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    a  r   X   i  v  :   1   1   0   7 .   5   8   4   9  v   3   [  q  u  a  n   t  -  p   h   ]   1   6   O  c   t   2   0   1   2 Formulating Quantum Theory as a Causally Neutral Theory of Bayesian Inference M. S. Leifer Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom  ∗ Robert W. Spekkens Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada, N2L 2Y5  † (Dated: June 17, 2012)Quantum theory can be viewed as a generalization of classical probability theory, but the anal-ogy as it has been developed so far is not complete. Specifically, the manner in which inferencesare made in classical probability theory is independent of the causal relation that holds betweenthe conditioned variable and the conditioning variable, whereas in the conventional quantum for-malism, there is a significant difference between how one treats experiments involving two systemsat a single time and those involving a single system at two times. In this paper, we develop theformalism of   quantum conditional states  , which provides a unified description of these two sorts of experiment. The analogies between quantum theory and classical probability theory are expressedsuccinctly within the formalism and concepts that are distinct in the conventional formalism —such as ensemble preparation procedures, measurements, and quantum dynamics — are shown toall be special cases of belief propagation. We introduce a quantum generalization of Bayes’ theoremand the associated notion of Bayesian conditioning. Conditioning a quantum state on a classicalvariable is the correct rule for updating quantum states in light of classical data, regardless of thecausal relationship between the classical variable and the quantum degrees of freedom, but it doesnot include the projection postulate as a special case. We show that previous arguments that pro- jection is the quantum generalization of conditioning are based on misleading analogies. Since ourformalism is causally neutral, conditioning provides a unification of the predictive and retrodictiveformalisms for prepare-and-measure experiments and leads to an elegant derivation of the set of states that a system can be “steered” to by making measurements on a remote system. PACS numbers: 03.65.Ca, 03.65.Ta, 03.67.-aKeywords: quantum conditional probability, quantum dynamics, quantum measurement, retrodiction, steer-ing I. INTRODUCTION Quantum theory can be understood as a non-commutative generalization of classical probability the-ory wherein probability measures are replaced by densityoperators. Much of quantum information theory, espe-cially quantum Shannon theory, can be viewed as thesystematic application of this generalization to classicalinformation theory.However, despite the power of this point of view, theconventional formalism for quantum theory is a poor ana-log to classical probability theory because, in quantumtheory, the appropriate mathematical description of anexperiment depends on its causal structure. For exam-ple, experiments involving a pair of systems at space-likeseparation are described differently from those that in-volve a single system at two different times. The formerare described by a joint state on the tensor product of two Hilbert spaces, and the latter by an input state and adynamical map on a single Hilbert space. Classical prob- ∗ Electronic address: matt@mattleifer.info;URL:  http://mattleifer.info † Electronic address: rspekkens@perimeterinstitute.ca;URL:  http://www.rob.rwspekkens.com ability works at a more abstract level than this. It spec-ifies how to represent uncertainty prior to, and indepen-dently of, causal structure. For example, our uncertaintyabout two random variables is always described by a jointprobability distribution, regardless of whether the vari-ables represent two space-like separated systems or theinput and output of a classical channel. Although chan-nels represent time evolution, they are described math-ematically by conditional probability distributions. Theinput state specifies a marginal distribution, and thus wehave the ingredients to define a joint probability distri-bution over the input and output variables. This jointprobability distribution could equally well be used to de-scribe two space-like separated variables. Therefore, wedo not need to know how the variables are embedded inspace-time in advance in order to apply classical prob-ability theory. This has the advantage that it cleanlyseparates the concept of correlation from that of causa-tion. The former is the proper subject of probabilisticinference and statistics. Within the subjective Bayesianapproach to probability, independence of inference andcausality has been emphasized by de Finetti ([1], Prefacepp. x–xi):Probabilistic reasoning—always to be un-derstood as subjective—merely stems fromour being uncertain about something. It  2makes no difference whether the uncertaintyrelates to an unforeseeable future, or to anunnoticed past, or to a past doubtfully re-ported or forgotten; it may even relate tosomething more or less knowable (by meansof a computation, a logical deduction, etc.)but for which we are not willing to make theeffort; and so on.Thus, in order to build a quantum theory of Bayesianinference, we need a formalism that is even-handed