Complex joint probabilities as expressions of determinism in quantum mechanics
aa r X i v : . [ qu a n t - ph ] D ec Complex joint probabilities as expressions of determinism in quantum mechanics
Holger F. Hofmann ∗ Graduate School of Advanced Sciences of Matter, Hiroshima University,Kagamiyama 1-3-1, Higashi Hiroshima 739-8530, Japan andJST, CREST, Sanbancho 5, Chiyoda-ku, Tokyo 102-0075, Japan
The density operator of a quantum state can be represented as a complex joint probability of anytwo observables whose eigenstates have non-zero mutual overlap. Transformations to a new basisset are then expressed in terms of complex conditional probabilities that describe the fundamentalrelation between precise statements about the three different observables. Since such transformationsmerely change the representation of the quantum state, these conditional probabilities provide astate-independent definition of the deterministic relation between the outcomes of different quantummeasurements. In this paper, it is shown how classical reality emerges as an approximation to thefundamental laws of quantum determinism expressed by complex conditional probabilities. Thequantum mechanical origin of phase spaces and trajectories is identified and implications for theinterpretation of quantum measurements are considered. It is argued that the transformation lawsof quantum determinism provide a fundamental description of the measurement dependence ofempirical reality.
PACS numbers: 03.65.Ta, 03.65.Wj, 03.67.-a, 03.65.Vf
I. INTRODUCTION
Advances in quantum information technology have established the complete reconstruction of quantum states fromexperimental data as a standard procedure for the characterization of quantum devices [1–4]. At the heart of theseprocedures lies the insight that the quantum state of a system with a d -dimensional Hilbert space provides a summaryof all possible measurement statistics in terms of d − d possible outcomes - the equivalent of a jointprobability for the eigenstates of two observables of the system.In the light of such complete measurement-based descriptions of quantum states, there has been a growing interestin the identification of fundamental measurement strategies that could serve as a new standard for the evaluationof quantum information encoded in arbitrary states [6–13]. Although this research has succeeded in revealing moreof the richness of Hilbert space topologies, the formulation of a single standard representation is difficult, since noefficient characterization of quantum states by a discrete set of measurement operators can reflect the continuoussymmetry of Hilbert space with regard to unitary transformations between different measurements. It seems that allattempts to identify fundamental measurements must necessarily introduce a bias that is not found in the originalHilbert space formalism with its equivalent representation of all projective measurements as orthogonal basis systemsof an isotropic vector space. However, the conventional Hilbert space representation of quantum statistics in termsof a single orthogonal basis set is even more biased, as it represents only half of the physics in terms of measurementresults, while the other half is encoded in terms of abstract quantum coherences. Since quantum coherences of onebasis show up as probabilities in another basis, it may be desirable to find a more symmetric description of thedensity operator that expresses the coherences of a quantum state in terms of joint probabilities for non-orthogonalmeasurement outcomes.Since the number of elements needed to describe the complete density operator is exactly equal to the number ofeigenstate combinations of two observables, it would seem natural to represent the quantum state as a joint probabilityof only two measurements. Intuitively, any pair of basis sets with non-zero mutual overlap should represent twoindependent pieces of information distinguishing d × d = d elements of the statistics. Different choices of observableswould then correspond to different parameterizations of the two-dimensional phase space topology defined by anypair of observables with mutually overlapping eigenstates. Interestingly, such a joint probability was already proposedvery early in the history of quantum mechanics, as an alternative to the Wigner function in phase space [14, 15].Essentially, this joint probability is obtained by multiplying the projection operators for the two measurements andtaking the expectation value of the resulting complex-valued operator. It is therefore the most natural definition of ∗ Electronic address: [email protected] joint probability that the quantum formalism provides for measurements that cannot actually be performed jointly.One of the reason that Kirkwood’s approach to joint probabilities in quantum mechanics received disappointinglylittle attention may be the lack of practical applications of the theory. After all, the Copenhagen interpretation ofquantum mechanics implies that questions about events that do not happen are inherently meaningless. However,quantum paradoxes clearly show that quantum mechanics makes non-trivial statements about the statistical relationsbetween measurements that cannot be performed at the same time. In fact, the paradoxes show that quantummechanics cannot be understood in terms of positive joint probabilities for measurement outcomes represented by non-commuting measurement operators. Recently, it has been shown that the paradoxical aspects of quantum statisticsare consistent with negative conditional probabilities determined in weak measurements [16–21]. The consistency ofthese results strongly suggests that the statistics of weak measurements is a fundamental element of the Hilbert spaceformalism. In particular, it is possible to develop a consistent explanation of weak measurement statistics in terms ofcomplex conditional and joint probabilities [22–25].Interestingly, the joint probabilities derived from weak measurement are identical with the joint probability originallyintroduced by Kirkwood on purely mathematical grounds [15]. It is therefore possible to express any quantum statein terms the joint probability distribution obtained from the weak measurement of | a ih a | , followed by a finalmeasurement of | b ih b | , where h a | b i 6 = 0. Alternatively, weak measurement statistics can also be obtained fromthe measurement back-action of projective measurements [26], or from the correlations between optimal quantumclones of the input state [27]. Complex joint probabilities thus provide a surprisingly consistent description of thecorrelations between pairs of measurements that cannot be performed jointly.However, there remains an important question