Violating the assumption of Measurement Independence in Quantum Foundations
VViolating the assumption of Measurement Independencein Quantum Foundations
A Project Reportsubmitted by
INDRAJIT SEN in partial fulfilment of the requirementsfor the award of the degree of
Master of Science
DEPARTMENT OF PhysicsINDIAN INSTITUTE OF TECHNOLOGY MADRAS.April 2016 a r X i v : . [ qu a n t - ph ] M a y ROJECT CERTIFICATE
This is to certify that the project report titled
Violating the assumption of Measure-ment Independence in Quantum Foundations , submitted by
Indrajit Sen , to theIndian Institute of Technology, Madras, in partial fulfilment of the requirements for theaward of the degree of
Master of Science , is a bona fide record of the research workdone by him under our supervision. The contents of this report, in full or in parts, havenot been submitted to any other Institute or University for the award of any degree ordiploma.
Sibasish Ghosh Prabha Mandyam
Project Guide Project GuideAssociate Professor Assistant ProfessorDept. of Physics Dept. of PhysicsIMSc Chennai IIT MadrasPlace: ChennaiDate: 25 April 2016
CKNOWLEDGEMENTS
I’d like to express my gratitude to Prof. Sibasish Ghosh, with whom I have discussedmy project work almost every week since the past one year. I thank him for the enor-mous time he has given me, and for the ease with which I could discuss any doubt I had,no matter how small. I am thankful to him for the freedom with which I could pursuea controversial topic in Quantum foundations, the viability of which seemed dim at thebeginning, and for guiding me throughout the project.I’d like to thank Prof. Prabha Mandyam for introducing me to the world of Quantumfoundations when I knew little about the people and literature in this field. I want tothank her for giving me the opportunity to do my Master’s project in foundations, andfor her considerable inputs in editing the thesis to its current form.I’d also like to express my gratitude to Prof. MJW Hall who was extremely gener-ous in entertaining all my doubts on measurement independence and related topics. Heproof read and checked several results(right or wrong) that I had, pointed out connect-ing ideas that I had missed, and was in general very kind to give so much of his time tome. His revolutionary work in the past few years opened up the topic of measurementindependence from apathy, and this thesis would not have been possible without hisbold paper of 2010.I’d like to thank my parents and my friends at IIT Madras for their wonderful com-pany.-Indrajit Sen i
BSTRACT
KEYWORDS: Quantum foundations, Measurement IndependenceThough Quantum Mechanics is one of the most successful theories in Physics,experimentally verified to a very high degree of accuracy, there is still little consen-sus among physicists on a number of fundamental conceptual problems that the the-ory is plagued with since inception. Richard Feynman once remarked, "I think I cansafely say that nobody understands quantum mechanics". Some of the founding fathersthemselves- Erwin Schroedinger, Louis de Brogelie and Albert Einstein - were not sat-isfied with the theory . The field of Quantum foundations strives to resolve these issuesby reformulating the old discussions mathematically; proposing hidden variable theo-ries that reproduce Quantum Mechanics; proposing experimental tests that can be usedto decide between various standpoints; and by working on the deeper theory of Quan-tum Gravity from the perspective of foundations.Measurement Independence is an assumption that has been used in such founda-tional arguments since the time of EPR thought experiment in 1935, and assumed in anoverwhelming majority of hidden variable models of Quantum Mechanics since then.And yet, the term "measurement independence" is very recent - introduced only in 2010.Before this, it was known vaguely as the "free will assumption" or "lack of retrocausal-ity". Compared to other assumptions in foundations, like locality or contextuality, it hasreceived little attention. This is because most researchers