in itstreatment of different causal scenarios. There are someclues that this might be possible. Several authors havenoted that that there are close connections, and often iso-morphisms, between the statistics that can be obtainedfrom quantum experiments with distinct causal arrange-ments [2–8]. Time reversal symmetry is an example of  this, but it is also possible to relate experiments involvingtwo systems at the same time with those involving a sin-gle system at two times. The equivalence [9] of prepare-and-measure [10] and entanglement-based [11] quantum key distribution protocols is an example of this, and pro-vides the basis for proofs of the security of the former[12]. Such equivalences suggest that it may be possibleto obtain a causally neutral formalism for quantum the-ory by describing such isomorphic experiments by similarmathematical objects.One of the main goals of this work is to provide thisunification for the case of experiments involving two dis-tinct quantum systems at one time and those involvinga single quantum system at two times, and to providea framework for making probabilistic inferences that isindependent of this causal structure. Both types of ex-periment can be described by operators on a tensor prod-uct of Hilbert spaces, differing from one another only bya partial transpose. Probabilistic inference is achievedusing a quantum generalization of Bayesian condition-ing applied to  quantum conditional states  , which are themain objects of study of this work.Quantum conditional states are a generalization of classical conditional probability distributions. Condi-tional probability plays a key role in classical probabil-ity theory, not least due to its role in Bayesian infer-ence, and there have been attempts to generalize it tothe quantum case. The most relevant to quantum infor-mation are perhaps the quantum conditional expectation[13] (see [14, 15] for a basic introduction and [16] for a review) and the Cerf-Adami conditional density operator[17–19]. To date, these have not seen widespread appli- cation in quantum information, which casts some doubton whether they are really the most useful generalizationof conditional probability from the point of view of prac-tical applications. Quantum conditional states, whichhave previously appeared in [4, 20, 21], provide an al- ternative approach to this problem. We show that theyare useful for drawing out the analogies between classi-cal probability and quantum theory, they can be used todescribe both space-like and time-like correlations, andthey unify concepts that look distinct in the conventionalformalism. For example, the descriptions of the prepara-tion of an ensemble of quantum states, the probabilitiesfor the outcomes of measurements, and quantum dynam-ics can all be written as a generalization of the classical belief propagation   rule (also called the law of total prob-ability) [74], which is given by P  ( S  ) =  R P  ( S  | R ) P  ( R ) .  (1)The three cases differ only in the choice of which vari-ables;  R ,  S   or neither; remain classical in the generaliza-tion. We also show that the ensemble of states generatedby the most general update rule for a quantum systemafter a measurement — a quantum instrument — can bedescribed by belief propagation with respect to a quan-tum conditional state.We introduce a quantum version of Bayes’ theorem,which generalizes a rule previously advocated by Fuchs[22]. Several well-known constructions in quantum infor-mation are instances of this theorem, including the cor-respondence between Positive Operator Valued Measures(POVMs) and ensemble decompositions of a density op-erator [3, 4, 23], the “pretty good” measurement [24–26], and the Barnum-Knill recovery map for approximate er-ror correction [27].Finally, we discuss conditioning a quantum state ona classical variable. This is the correct way to updatequantum states in light of classical data, regardless of the causal relationship between the classical variable andthe quantum degrees of freedom. The causal neutral-ity of our formalism unifies the treatment of predictionswith that of retrodictions (inferences about the past), inanalogy with the unification found in classical Bayesianinference. The retrodictive formalism we devise coincideswith the one introduced in [28–30] in the case of unbi- ased sources, but differs in the general case, retaininga closer analogy with classical Bayesian inference. Theformalism also describes the case of conditioning on theoutcome of a remote measurement, such as in the EPRexperiment or more generally in “quantum steering”. Al-though our notion of conditioning does not include theprojection postulate as a special case, we argue that it isnevertheless the correct way to update states in the lightof measurement results, and that the assertion that theprojection postulate is analogous to Bayesian condition-ing [31, 32] is based on a misleading analogy. The latter is best described as the application of a belief propaga-tion rule (a non-selective update map), followed by con-ditioning (the selection). The remote measurement caseprovides an elegant derivation of the formula for the setof ensembles to which a remote system may be steered,previously obtained by conventional methods in [33]. A. Causal Neutrality Unifying the quantum description of