that needs to be addressed: complex probabilities cannot be inter-preted as relative frequencies of microscopic realities. Therefore, they cannot be identified with classical phase spacepoints. In particular, the measurement outcomes for a third measurement c cannot be related to well-defined pairs ofmeasurement outcomes ( a, b ), as the classical phase space analogy would suggest. Nevertheless, a description of thequantum state in terms of joint probabilities for b and c is just as complete as a description based on a and b . There-fore, the transformation between the two representations is reversible and deterministic. In the following, I will takea closer look at this relation between different joint probabilities. It is shown that the deterministic transformation isgiven by the complex conditional probabilities p ( c | a, b ) that characterize weak measurement statsitics [25]. Reversibil-ity of the transformation requires that the information about a can be recovered completely from the informationabout c after the transformation. For positive probabilities, this condition requires that c is a well-defined functionof a and b . In quantum mechanics, the same mathematical relation is fulfilled as a result of the orthogonality of theHilbert space vectors {| a i} . The structure of Hilbert space can then be understood as a modification of determinismthat reconciles continuous transformations with discrete measurement results at the expense of microscopic realism.The fact that the fundamental expression of determinism in quantum mechanics can be represented by complexconditional probabilities has significant implications for the formulation of the classical limit that represents theconventional notion of determinism in physics. Specifically, this classical determinism only emerges as a macroscopicapproximation to the microscopic quantum description. To illustrate this emergence of classical realism, it is necessaryto introduce the concept of measurement resolution, based on a sequence of quantum states that defines the distancebetween two orthogonal states. With this metric, the complex phases of the conditional probabilities p ( c | a, b ) can beidentified with phase space distances [25]. Since large phase space distances correspond to rapid phase oscillations in c , coarse graining rapidly reduces the precise expression of quantum determinism in terms of complex probablities toa single-peaked function centered around a single value of c , as expected from classical determinism. Quantum deter-minism can thus explain how the classical notion of a measurement independent reality emerges as an approximationto the more accurate description of context dependent realities in quantum mechanics.The rest of the paper is organized as follows. In section II, the representation of quantum states as complex jointprobabilities of observables with mutually overlapping eigenstates is introduced and the operator algebra is defined. Insection III, it is shown that transformations between different measurements are expressed by the complex conditionalprobabilities corresponding to the weak values of the projection operators for the new basis. The general criterionfor quantum determinism is derived and the differences between classical determinism and quantum determinism arediscussed. In section IV, it is shown that a classical phase space topology emerges in higher dimensional Hilbert spaces.Quantum determinism is still fundamentally different from classical determinism, but they become indistinguishablewhen the resolution of a measurement result is limited by Gaussian noise. Measurement independent phase spacepoints therefore emerge as approximate realities in the limit of low measurement resolution. In section V, quantumdeterminism is applied to unitary dynamics and different representations of causality are considered. It is pointedout that the identification of quantum dynamics with paths or histories described by a sequence of measurementresults may be a misinterpretation of quantum determinism based on the extrapolation of realist notions beyond theirnatural limit of validity. In section VI, it is pointed out that the imaginary part of complex joint probability doeshave a classical limit, represented by the gradients of the classical phase space distribution. Quantum corrections toclassical determinism become relevant when the imaginary part of the joint probability becomes comparable to thereal part. In section VII, the empirical foundations of quantum determinism are reviewed and consequences for theinterpretation of quantum mechanics are considered. It is emphasized that complex joint probabilities do not representrelative frequencies of quasi-realities, but should be understood as the fundamental deterministic relations betweenmeasurements that can never be performed jointly. Quantum determinism therefore highlights the dependence ofempirical reality on the measurement context. II. JOINT PROBABILITY REPRESENTATION OF QUANTUM STATES
A complete description of quantum statistics in terms of measurement probabilities is not a straightforward matter,because the uncertainty principle generally prevents the joint performance of separate quantum measurements. It istherefore impossible to simultaneously measure two observables with different eigenstates. In principle, it is possibleto perform a sequence of measurements on the same system, but then the measurement interaction of the firstmeasurement will change the result of the second measurement, so that the outcome of the second measurementcannot be identified with the value of the observable before the first measurement.Interestingly, there exist situations where sequential measurements can be interpreted as joint measurements. Thisis the limit of weak measurements [16], where the measurement interaction of the first measurement is so low thatits effect on the second measurement is negligible. Although the signal-to-noise ratio of weak measurements ismuch smaller than one, the average measurement results are consistent with the expectation values of the measuredobservables. A final measurement can then identify the conditional expectation values, also known as weak values.The complex joint probabilities obtained from weak measurements have a particularly simple mathematical form.In general, they correspond to the expectation value obtained for the operator product of the two measurementoperators [24]. For two observables with mutually overlapping sets of eigenstates {| a i} and {| b i} , the complex jointprobabilities representing the density operator ρ of an arbitrary quantum state are therefore given by the expectationvalue of the ordered product of the projection operators | b ih b | and | a ih a | , ρ ( a, b ) = h b | a ih a | ˆ ρ | b i . (1)As pointed out by Johansen [15], this is identical to the joint probability introduced by Kirkwood in 1933 [14].Johansen also showed that the complex probabilities provide a complete expansion of the density operator, with veryconvenient mathematical properites [15].An essential advantage of the joint probability representation given in Eq.