still consider the assumptiontoo natural to expect its violation. In the past few years however, a lot of work has Some of the problems are:1. The Measurement problem: when and how does the collapse of the wavefunction occur?2. The Quantum to Classical transition problem: exactly how many microscopic(quantum) objects mustbe collected together for the composite object to be macroscopic(classical)?3. The problem of Reality: whether there exists an objective real world independent of our observations.4. The problem of Determinism: whether Nature is fundamentally random.5. The problem of Completeness: In its domain Quantum Mechanics is correct, but is it a complete,description of Nature? iieen done in analysing the assumption. It has been quantified, and several interestingconsequences of its violation have been derived.In this thesis we study the various contexts in which the assumption is used, re-view theorems about hidden variable models in light of relaxing measurement inde-pendence, develop several new measurement dependent hidden variable models whichhave interesting properties, finding application in the foundational question of "realityof wavefunction", and in classical simulation of quantum channels.iii
ABLE OF CONTENTS
ACKNOWLEDGEMENTS iABSTRACT iiLIST OF FIGURES viABBREVIATIONS vii1 Introduction 1 et al’s results[38] . . . . . . . . 263.2 The Brans model[33] . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Correlation between the particles and measurement choices 293.3 A maximally epistemic model in d N - The generalised Brans Model 303.3.1 How epistemic is the model? . . . . . . . . . . . . . . . . . 313.3.2 Does the model satisfy Preparation Independence[23]? . . . 323.3.3 What is the randomness in the model? . . . . . . . . . . . . 333.4 Simulating Quantum Channels using Measurement Dependent models 343.4.1 Modified Kochen Specker model II . . . . . . . . . . . . . 363.4.2 Protocol to simulate quantum channels using modified KochenSpecker model II . . . . . . . . . . . . . . . . . . . . . . . 363.4.3 Calculation of Communication cost . . . . . . . . . . . . . 373.4.4 Advantage of using Measurement Dependent models for simu-lation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 A Measurement Dependent model that cross correlates particles andmeasurement choices in EPR scenario . . . . . . . . . . . . . . . . 393.5.1 Modified Hall Model . . . . . . . . . . . . . . . . . . . . . 393.5.2 Foundational implications of such a model . . . . . . . . . 403.6 Can a ψ ontic ontological model be converted to ψ epistemic by intro-ducing Measurement Dependence? . . . . . . . . . . . . . . . . . . 423.6.1 Modified Bell Mermin model . . . . . . . . . . . . . . . . 42 IST OF FIGURES
BBREVIATIONS MI Measurement Independence MD Measurement Dependent PI Preparation Independence vii
HAPTER 1Introduction1.1 Does Quantum Foundations matter? "I am a Quantum Engineer, but on Sundays I have principles." - John Bell ( as quotedin [1])
Bohmian Mechanics [2] is a completely deterministic, non-local, contextual hiddenvariable reformulation of standard Quantum Mechanics, where particles have classicaltrajectories. It reproduces all known experimental predictions of Quantum Mechanics,and has also been generalised to Quantum Field Theories [3]. Thus, explanation ofexperimental phenomena does not require us to abandon our classical notions, say ofdeterminism, to explain events on microscopic scale. Infact, there are several hiddenvariable theories[4] [5] [6] which exactly reproduce the predictions of Quantum Me-chanics, implying that no unique interpretation can be drawn from the experimentaldata. Recently, hidden variable theories have also predicted phenomena which contra-dict Quantum Mechanics in early universe and around black holes [7], obviating thecriticism of hidden variable theories that they contain no new Physics.In light of this, it becomes important to ask whether the concepts used in standardQuantum Mechanics : randomness, wave-function collapse, lack of causal explanation ,are inevitable