experiments in-volving two distinct systems at one time with the de-  3scription of those involving a single system at two dis-tinct times requires some modifications to the way thatthe Hilbert space formalism of quantum theory is usuallyset up. Conventionally, a Hilbert space  H A  describes asystem, labelled A , that persists through time. Given twosuch systems,  A  and  B , the joint system is described bythe tensor product  H AB  =  H A  ⊗ H B . In the presentwork, a Hilbert space and its associated label shouldrather be thought of as representing a localized region of space-time. Specifically, an  elementary region   is a smallspace-time region in which an agent might possibly makea single intervention in the course of an experiment, forexample by making a measurement or by preparing aspecific state. Each elementary region is associated witha label,  A , and a Hilbert space  H A .Generally, a  region   will refer to a collection of elemen-tary regions. A region that is composed of a pair of disjoint regions, labelled  A  and  B , is ascribed the tensorproduct Hilbert space  H AB  =  H A  ⊗H B . In contrast tothe usual formalism, this applies regardless of whether  A and  B  describe independent systems or the same systemat two different times. Because of this, if an experimentinvolves a system that does persist through time, thena different label is given to each region it inhabits, e.g.,the input and output spaces for a quantum channel areassigned different labels.As discussed in the introduction, we will make use of a conditional quantum state to achieve a unified descrip-tion of the spatial and temporal scenarios. In fact, al-though we motivate our work by the distinction betweenspatial and temporal separation, we find that it is  not   thespatio-temporal relation between the regions that is rel-evant for how they ought to be represented in our quan-tum generalization of probability theory. Rather, it isthe  causal   relation that holds between them which is im-portant.More precisely, what is important is the distinctionbetween two regions that are  causally-related  , which is tosay that one has a causal influence on the other (perhapsvia intermediaries), and two regions that are  acausally-related  , which is to say that neither has a causal influenceon the other (although they may have a common causeor a common effect, or be connected via intermediariesto a common cause or a common effect).The causal relation between a pair of regions cannotbe inferred simply from their spatio-temporal relation.Consider a relativistic quantum theory for instance. Al-though a pair of regions that are space-like separatedare always acausally-related, a pair of regions that aretime-like separated can be related causally, for instanceif they constitute the input and the output of a chan-nel, or they can be related acausally, for instance if theyconstitute the input of one channel and the output of another. Although time-like separation implies that acausal connection is  possible  , it is whether such a con-nection  actually holds   that is relevant in our formalism.The distinction can also be made in non-relativistic theo-ries, and in theories with exotic causal structure. Indeed,causal structure is a more primitive notion than spatio-temporal structure, and it is all that we need here.Typically, we shall confine our attention to twoparadigmatic examples of causal and acausal separation(which can be formulated in either a relativistic or a non-relativistic quantum theory). Two distinct regions atthe same time, the correlations between which are con-ventionally described by a bipartite quantum state, areacausally related. The regions at the input and output of a quantum channel, the correlations between which areconventionally described by an input state and a quan-tum channel, are causally-related (although there are ex-ceptions, such as a channel which erases the state of thesystem and then re-prepares it in a fixed state) [75].We unify the description of Bayesian inference in thetwo different causal scenarios in the sense that variousformulas are shown to have precisely the same form, inparticular, the relation between joints and conditionals,the formula for Bayesian inversion and the formula forbelief propagation.Nonetheless, the different causal scenarios continue tobe distinguished insofar as the set of operators that canrepresent a possible state in one scenario is different fromthe set that does so in the other scenario. This latter factdoes not constitute a failure to achieve causal neutralityin the formalism for Bayesian inference because a similarphenomenon occurs classically. For instance, the causalrelations among a triple of variables are significant forthe sort of probability distribution that can be assignedto them. Specifically, if variable  R  is a common cause of variables  S   and  T  , while there is no direct causal con-nection between  S   and  T  , then  S   and  T   should be con-ditionally independent given  R , which is to say that the joint distribution over these variables is not arbitrary, buthas the form  P  ( R,S,T  ) =  P  ( S  | R ) P  ( T  | R ) P  ( R ). In ourframework, the operator describing the state of some setof regions may also depend on the causal relations amongthose regions.There is, however, one sense