(1) is that it stays very close to the originalHilbert space formalism, where the density matrix is defined in terms of a single measurement basis. Effectively, thejoint probability can be understood as a partial transformation of the right side of the a -basis density matrix to the b -basis, followed by an adjustment with a complex overlap factor of h b | a i . This transformation is obviously reversiblefor all transformations with non-zero overlap h b | a i . The complex joint probability of a and b thus provides a completeexpression of quantum coherences without the need for interferences between mutually exclusive alternatives.Classical joint probabilities refer to joint realities of a and b . In the quantum formalism, this corresponds to anormalized contribution to the density operator with simultaneous probabilities of one for both | a i and | b i . For thecomplex joint probabilities of Eq.(1), this set of basis operators is given byˆΛ( a, b ) = | a ih b |h b | a i . (2)The operators Λ( a, b ) are orthogonal with regard to the adjoint product trace, where the norm of the operators isgiven by the inverse overlap of | a i and | b i ,Tr (cid:16) ˆΛ( a, b )ˆΛ † ( a ′ , b ′ ) (cid:17) = 1 |h b | a i| δ a,a ′ δ b,b ′ . (3)Using this d -dimensional orthogonal operator basis, any density operator can be expressed in terms of the complexjoint probabilities of a and b , ˆ ρ = X a,b |h b | a i| Tr (cid:16) ˆ ρ ˆΛ † ( a, b ) (cid:17) ˆΛ( a, b )= X a,b ρ ( a, b ) ˆΛ( a, b ) . (4)Eq.(4) shows that complex joint probabilities are a complete representation of quantum statistics, regardless ofmeasurement context. It is therefore possible to represent all measurement statistics in terms of the statistics relatingto the measurements of a and b . In particular, the expectation values of all self-adjoint operators ˆ M can be definedin terms of a and b by simply expanding the operators in the adjoint operator basis { Λ † ( a, b ) } . The coefficients ofthis expansion are given by Tr (cid:16) ˆΛ( a, b ) ˆ M (cid:17) = h b | ˆ M | a ih b | a i . (5)Since these are the weak values of the operator ˆ M observed for an initial state of | a i and a final state of | b i , thecomplex joint probabilities ρ ( a, b ) appear to describe the density matrix as a mixture of transient quantum states { ˆΛ( a, b ) } defined by the respective combinations of initial and final states [24, 28]. Consequently, the expectationvalue of ˆ M corresponds to the average weak value given by h ˆ M i = X a,b ρ ( a, b ) h b | ˆ M | a ih b | a i . (6)In the light of the formal similarity to classical statistics, it may be important to remember that this expectationvalue describes the results of a direct measurement of ˆ M , and not the results of weak measurements. The weak valueof ˆ M is therefore not just an experimental result, but also a fundamental element of the operator algebra, similar tothe values of operators obtained for a point in phase space in the Wigner transformation of an operator. The complexweak values of ˆ M conditioned by a and b thus provide a complete mathematical expression of the operator ˆ M .The possibility of constructing joint probability representations for nearly arbitrary pairs of observables raises a fewinteresting questions about the relation between observables and the structure of Hilbert space. Specifically, any pairof observables can now serve as a “parameterization” of quantum states. If the results could be interpreted in termsof classical joint probabilities, each pair of values ( a, b ) would designate a microstate defining a phase space point.Keeping this analogy in mind, the transformation between different measurement bases corresponds to a change ofcoordinates in the effective phase space. In the following, I will examine the quantum mechanical expressions thatdescribe such transformations in the extreme quantm limit. III. TRANSFORMATIONS OF COMPLEX JOINT PROBABILITIES
Complex joint probabilities can be formulated for any pair of observables whose eigenstates have non-zero mutualoverlap. It is therefore possible to transform complex joint probabilities between different basis sets representingdifferent measurements. If we consider the transformation from a basis set ( {| a i} , {| b i} ) to a basis set ( {| c i} , {| b i} ),the transformation is given by ρ ( c, b ) = X a p ( c | a, b ) ρ ( a, b ) , (7)where the coefficients of the transformation are given by the weak conditional probabilities p ( c | a, b ) with p ( c | a, b ) = h b | c ih c | a ih b | a i . (8)These conditional probabilities are equal to the weak values of the projection operators of | c i and thus correspondto the probability of finding c conditioned by an initial value of a and a final value of b . A statistical interpretationof this transformation would suggest that the relation between a and c is random, corresponding to an irreversiblescattering of inputs a into different outputs c . However, the transformation is merely a change of representation anddoes not change the physical properties of the state. It should therefore be fully deterministic.A formal definition of determinism can be obtained from the reversibility of the transformation. If the transfor-mation from a to c is deterministic, the original joint probability can be recovered by the inverse transformationrepresented by the conditional probabilities p ( a ′ | c, b ). Therefore, conditional probabilities can only describe a de-terministic transformation between different representations of the same probability distribution if they satisfy therelation X c p ( a ′ | c, b ) p ( c | a, b ) = δ a,a ′ . (9)For classical statistics, where probabilities are real and positive, the above relation can only be satisfied if the condi-tional probabilities assign a specific value of c to each value of a , so that the conditional probabilities are one for thecorrect assignment and zero for all other assignments. In the quantum limit, the relation is still valid, but insteadof taking only values of zero or one, the complex conditional probabilities reflect the structure of Hilbert space, asshown by the contributions from each value of c , p ( a ′ | c, b ) p ( c | a, b ) = h b | a ′ ih b | a i h a ′ | c ih c | a i . (10)Thus, the quantum limit of determinism is obtained from the orthogonality of | a i and | a ′ i , even though there is noconditional assignment of a fixed value of c to each pair of values ( a, b ). In fact, quantum determinism as defined bythe conditional probabilities in Eq. (8) not only fails to assign a specific value of c to each pair ( a, b ), but actuallyassigns a non-zero value to the complex probability of each state | c i that is not orthogonal to either | a i or | b i .For basis sets with non-zero mutual overlap, the relation between c and ( a, b ) is therefore spread out over all possiblecombinations of a , b , and c . Determinism only emerges because of the complex phases of the conditional probabilities.To recognize the significance of the difference between classical determinism and quantum determinism, it is usefulto consider the classical interpretation of joint probabilities as relative frequencies of microstates defined by the phasespace point ( a, b ). In this case, deterministic transformations can only correspond to an exchange of labels denoting thefundamental representation independent reality of the phase space point ( a, b ) = ( c, b ). On the other hand, quantumdeterminism prevents the identification of such representation independent realities. The statistical relations definedby the Hilbert space structure of quantum mechanics imply that the mathematical points ( a, b ) are fundamentallydifferent from the mathematical points ( c, b ). Quantum determinism is therefore completely detached from classicalrealism. In the next section, I will illustrate the transition between quantum determinism and classical determinismby constructing a phase space over a sufficiently large Hilbert space. It is then possible to see how the classical notionof a measurement independent reality can emerge as an approximation of the more accurate relations of contextualquantum determinism. IV. EMERGENCE OF PHASE SPACE TOPOLOGIES FROM QUANTUM DETERMINISM
A fundamental contradiction between classical determinism and quantum determinism arises in discrete systems,where quantum determinism allows continuous transformations, whereas classical determinism only allows discreteexchanges of points. As a result, it is difficult to construct a phase space topology for few level systems. Even in thelimit of high dimensional Hilbert spaces, it is not immediately clear how to identify quantum states with parameters. Inpractical systems, this parameterization usually emerges from the interactions with the environment, which introducesa sequence of states, so that the distance between two orthogonal states | a i and | a ′ i can be expressed as a numericaldifference of a − a ′ . Continuous phase space topologies then emerge when the discrete steps of ± a or b can beconsidered microscopically small. For basis sets with non-zero overlap, the conditional probabilities p ( c | a, b ) can thenbe given by continuously varying functions of a , b and c . If the absolute values of the overlaps between the statesvary only slowly, the phase of the complex conditional probability can be expanded in a Taylor series up to secondorder around an extremum, resulting in a complex Gaussian with an imaginary variance of iV q , p ( c | a, b ) = 1 p πV q exp( i ( c − f c ( a, b )) V q − i π . (11)Since the phase also varies slowly in a and b , f c ( a, b ) can be approximated by a linear function of a and b . Comparisonwith Eq.(8) shows that the imaginary variance is given by V q = |h b | a i| π |h b | c i| |h c | a i| . (12)The gradients of f c ( a, b ) can be determined by considering the normalizations of p ( a | c, b ) and p ( b | a, c ). The resultsread ∂∂a f c ( a, b ) = |h b | a i| |h b | c i| ∂∂b f c ( a, b ) = |h b | a i| |h c | a i| . (13)The conditional probability p ( c | a, b ) is therefore completely determined by the Hilbert space overlaps of the basisstates. At the same time, a , b and c correspond to phase space coordinates, where f c ( a, b ) defines the correspondingclassical coordinate transformation.Since quantum determinism requires that the absolute values of all conditional probabilities are non-zero, it isfundamentally different from classical determinism, where conditional probabilities of zero are assigned to all combi-nations of a , b and c that do not fulfill the functional dependence given by c = f c ( a, b ). Instead, quantum determinismrepresents the relation between a , b and c in terms of complex phase oscillations. Specifically, the functional depen-dence given by f c ( a, b ) defines the values of c for which the complex phase of p ( c | a, b ) achieves its minimum. Classicalrealism emerges if this phase minimum can be identified with the only relevant value of c . In this case, p ( c | a, b ) canbe replaced by a delta function, δ ( c − f c ( a, b )). To see how well classical realism can approximate the more precisequantum results, it is possible to compare the predictions of quantum determinism and classical realism for coarsegrained probabilities, e.g. by folding the conditional probabilities p ( c | a, b ) with a Gaussian of variance σ . For theclassical probability δ ( c − f c ( a, b )), the result is a Gaussian with variance σ around ( c − f c ( a, b )). The precise resultobtained from the complex conditional probability in Eq.(11) can be written as p ( c ; σ ) = 1 p πσ (1 + iǫ ) exp (cid:18) ( c − f c ( a, b )) σ (1 + ǫ ) (1 − iǫ ) (cid:19) , (14)where ǫ = V q /σ describes the relative deviation from the classical probability distribution. Clearly, quantum de-terminism converges on classical determinism for small values of ǫ . This means that quantum determinism is in-distinguishable from classical determinism at resolutions of c much lower than p V q . Fig. 1 illustrates this rapiddisappearance of experimentally observable contradictions between the predictions of classical realism and quantumdeterminism. Since the low resolution limit characterizes almost all of our actual experience, our intuitive notion ofrealism may well be explained as a product of this classical approximation to quantum determinism. p ( c ; σ ) - (a) σ = 0 . c p ( c ; σ ) - - (b) σ = 0 . cp ( c ; σ ) - - (c) σ = 1 c p ( c ; σ ) - (d) σ = 2 c FIG. 1: Comparison between the complex conditional probabilities of quantum determinism and the corresponding classicalpredictions for an imaginary variance of V q = 1 in c for different Gaussian resolutions σ . Thick lines show the real part of thecomplex probability p ( c ; σ ), thick dashed lines show the corresponding classical probability distribution, and thin dashed linesshow the imaginary part of p ( c ; σ ). (a) illustrates the difference between quantum determinism and classical predictions at ahigh resolution of σ = 0 .
25, (b) shows the transition to low resolution at σ = 0 .