or whether, as suggested by Einstein [8], are indicative only of QuantumMechanics being an incomplete and provisional theory, or whether, as is commonlyheld, a question of Semantics or Philosophy than Science.The result of such investigations have relevance in Quantum Information, QuantumGravity and Cosmology. Bell’s theorem has applications in Quantum Cryptography[9]and Quantum random number generators[10] Certain classes of hidden variable mod-els can be used to classically simulate quantum channels and define communicationomplexity[11]. The problem of defining time, and causal structures in Quantum Grav-ity have connections to foundations. York parameter as a candidate for time parameteris suggested by de Brogelie-Bohm theory[12]. Indefinite causal structure, where causalrelationships are dynamic as well as probabilistic, has been proposed for gravity by re-searchers in foundations [13]. de Brogelie Bohm theory also predicts[7] signature ofviolation of Born’s rule in the cosmic microwave background, relic cosmological par-ticles, Hawking radiation, photons with entangled partners inside black holes, neutrinooscillations and particles from very distant sources. "It is sometimes said that quantum theory saves free will. In the context of this pa-per.....free will saves quantum theory....in the sense of eliminating hidden variable al-ternatives." -C.H. Brans [33]
Measurement Independence(MI) is the condition of non-correlation between the hy-pothetical hidden variables and measurement choices made in an experiment. This as-sumption has been widely assumed in Quantum foundation literature. It is present in theEPR paper[8], in Bell’s theorem [15], and in the framework of Ontological models[16].An early justification for this assumption was given by Bohr, in his reply to the EPR pa-per: "our freedom of handling the measuring instruments [is] characteristic of the veryidea of experiment"[11]. Measurement Independence is considered a very "reasonable"assumption by most researchers in Quantum Foundations[17]. Not much work has beendone on this assumption, though it has existed since the early days of Quantum founda-tion. Infact the term "Measurement Independence" itself has been coined very recently,in 2010 by MJW Hall[18]. Also pointed out by Hall [18] against the general compla-cency was that reasonableness alone is not sufficient - locality was very reasonable toEinstein[19].Correlation between the hidden variables and the choice of measurement made by2n experimenter can be due to several factors. Correlation can arise if measurementchoices are no longer assumed to be acts of "free will" [20] and thus uncorrelated withall events in their backward light-cones. The correlations may also result from retro-causality, wherein the event of making a choice in future affects the hidden variablestate of the system in past [21].
Chapter 2 introduces general hidden variable theories, and the necessary definitions thatare required to understand the subsequent chapters.Chapter 3 contains the bulk of work done as part of thesis. First we review someimportant theorems for ontological models [16] in the light of relaxing MI, and con-sider if they still retain validity. Then we introduce the Brans [33] model for singletstate correlations, which we generalise. We study the important properties of both themodels, which leads us to a new result. Next, we introduce a MD model for qubits, andintroduce a protocol to simulate quantum channels using it, noting the possible advan-tages of using MD models over ontological ones. We further introduce a MD modelwhich has important properties relevant to EPR scenario, and discuss its foundationalimplications. Lastly, we show how a ψ ontic model can be converted to epistemic byintroducing measurement dependence in it.Chapter 4 contains some observations about a different assumption, Preparation In-dependence, used in ref. [23] to derive an important result.Chapter 5 concludes with list of new results derived, and some questions that areleft unanswered, serving as future directions for research.3 HAPTER 2An introduction to hidden variable theories2.1 Formulation of hidden variable theories
