic probabilities bound2
up with limitations on state preparation. A natural alternative is to conclude that thosefeatures that can be reproduced in an essentially classical setting ought not to have beenconsidered distinctively quantum in the ﬁrst place. This, is, I think, the right lesson todraw. If this is right, then we must seek deep distinctions between the classical and thequantum elsewhere. To anticipate a conclusion to be drawn below, a case can be made thatSchr¨odinger (1936) was right to locate the essential diﬀerence between the classical and thequantum in its treatment of combined systems. However, Schr¨odinger’s conclusion that itis entanglement that distinguishes the quantum from the classical requires qualiﬁcation—aswe shall see.In this paper, I will compare and contrast two approaches to the construction of neutral frameworks in which theories can be compared. The ﬁrst is the algebraic framework,which begins with an algebra, among the elements of which are included the observables of the theory. The second is the operational approach, which motivates the introduction of theconvex sets framework. I will end with a discussion of one particular model, the toy theorydevised by Rob Spekkens (2001, 2007), which, with minor modiﬁcations, ﬁts neatly withinthe convex sets framework and which displays, in an elegant manner some of the similaritiesand diﬀerences between classical and quantum theories.
2 Algebraic frameworks
Clifton, Bub, and Halvorson (2003) (henceforth CBH) undertook the task of characterizingquantum mechanics in terms of information-theoretical constraints. They adopted a frame-work in which a physical theory is associated with a
C
∗
-algebra, the self-adjoint elements of which represent the bounded observables of the theory. For the deﬁnition of a
C
∗
-algebra,see the appendix; for our purposes it suﬃces to know that the set of all bounded operatorson a Hilbert space is a
C
∗
-algebra, as is any subalgebra of these that is closed under theoperation of taking adjoints, and is complete in the operator norm. Moreover, the set of allbounded, continuous complex-valued functions on a classical phase space is a
C
∗
-algebra, asis the set of all bounded, measurable complex-valued functions on a classical phase space.Thus, classical mechanics also admits of a
C
∗
-algebra representation, as was shown by Koop-man (1931) . The diﬀerence between the quantum case and the classical case is that, in theclassical case, the algebra is abelian.A
state
on a
C
∗
-algebra
A
is a positive linear functional
ρ
:
A
→
C
, normalized sothat
ω
(
I
) = 1. For self-adjoint
A
, the number
ω
(
A
) is to be interpreted as the expectationvalue of the observable corresponding to
A
, in state
ω
. The set of states is a convex set: forany states
ρ,σ
, and any real
λ
∈
(0
,
1), the functional deﬁned by
ω
(
A
) =
λ ρ
(
A
) + (1
−
λ
)
σ
(
A
)is also a state, a mixture of
ρ
and
σ
. A state that is not a mixture of any two distinct statesis called
pure
. General state evolution is represented by completely positive norm-preservinglinear maps, also known as non-selective operations.3
A state is
dispersion-free
iﬀ
ρ
(
A
2
) =
ρ
(
A
)
2
for all self-adjoint
A
. Any dispersion-free state is pure. It can be shown that a
C
∗
-algebra is abelian iﬀ all its pure states aredispersion-free. It can be also shown that a theory involving an abelian
C
∗
-algebra admitsof an essentially classical representation, in which the states are probability distributions onthe set of its pure states (see Kadison and Ringrose 1983, Thm. 4.4.3). Thus within the
C
∗
-algebraic framework, the classical theories are those whose algebras are abelian.Within this framework, CBH characterize quantum theory via three properties of thealgebra:i). Algebras associated with distinct physical systems commute.ii). Any individual system’s algebra of observables is noncommutative.iii). Spacelike separated systems at least sometimes occupy entangled states.As CBH (2003, 1570) point out, the ﬁrst two conditions entail that the state spaceof a composite systems contains nonlocally entangled states. What the third requirement ismeant to do is to guarantee that these states are physically accessible. Thus, CBH allow fortheories in which the set of preparable states is a proper subset of the full state space of thealgebra of observables. If the set of preparable states is taken to be the full set of states of a
C
∗