5, (c) shows the similarity of quantum statisticsand classical statistics at σ = 1, and (d) shows the small deviations from the classical limit that remain at σ = 2. In addition to the classical functional relations c = f c ( a, b ) that relate different parameterizations of phase space toeach other, classical phase space also has a well-defined metric that ensures the conservation of phase space volumeunder all canonical transformations. In quantum mechanics, this metric corresponds to the density of states in thephase space volume defined by changes of a and b . In the discussion above, a and b are integers that number thediscrete basis states of a d -dimensional Hilbert space. For this quantum mechanical parameterization, the metric ofphase space emerging in the classical limit is found by replacing the sum over all values of a and b with approximateintegrals, so that the total number of states is given by Z d Z d |h a | b i| da db = Z d db ≈ d. (15)The metric emerging from a derivation of phase space from a Hilbert space parametrized by numbering the statestherefore has a metric that is given in terms of the density of quantum states, which is equal to |h a | b i| near thephase space point ( a, b ). A canonical parameterization of phase space is obtained for h a | b i = 1 √ d exp (cid:18) i πd ab (cid:19) . (16)In this case, unitary phase shifts in a generates shifts in b , and vice versa. The parameters a and b can be re-scaledin units of position x and momentum p , so that the phase of h x | p i is given by xp/ ¯ h . This re-scaling shows howthe classical action emerges from the quasi-continuous limit of joint probability representations in sufficiently largeHilbert spaces. V. CAUSALITY AS QUANTUM DETERMINISM
According to classical causality, a single point in phase space defines the properties of a closed system at all times.In this sense, the canonical phase space coordinates of position and momentum can be interpreted as parametrizationbased on a specific reference time, and the time evolution of the coordinates represents transformations to differentparameterizations of the same phase space. In general, it is therefore possible to define phase space parameterizationsreferring to multiple times and even to weighted averages over time.In classical determinism, this ambiguity of phase space concepts is not particularly relevant, since it is always possibleto identify the continuous time-evolution of observable properties in terms of well-defined time-dependent functions.However, the situation is quite different in the limit of quantum determinism. Here, simultaneous statements about thesame property at different times do not usually commute. Therefore, it is not correct to assign reality to a continuoustrajectory describing the dynamics of the system. It may well be the case that the focus on dynamics and timeevolution in traditional physics has unnecessarily complicated the picture we have of quantum mechanics. Quantumdeterminism addresses this problem by describing the time evolution of closed systems as a re-parameterization ofan unchanged quantum state ˆ ρ . Causality in quantum mechanics is then described by the complex conditionalprobabilities of quantum determinism for statements associated with different times.The conventional representation of deterministic causality in quantum mechanics is given by the unitary transfor-mation ˆ U ( t j − t i ) that defines the relation between states at time t j with states at time t i . If a quantum state ˆ ρ isexpressed by the complex joint probability ρ ( a , b ) of the properties a and b at time t , the transformation to a and b at time t should proceed in two steps, since elementary quantum determinism describes the relations betweensets of three observables. For example, determinism defines the value of a at time t as a function of both a and b . Therefore, either a or b can be replaced by a . For reasons of symmetry, the natural choice seems to be atransformation to ( a , a ), ρ ( a , a ) = X b p ( a | a , b ) ρ ( a , b ) . (17)This two-time representation of the quantum state reflects the fact that trajectories can be defined by the positionsat two different times. Since this representation is in principle equivalent to any other, the evolution of a i up to athird time t can be evaluated directly from the complex joint probabilities of a and a , ρ ( a , a ) = X a p ( a | a , a ) ρ ( a , a ) . (18)Here, the conditional probability p ( a | a , a ) corresponds to the probability of finding the system in a at time t ,when it was initially in a and finally arrived in a . For the positions of a free particle, the complex phase of thisconditional probability is given by the action of the trajectory a → a → a , so that the classical result for a corresponds to the path of least action [25].The connections between extended probabilities and path integrals or quantum histories have already been notedin other works [29, 30]. However, the explanations given there seem to be at odds with determinism, since therepresentations appear to assign a non-deterministic time evolution to a single quantum object. Nevertheless quantumdeterminism can reproduce the same results in terms of a gradual transformation from a at t to a n at t n in a numberof steps evolving a i at t i to a i +1 at t i +1 . The total conditional probability for the transformation is then given by p ( a n | a , a ) = X { a i } p ( a n | a , a n − ) p ( a n − | a , a n − ) . . .. . . p ( a | a , a ) p ( a | a , a ) , (19)which converges on the path integral for the evolution of a ( t ) in the limit of continuous times. Specifically, the phaseof each contribution to the sum over the paths { a i } is defined by a sum corresponding to the total action of that path.Since sums over rapidly oscillating phases cancel out, the end result can be obtained by summing over only a finiteinterval around the classical trajectory given by the path of least action.Although Eq.(19) shows that the conditional probabilities of quantum determinism can be expressed in termsof path integrals, it seems significant that these path integrals do not describe the evolution of a quantum state.Instead, they describe a sequential transformation of state-independent conditional probabilities that describe thefundamental deterministic relations between the non-commuting observables a i . Quantum determinism thus providesan alternative explanation for the role of path integrals in the description of the dynamics of a system. Specifically, thetransformations in Eq.