Consider an ensemble of identically prepared quantum systems, specified by a purestate. A general hidden variable theory specifies additional variables to each of thesesystems, usually different for each individual system. Thus, in a hidden variable the-ory, the systems that are identical at the quantum mechanical level differ at the hiddenvariable level, specified by different hidden variable configurations. A useful analogyis with that of classical statistical mechanics, where systems that are identical at themacroscopic level, say in a microcanonical ensemble each having the same energy, aredifferent at the microscopic level, with different positions and momenta.Let us call the hidden variable λ . Since we are considering hidden variable theo-ries in general, λ can be anything: complex numbers, vectors, matrices etc. We nextadd the constraint upon the hidden variable theory that it reproduces the predictions ofQuantum Mechanics, as we know empirically that Quantum Mechanics is correct(innon-relativistic domain). Since Quantum Mechanics gives probabilistic results whichcan be verified only statistically, that is over an ensemble of identically prepared sys-tems, we require a hidden variable theory to reproduce the same probabilities by aver-aging over λ . That is, specification of λ associated with each system in the ensemblewill first determine the result of measurement on that particular system(which QuantumMechanics does not give), and then we integrate over all systems in the ensemble.Let a certain fraction of the ensemble have the hidden variable configuration λ , acertain fraction have the hidden variable configuration λ ...and so on. Thus, one candefine the probability distribution of λ over the entire ensemble. In general the distri-bution can depend on many different factors, but we will consider only the distributionconditioned over the ensemble chosen and the measurement chosen, as these two arethe only variables controlled by an experimenter, i.e p ( λ || ψ (cid:105) , M ) where the ensembles specified by the ket | ψ (cid:105) and M = { E i } is a collection of POVMs defining the mea-surement that will be performed on the ensemble. To be a valid probability distribution,it must be normalised (cid:90) Λ dλp ( λ || ψ (cid:105) , M ) = 1 (2.1)where Λ is the set of all possible λ s.We do not however, impose the requirement of determinism at the hidden variable, sothat specifying the hidden variable state λ of the system does not necessarily give us aresult with certainty. Let the result of measurement for each individual system in theensemble be characterised by p ( k | λ, | ψ (cid:105) , M ) . If p ( k | λ, | ψ (cid:105) , M ) = p ( k | λ, | ψ (cid:105) , M ) forall possible combinations of k , λ , | ψ (cid:105) and M , the hidden variable theory is deterministic.Since upon measurement on a particular system, we are certain to get one of the possibleresults, we have (cid:88) k p ( k | λ, | ψ (cid:105) , M ) = 1 (2.2)Finally, the sets {| ψ (cid:105)} and { M } for which the hidden variable theory is valid mustbe specified if it is not a general theory. For example hidden variable theories are oftenrestricted to certain number of dimensions in Hilbert space or to projective measure-ments.Given these, the probability of getting an outcome k upon having prepared a purestate ensemble denoted by | ψ (cid:105) and performing a measurement M , is p ( k || ψ (cid:105) , M ) = (cid:90) Λ dλp ( k || ψ (cid:105) , M, λ ) p ( λ || ψ (cid:105) , M ) (2.3)which is the sum over the entire ensemble.Hence we have the following definition, Definition 2.1.0.1. A hidden variable reformulation of Quantum Mechanics definesthe following:1. Λ which is the set of all possible λ s, called the ontic space [16].2. The probability distribution p ( λ || ψ (cid:105) , M ) , called the density function [16], satisfyingconstraint 2.1. . The probability distribution p ( k | λ, | ψ (cid:105) , M ) , called the response function [16], satis-fying constraint 2.2.over a set of preparations defined by {| ψ (cid:105)} and measurements { M } such that the av-erage probability of getting an outcome over the ensemble using relation 2.3 matcheswith that predicted by Quantum Mechanics. Note that mixed states have not been considered yet, but the extension is simple.Consider a mixed state ρ = (cid:80) i c i | a i (cid:105)(cid:104) a i | where (cid:80) i c i = 1 . Then, we have p ( λ | ρ, M ) = (cid:88) i p ( λ | ρ, | a i (cid:105) , M ) p ( | a i (cid:105)| ρ, M ) (2.4) = (cid:88) a i p ( λ | ρ, | a i (cid:105) , M ) c i (2.5) = (cid:88) a i p ( λ || a i (cid:105) , M ) c i (2.6)where, from the second line to third, we have assumed p ( λ | ρ, | a i (cid:105) , M ) = p ( λ || a i (cid:105) , M ) ,that is, the distribution of hidden variables in a pure state ensemble is independent ofwhich ρ the pure state is a part of. This was already implicitly assumed in eqn. 2.1, forthe expression p ( λ || ψ (cid:105) , M ) to make meaning.Further, we have p ( k | λ, ρ, M ) = (cid:88) a i p ( k | λ, ρ, | a i (cid:105) , M ) p ( | a i (cid:105)| λ, ρ, M ) (2.7) = (cid:88) a i p ( k | λ, | a i (cid:105) , M ) p ( | a i (cid:105)| λ, ρ, M ) (2.8)where in the second line we have assumed p ( k | λ, ρ, | a i (cid:105) , M ) = p ( k | λ, | a i (cid:105) , M ) . Thiswas implicitly assumed in eqn. eqn. 2.2, that the probability of getting a result given thehidden variable description of an individual system which is part of a pure state ensem-ble, is independent of which ρ the pure state is a part of. The expression p ( | a i (cid:105)| λ, ρ, M ) depends on the properties of the particular hidden variable theory.6sing eqns. 2.6 and 2.8, we have p ( k | ρ, M ) = (cid:90) Λ dλp ( k | ρ, M, λ ) p ( λ | ρ, M ) (2.9) = (cid:90) Λ dλ (cid:88) a i ,a j a i p ( λ || a i (cid:105) , M ) p ( k | λ, | a j (cid:105) , M ) p ( | a j (cid:105)| λ, ρ, M ) (2.10)For the hidden variable theory to match experimental results, the LHS of eqn. 2.9must be equal to the value given by Quantum Mechanics. MI is the assumption of noncorrelation between the hidden variables λ and the mea-surement M chosen by the experimenter. So all expressions containing terms like p ( λ || ψ (cid:105) , M ) in section 2.1 for theories satisfying this assumption should be replaced by p ( λ || ψ (cid:105) ) . Historically, the EPR argument [8] made the assumption that experimenterswere "free" to choose whichever measurement to perform on their test system, regard-less of the past. When John Bell derived his theorem[15], he replaced the "freedom ofexperimenter" assumption with non-correlation of hidden variables and measurementchoices. This latter assumption is MI, and is infact stronger than the assumption of"experimenter’s freedom". It also rules out retrocausality [21].It is worth having a look at how the assumption enters the Quantum foundation lit-erature, namely in Bell’s theorem, which reveals how crucial it is to the central resultsin the field. Bell derived his inequalities twice - first in 1964[15], under the assumptionof locality, determinism and MI, and again in 1976[21], under the assumption of localcausality and MI. In the next subsection we explain the standard Bell scenario commonto both the theorems, and how the assumption of MI is formulated; for the completederivations please refer to [15][21].Finally, the formulation of MI in Ontological models framework [16] which hasbeen extensively used to prove various theorems recently [23] [24] [25] [26], is dis-cussed. 