in which the formalism forquantum Bayesian inference that we develop here is  more sensitive   to the causal structure than the formalism forclassical Bayesian inference. In the latter, if we considerall the possible joint distributions over a pair of variables, R  and  S  , we find that the set of possibilities is the samefor the case where  R  and  S   are  causally  -related as it isfor the case where  R  and  S   are  acausally  -related. So, thefact that the set of possible states that can be assignedto a set of regions is constrained by the causal relationbetween those regions is  common   to the classical andquantum theories of inference. What is particular to thetheory of quantum inference is that even in the case of a pair   of regions, the causal relation between the regions isrelevant for the set of possible states that can be assignedto those regions [76].  4 II. CLASSICAL CONDITIONAL PROBABILITY In this section, the basic definitions and formalism of classical conditional probability are reviewed, with a viewto their quantum generalization in  § III.Let  R  denote a (discrete) random variable,  R  =  r  theevent that  R  takes the value  r ,  P  ( R  =  r ) the probabilityof event  R  =  r , and  P  ( R ) the probability that  R  takes anarbitrary unspecified value. Finally,   R  denotes a sumover the possible values of   R .A conditional probability distribution is a function of two random variables  P  ( S  | R ), such that for each value r  of   R ,  P  ( S  | R  =  r ) is a probability distribution over  S  .Equivalently, it is a positive function of   R  and  S   suchthat  S  P  ( S  | R ) = 1 (2)independently of the value of   R .Given a probability distribution  P  ( R ) and a condi-tional probability distribution  P  ( S  | R ), a joint distribu-tion over  R  and  S   can be defined via P  ( R,S  ) =  P  ( S  | R ) P  ( R ) ,  (3)where the multiplication is defined element-wise, i.e. forall values  r,s  of   R  and  S  ,  P  ( R  =  r,S   =  s ) =  P  ( S   = s | R  =  r ) P  ( R  =  r ).Conversely, given a joint distribution  P  ( R,S  ), themarginal distribution over  R  is defined as P  ( R ) =  S  P  ( R,S  ) ,  (4)and the conditional probability of   S   given  R  is P  ( S  | R ) =  P  ( R,S  ) P  ( R )  .  (5)Note that eq. (5) only defines a conditional probabilitydistribution for those values  r  of   R  such that  P  ( R  = r )   = 0. The conditional probability is undefined for othervalues of   R .The chain rule for conditional probabilities statesthat a joint probability over  n  random variables R 1 ,R 2 ,...,R n  can be written as P  ( R 1 ,R 2 ,...,R n ) =  P  ( R n | R 1 ,R 2 ,...,R n − 1 ) × P  ( R n − 1 | R 1 ,R 2 ,...,R n − 2 ) ...P  ( R 2 | R 1 ) P  ( R 1 ) .  (6)Finally, note that the process of marginalizing a distri-bution over a set of variables commutes with the processof conditioning on a disjoint set of variables, as illustratedin the following commutative diagram. P  ( R,S,T  )  R −−−−→  P  ( S,T  )  × P  ( T  ) − 1  × P  ( T  ) − 1 P  ( R,S  | T  )  R −−−−→  P  ( S  | T  )(7) III. QUANTUM CONDITIONAL STATES In this section, the quantum analog of conditionalprobability — a conditional state — is introduced. Wealso discuss how the states assigned to disjoint regionsare related via a quantum analog of the belief propaga-tion rule  P  ( S  ) =   R P  ( S  | R ) P  ( R ). There is a small dif-ference between conditional states for acausally-relatedand causally-related regions. The acausal case is dis-cussed in  § IIIA- § IIIB.  § IIIC-IIIK mainly concern thecausal case, wherein we find that quantum dynamics, en-semble averaging, the Born rule, Heisenberg dynamics,and the transition from the initial state to the ensembleof states resulting from a measurement can all be rep-resented as special cases of quantum belief propagation.Acausal analogs of some of these ideas are also developedin these sections. A. Acausal Conditional States In this section, the quantum analog of a conditionalprobability distribution, a  conditional state  , is developedas it applies to acausally-related regions. This causal sce-nario, and its classical analog, are depicted in fig. 1. Thedefinition proceeds in analogy with the classical treat-ment given in  § II. The convention of using  A,B,C,... to label quantum regions that are analogous to classicalvariables  R,S,T,...  is adopted throughout. The labels X,Y,Z,...  are reserved for classical variables associatedwith preparations and measurements, which remain clas-sical when we pass from probability theory to the quan-tum analog. A B (a) R S  (b) FIG. 1: Acausally-related quantum and classical regions.Classical variables are denoted by triangles and quantum re-gions by circles (this convention is suggested by the shapeof the convex set of states in each theory). The dotted linerepresents acausal correlation. (a) Two quantum regions inan arbitrary joint state (possibly correlated). (b) Two classi-cal variables with an arbitrary joint probability distribution(possibly correlated). The analog of a probability distribution  P  ( R ) assignedto a random variable  R  is a quantum state (density oper-ator)  ρ A  acting on a Hilbert space  H A . When there aretwo disjoint regions with Hilbert spaces  H A  and  H B , the
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