(19) are merely a change of representation. It is therefore difficult to justify the interpretation ofan individual path as the history of an individual system, even though the formal assignment of a complex probabilityto each path is indeed possible [29, 30]. Clearly, each path is merely a sequence of statements, each of which can betranslated to equivalent statements at other times. Since a pair of statements is in principle sufficient to define thestatistics of all other statements, the paths are merely redundant representations of the fully deterministic evolutionof the physical properties that characterize the system. The misleading impression that a quantum system could“choose” between alternate paths or histories arises from a misinterpretation of joint probabilities with joint realities.As we saw in the previous section, such an identification represents an approximation valid only in the classical limitof low measurement resolution.In Hilbert space, the time evolution of quantum states is represented by unitary transformations ˆ U ( t j − t i ) generatedby the Hamilton operator ˆ H . If only the time evolution of a single measurement outcome a is of interest, it maytherefore be convenient to express the quantum state as a complex joint probability of | a ( t ) i = ˆ U ( t ) | a i and aneigenstate | n i of the Hamiltonian ˆ H with an energy eigenvalue of E n . The time evolution can then be expressedin terms of the complex conditional probability p ( a ( t ) | n, a ′ ). The time dependence of this conditional probabilitycorresponds to the formulation of the time dependent Schroedinger equation in the {| a i} -basis, ddt ( p ( a ( t ) | n, a ′ ) h a ′ | n i ) = − i ¯ h X a ′′ h a ′ | ( ˆ H − E n ) | a ′′ i ( p ( a ( t ) | n, a ′′ ) h a ′′ | n i ) . (20)Essentially, the re-scaled conditional probabilities p ( a ( t ) | n, a ′ ) h a ′ | n i evolve just like the a ′ -components of a statevector. In the limit of smoothly varying phases, these dynamics therefore correspond to the well known dynamicsof dispersion in wave propagation. Quantum determinism thus reproduces the formal aspects of the wave-particledualism implied by the conventional formulation of the Schroedinger equation. However, the re-formulation in termsof conditional probabilities for measurements at different times shows that the object of the dynamical evolution isnot a physical wave, but the statistics of statements about a property of the quantum system at different times. Thedeeper meaning of the formal analogy between the elastic properties of physical waves and the conditional statisticsof post-selected measurements is therefore far from obvious, and related measurement results such as [21] should notbe misinterpreted in terms of a “realism” of the wavefunction.The analysis of Hamiltonian dynamics also reveals a highly non-classical relation between transformation dynamicsand statistics that can be expressed in the form of complex probabilities [25]. In its most simple form, this relationis expressed by the definition of imaginary weak values as logarithmic derivatives of the post-selected probabilitiesfor a weak unitary transformation generated by the respective observable [31]. The time evolution of measurementprobabilities can therefore be expressed in terms of imaginary weak values of energy, ddt h a | ˆ ρ | a i = − i ¯ h (cid:16) h a | ˆ H ˆ ρ | a i − h a | ˆ ρ ˆ H | a i (cid:17) = X n E n ¯ h Im ( ρ ( E n , a )) . (21)This expression provides a direct interpretation of imaginary probabilities that is consistent with classical theoriesof phase space transformations. In the following, I will use this analogy to provide a classical definition of complexprobability that corresponds to the low resolution limit of the quantum mechanical values. VI. COMPLEX PROBABILITY IN THE CLASSICAL LIMIT
As shown in section IV, classical phase space features emerge as soon as Hilbert space is sufficiently large to allowa representation of quantum phases and amplitudes as smooth continuous functions of the variables a and b . for adiscussion of classical limits, it is therefore often sufficient to focus on a continuous variable phase space defined interms of position ˆ x and momentum ˆ p . The complex joint probability of a quantum state is then given by ρ ( x, p ) = h p | x ih x | ˆ ρ | p i , (22)where h p | x i = exp( − ipx/ ¯ h ) / √ π ¯ h . Incidentally, this is precisely the form in which Kirkwood originally introducedthe complex probability distribution in 1933, as an alternative to the Wigner function [14]. However, it gained muchless recognition than the Wigner function, probably mostly because the complex phases appear to complicate thecomparison with classical statistics. It is therefore a somewhat ironic twist that the Kirkwood distribution actuallydescribes the measurement statistics observed in weak measurements, to the point where its discrete versions canresolve quantum paradoxes. The Kirkwood distribution thus provides the correct continuous variable limit of themore general discrete quantum statistics that can be observed and verified by weak measurements.In general, the imaginary part of complex probabilities can be defined operationally as logarithmic derivativesof measurement probabilities in response to weak transformations generated by the observable in question [31]. Inparticular, Eq.(21) shows how the time evolution of a measurement distribution depends on the imaginary parts ofthe joint probability with the eigenstates of the Hamiltonian. This relation can be applied to position and momentumby considering the change in a momentum distribution ρ ( p ) caused by a potential V ( x ), ddt ρ ( p ) = 2¯ h Z V ( x )Im ( ρ ( x, p )) dx. (23)In the classical limit, the change of momentum is given by dp/dt = − ∂V /∂x , so the relation between the change of ρ ( p ) and the real-valued joint probability reads ddt ρ ( p ) = Z ∂∂x V ( x ) ∂∂p Re ( ρ ( x, p )) dx. (24)Integration in parts can be used to identify the imaginary probability in Eq.(23) with the real probability in Eq.(24).The classical limit of imaginary joint probabilities is then given byIm ( ρ ( x, p )) = ¯ h ∂ ∂x∂p Re ( ρ ( x, p )) . (25)The appearance of ¯ h in this classical definition of imaginary probability indicates that, in the classical limit, theimaginary part will be much smaller than the real part. Oppositely, a joint probability can only be considered classicalif the action given by the ratio of the joint probabilities and its second order derivative in x and p is sufficiently smallerthan ¯ h .Although the result above has been derived for the Kirkwood distribution in phase space, its generalization to theclassical limit of high-dimensional discrete Hilbert spaces is straightforward. For slowly varying phases and amplitudesof h a | b i , the corresponding expression can be obtained by replacing ¯ h = 1 / (2 π |h p | x i| ) with 1 / (2 π |h a | b i| ). Theresult reads Im ( ρ ( a, b )) = 14 π |h a | b i| ∂ ∂a∂b Re ( ρ ( a, b )) . (26)In general, the classical limit of imaginary probabilities can be represented by the gradients of the phase space distri-bution associated with general transformations of the parameters. The complex probabilities of quantum mechanicstherefore represent a unification of statistics with the dynamics of transformations [31]. The quantum of action definesthe point at which the classical separation between dynamics and (static) information breaks down. At that point,it is necessary to include the topology of transformations in the definition of joint statistics, a task that is achievedmost naturally by expressing quantum mechanics in terms of complex probabilities.0 VII. ON THE EMPIRICAL FOUNDATIONS OF QUANTUM DETERMINISM