-algebra, then the third condition is redundant.Though CBH describe their conditions as “deﬁnitive of what it means to be a quantumtheory in the most general sense” (1563), it should be noted that these three properties donot suﬃce to characterize quantum mechanics. In a quantum theory, there are no states thatare dispersion-free in all observables. This is not entailed by CBH’s conditions, as can beshown by the following simple example. Let
A
,
B
be two separated systems, and associatewith each of these the algebra
M
(
C
)
⊕
M
(
C
2
)— that is, the algebra of 3
×
3 complex matricesof the form:
α
0 00
β γ
0
δ
Associate with the composite system the algebra that is the tensor product of the algebrasassociated with
A
and
B
.All three of CBH’s conditions are satisﬁed. However, unlike either quantum mechanicsor classical mechanics, some of the pure states are dispersion-free, and some are not. Thestate corresponding to the vector,
100
is an eigenvector of every observable. To ensure that the state space of our theory is quantummechanical, some additional condition is needed. Plausible candidates are symmetry condi-tions; one might impose, for example, the condition that, for any pair of pure states
ρ
,
σ
on4
A
(or
B
), there is an automorphism of the algebra
A
taking
ρ
into
σ
. This condition wouldentail that either all pure states are dispersion-free (in which case the algebra is abelian, andthe state space is a classical simplex), or that none are (as in the quantum case).The virtues of the
C
∗
-algebraic formulation are that there is a rich and well-workedout theory of
C
∗
-algebras, and that both classical and quantum theories—including quantummechanics and quantum ﬁeld theories—are readily formulated in such terms. This rich theorycomes at a price, however. The assumption that the set of observables be the self-adjoint partof a
C
∗
-algebra requires that sums and products of observables be well-deﬁned even whenthe observables are incompatible ones. When self-adjoint operators
A
,
B
fail to commute,the product
AB
will not correspond to any observable; nevertheless, our deﬁnition of staterequires that states assign numbers to such products, numbers that are not interpretable asexpectation values of the results of measurement. The embedding of our observables intoan algebra imposes non-trivial algebraic relations between expectation values assigned toobservables.It would be diﬃcult to argue—or at least, at this point nobody knows how to argue—that any plausible physical theory would admit of a
C
∗
-algebraic formulation. If we furtherrequire that the set of preparable states be the full state space of some
C
∗
-algebra, thenthe
C
∗
-algebraic framework becomes decidedly too restrictive. Halvorson (2004) discussesthe case of a theory that he calls the Schr*dinger theory, in which elementary systems arelike quantum systems, but in which entangled states decay into mixtures when the systemsare separated. Such a theory is locally quantum, but admits of no Bell-inequality violatingcorrelations. Such a theory was suggested by Schr¨odinger (1936), who pointed out that atthe time there was little in the way of experimental evidence of nonlocally entangled states.Though there is now abundant evidence of nonlocal entanglement, it does not seem that theSchr*dinger theory is one that could, or should, have been ruled out in advance of experiment.Moreover, we would like a framework in which we can consider some admittedly artiﬁcialconstructions, such as Spekkens’ toy theory, discussed in section 5, below. As Halvorson(2004) has shown, the state space of the Spekkens theory is not the state space of a
C
∗
-algebra.Within the algebraic approach, one can also consider weakening of the algebraicassumptions. One can, for example, consider Jordan-Banach (JB) algebras, or Segal algebras(see Halvorson (2004) for deﬁnition and discussion), both of which contain
C
∗
-algebras asspecial cases. Seeking a further widening of the algebraic framework, with constraints limitedto those that, arguably, any reasonable physical theory must share, one is led naturally toeﬀect algebras.
1
An eﬀect algebra is meant to represent the set of yes-no tests that can be performed ona physical system. It contains distinguished elements 0 and
u
(the unit element), representing
1
The notion of an eﬀect algebra has occurred in the writings of many authors working on the foundationsof quantum mechanics. The presentation here is based on that of Beltrametti and Bugajski (1997). It shouldbe pointed out that, despite the name, an eﬀect algebra is not an algebra: mutiplication is not deﬁned, andaddition is a partial operation.
5

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