Quantum determinism might have far reaching consequences for our understanding of quantum physics. However,the possibility of addressing seemingly counterintuitive properties of quantum mechanics in a new light may also causenew misunderstandings. In fact, the difficulty of identifying the precise physics behind useful mathematical conceptsseems to be the very reason why there is so much fundamental disagreement on the proper interpretation of quantummechanics. It may therefore be justified to take an extra sharp look at the physics that support and justify the useof complex joint probabilities.As mentioned in the introduction, it is fundamentally impossible to perform two quantum measurements jointly.Nevertheless, all measurements can be performed in parallel, on separate representatives of the same system. Thatis why quantum theory does define the relations between completely different measurements, and physicists shouldtry to make these relations as clear as possible. Unfortunately, previous constructions of joint probabilities such asthe Wigner function or the one used in Feynman’s explanation of quantum computation all exploited the ambiguityof partial measurement results, filling the gaps by convenient but necessarily arbitrary assumptions [30, 32–34]. Itis therefore important to emphasize that the present approach is firmly rooted in the experimentally observableproperties of quantum statistics.Firstly, weak measurements can confirm complex joint probabilities directly. The only assumption used in the weakmeasurement is that the probabilities of the actual measurement outcomes of the weak measurement are proportionalto the probability of the precise measurement result. Since this assumption clearly holds when no final measurementis performed (or when the final measurement confirms the weak measurement), it seems to be difficult to avoid theconclusion that the complex value obtained in a post-selected weak measurement represents the correct conditionalprobability. Moreover, weak measurement statistics can be observed directly in the back-action of strong measurements[15] and in the correlations between optimally cloned quantum systems [31].An essential point in the experimental evaluation of joint probabilities is the requirement of consistency. Weakmeasurement statistics require no implicit assumptions about correlations between different observables, since causalityensures that the weak measurement is not affected by the post-selection process, and the weakness of the measurementensures that the final outcome is not influenced by the intermediate measurement. In contrast, the construction of theWigner function from parallel measurements of linear combinations of ˆ x and ˆ p implicitly assumes that the eigenvaluesof ˆ x + ˆ p should be equal to the eigenvalues of ˆ x plus the eigenvalues of ˆ p - an assumption that is clearly inconsistentwith operator algebra.Secondly, the joint complex probabilities discussed here are a natural mathematical choice based on the properties ofoperator algebras in Hilbert space. That is the reason why they were actually discovered long before their usefulness forthe explanation of weak measurements and other paradoxical quantum statics were known. The definition of complexprobabilities as expectation values of the products of two measurement operators is a simple representation of the“AND” operation in classical logic, where the truth value is also given by a product of the individual truth values.It therefore provides a natural expression for the joint validity of two quantum statements, without interpretationalbias in favor of a specific type of measurement or physical system.One problem might be that quasi-probabilities have often been motivated by the assumption of quasi-realities,that is, by an understandable desire to return to some form of classical realism that defines objects in terms ofcompletely measurement independent concepts. However, the present approach does the opposite: it shows that suchan ersatz reality cannot be constructed from the mathematical objects that represent joint probabilities, and it explainshow the measurement independent reality of classical physics can emerge as an approximation to the measurementdependent reality of quantum physics. Specifically, the functional relation between two measurement outcomes anda third measurement outcome that characterize the measurement independent determinism of classical physics areonly approximations. Quantum mechanics does not provide a replacement for such classical determinisms. Instead,determinism is expressed in terms of statistical relations that should not be confused with the relative frequencies ofclassical statistics: a non-zero value of p ( c | a, b ) does not mean that sometimes, the system is accidentally describedby a,b, and c, but rather indicates that the separate frequencies of a, b, and c must be related to each other in aspecific way, so that complete knowledge of the statistics of a and b means that we can determine the statistics of cas well. The complex values of these conditional probabilities are a strong indication that realist interpretations arenot helpful. In fact, it seems that the present formulation of quantum statistics shows that determinism (and hencecausality) does not require realism and actually contradicts realist assumptions in the quantum limit.An empirical interpretation of quantum mechanics requires that realism be restricted to the outcomes of actualmeasurements. In the context of this empirical realism, each individual system is characterized by its preparation anda single measurement outcome, where the specific form of both fully defines a context dependent reality accessible fromthe “outside”. The complex joint probabilities discussed here apply to ensembles and indicate the statistical relationsbetween different systems from the same source, measured in different ways. Thus, complex joint probabilities supportand confirm the dependence of individual realities on the specific measurement context.1 VIII. CONCLUSIONS
Complex joint probabilities provide a representation of quantum states in terms of any pair of observables withmutually overlapping eigenstates. Such states can never be measured jointly, but their statistical connection can beobserved in weak measurements. The fundamental nature of this relation between incompatible quantum measure-ments is revealed when transformations between different joint probability representations are considered, since theserelations describe how the deterministic relation between two measurements and a third measurement is described inquantum theory. The classical notion of completeness associated with phase space points thus survives in quantummechanics. However, the complex probabilities associated with joint statements about non-commuting observablesrequire a modification of classical determinism, so that the simultaneous assignment of measurement outcomes corre-sponding to measurement independent phase space points is impossible. Instead, determinism is expressed in termsof complex phases relating to the properties of phase space transformations. For sufficiently smooth phase spacetopologies, quantum determinism can be expressed by Gaussian distributions with imaginary variance. Thus, the dif-ferences between classical determinism and quantum determinism become relevant when the measurement resolutionapproaches or exceeds the imaginary variance of quantum determinism.The discussion above shows that the classical notion of reality emerges naturally from quantum contextualitywhen the measurement resolution is sufficiently low. The idea of a measurement independent reality “out there”may therefore reflect a reasonable approximation, similar to the assumption of a flat space time in the absence ofstrong gravitational fields. Importantly, the lack of measurement independent realities can now be explained interms of precise deterministic relations between the different possible measurements. Hilbert space thus provides awell-defined quantum limit of phase space topologies. In the context of time evolution and causality, this meansthat a single pair of observables determines the complete history of a quantum object. However, this history cannotbe described by assigning a time-dependent value to a specific property, since such an assignment corresponds tosimultaneous measurements of multiple non-commuting properties. Instead, quantum determinism only providesprecise statements about the relation between the measurement statistics obtained for different representatives of thesame source measured at different times.Even from a merely technical viewpoint, quantum determinism should proof useful by providing a consistentmeasurement-based description of quantum mechanics. The reformulation of Hilbert space concepts in terms ofstatistical expressions may be particularly useful in the analysis of the quantum information content of states assuggested by related approaches to quantum statistics that contributed to the motivation for the present work [6–13].From my own perspective, however, the most surprising aspect of the present work is the possibility of defining de-terministic relations between different measurements that are independent of the assignment of simultaneous valuesto the measurements and actually contradict such assignments in all precisely defined cases. This means that thereis actually much less freedom in the interpretation of quantum mechanics than previously thought. In particular,quantum determinism appears to introduce a complete definition of the fabric of empirically accessible reality, repre-senting an entirely new framework for all experimentally accessible aspects of quantum physics. Once the topology ofquantum determinism is fully understood, it may finally be possible to explain quantum mechanics entirely in termsof empirical concepts, without the need for postulates in the form of unmotivated mathematical abstractions.
Acknowledgment
Part of this work has been supported by the Grant-in-Aid program of the Japanese Society for the Promotion ofScience, JSPS.
References [1] U. Leonhardt, Phys. Rev. Lett. , 4101 (1995).[2] A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, Phys. Rev. Lett. , 3103 (1999).[3] K. J. Resch, P. Walther, and A. Zeilinger, Phys. Rev. Lett. , 070402 (2005).[4] M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Korber, W. Hansel, H. Haffner, C. F. Roos, and R. Blatt,Phys. Rev. Lett. , 220407 (2006). [5] G. Mahler and V. A. Weberruß, Quantum Networks (Springer, Berlin, 1998).[6] C. Brukner and A. Zeilinger, Phys. Rev. Lett. , 3354 (1999).[7] J. Lawrence, C. Brukner, and A. Zeilinger, Phys. Rev. A , 032320 (2002).[8] T. Paterek, B. Dakic, and C. Brukner, Phys. Rev. A , 012109 (2009).[9] R. G. Diaz, J. L. Romero, G. Bj¨ork, and M. Bourennane, New J. Phys. , 256 (2005).[10] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, J. Math. Phys. , 2171 (2004).[11] C. A. Fuchs and R. Schack, e-print arXiv:0906.2187v1.[12] Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, Phys. Rev. A , 051801(R) (2011).[13] J. Rau, Ann. Phys. , 2622 (2009).[14] J. G. Kirkwood, Phys. Rev. , 31 (1933).[15] L. M. Johansen, Phys. Rev. A , 012119 (2007).[16] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. , 1351 (1988).[17] R. Mir, J. S. Lundeen, M. W. Mitchell, A. M. Steinberg, J. L. Garretson, and H. M. Wiseman, New J. Phys. , 287 (2007).[18] J. S. Lundeen and A. M. Steinberg, Phys. Rev. Lett. , 020404 (2009).[19] K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, New J. Phys. , 033011 (2009).[20] M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, Proc. Natl. Acad.Sci. U. S. A. , 1256 (2011).[21] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature (London) , 188 (2011).[22] A. M. Steinberg, Phys. Rev. A , 32 (1995).[23] A. Hosoya and Y. Shikano, J. Phys. A: Math. Theor. , 385307 (2010).[24] H. F. Hofmann, Phys. Rev. A , 012103 (2010).[25] H. F. Hofmann, New J. Phys. , 103009 (2011).[26] L. M. Johansen, Phys. Lett. A , 374 (2007).[27] H. F. Hofmann, e-print arXiv:1111.5910v2.[28] Y. Shikano and A. Hosoya, J. Phys. A: Math. Theor. , 025304 (2010).[29] D. Sokolovski and R. S. Mayato, Phys. Rev. A , 052115 (2006).[30] J. B. Hartle, Phys. Rev. A , 012108 (2008).[31] H.F. Hofmann, Phys. Rev. A , 022106 (2011).[32] R. P. Feynman, Int. J. Theor. Phys. , 467 (1982).[33] Y.D. Han, W. Y. Hwang, and I. G. Koh, Phys. Lett. A , 283 (1996)[34] H. F. Hofmann, J. Phys. A: Math. Theor.42