7 .2.1 Measurement Independence in Bell’s 1964 Theorem For this section as well as the next, the scenario is the following: Consider two partiesAlice and Bob, each possessing a particle entangled in spin singlet state | ψ (cid:105) singlet = | (cid:105)| (cid:105)−| (cid:105)| (cid:105)√ with the other, and spatially separated by large distances. Let Alice measurethe spin of her particle along ˆ a ( ˆ σ · ˆ a ) and Bob measure the spin of his particle along ˆ b ( ˆ σ · ˆ b ), where ˆ k is a unit vector. The question Bell asks is if correlation betweenthe measurement results generated by Alice and Bob upon repeatedly performing suchmeasurements, can be reproduced by a hidden variable theory which satisfies someplausible assumptions. The expectation value (cid:104) ˆ σ · ˆ a ⊗ ˆ σ · ˆ b (cid:105) is calculated as, (cid:104) ˆ σ · ˆ a ⊗ ˆ σ · ˆ b (cid:105) = (cid:90) dλA (ˆ a, λ ) B (ˆ b, λ ) ρ ( λ ) = − ˆ a. ˆ b (2.11)where A (ˆ a, λ ) , B (ˆ b, λ ) ∈ {− , +1 } are the values obtained by Alice and Bob uponmeasurement, given that the probability distribution ρ ( λ ) of the hidden variable λ sat-isfies (cid:90) ρ ( λ ) dλ = 1 (2.12)where, λ ∈ Λ (2.13)If ˆ a = ˆ b , (cid:104) ˆ σ · ˆ a ⊗ ˆ σ · ˆ b (cid:105) = − from eqn. 2.11. Adding to eqn. 2.12 (cid:90) dλρ ( λ )(1 + A (ˆ a, λ ) B (ˆ b, λ )) = 0 (2.14) ⇒ A (ˆ a, λ ) B (ˆ a, λ ) = − (2.15) ⇒ A (ˆ a, λ ) = − B (ˆ a, λ ) (2.16)Assuming this, and that λ and ˆ a , ˆ b are uncorrelated, we have for the general case ˆ a (cid:54) = ˆ b , as assumed in the paper: (cid:104) ˆ σ · ˆ a ⊗ ˆ σ · ˆ b (cid:105) = − (cid:90) dλA (ˆ a, λ ) A (ˆ b, λ ) ρ ( λ ) (2.17)If however, λ and ˆ a , ˆ b were correlated, then ˆ a = ˆ b would have corresponded to λ ∈ Λ d ,8igure 2.1: Local hidden variables in Bell scenariowhere Λ d ⊂ Λ . For a given λ ∈ Λ \ Λ d , it will not be possible for Alice and Bobto measure spins along ˆ a direction for their particles simultaneously, and the relation A (ˆ a, λ ) = B (ˆ a, λ ) will be ill-defined for such λ s. Further one will have to introducea distribution function for λ which is correlated with the measurement directions. Ingeneral, ρ ( λ | ˆ a, ˆ b ) (cid:54) = ρ ( λ | ˆ a, ˆ a ) (2.18) Bell considers local hidden variables, dividing them into non-hidden parts (a,b,c) whichdescribe the experimental setup, and ( µ, υ, λ ) , the local hidden variables that are hid-den(refer Fig. 2.1). c lists the non-hidden variables in the overlap of the backwardlight cones of Alice and Bob, and a and b list non-hidden variables in the remainderof the light cones. Similarly, λ lists the hidden variables in the overlap, and µ and υ list hidden variables in the remainders. Space-time regions A and B point to the mea-surement events taking place on 2 different instruments M A and M B respectively. Theassumption of MI is formulated as: p ( λ | a, b, c ) = p ( λ | a (cid:48) , b, c ) = p ( λ | a, b (cid:48) , c ) = p ( λ | a (cid:48) , b (cid:48) , c ) (2.19)The above equation says that the settings of instruments M A and M B , denoted by vari-ables a and b respectively, are uncorrelated with the hidden variable λ in the overlap of9ackward light cones of A and B. Assumption of non-correlation between other possi-ble pairs of hidden and non-hidden variables, for example between µ and (a,b,c), is notrequired. The Ontological model framework considers the experimental probability of getting anoutcome k , given a preparation ρ and measurement procedure M : p ( k | ρ, M ) = (cid:90) p ( k | ρ, M, λ ) p ( λ | ρ, M ) dλ (2.20)where, (cid:90) p ( λ | ρ, M ) dλ = 1 (2.21)which is simply summing up the probabilites over λ . Ontological model frameworkfurther assumes: p ( k | ρ, M, λ ) = p ( k | M, λ ) (2.22) p ( λ | ρ, M ) = p ( λ | ρ ) (2.23)The first assumption is simply to provide a framework for distinguishing ψ epistemicand ψ ontic theories [16]( discussed in section 2.3.2 ), while the second assumption isMI. The Ontological models framework [16] was introduced to tackle an unresolved de-bate since the early days in the subject, whether the wavefunction represents an experi-menter’s knowledge about the system, or is itself a property of the system. Considereda matter of philosophy by most physicists, it is a great achievement of the authors of[16] to have provided a clear mathematical formulation of the issue, which was sub-sequently taken up in [23] to give what is considered by many[27] a central result inQuantum Foundations, the PBR theorem. 10he Ontological models framework was introduced in section 2.2.3. In particular,it assumes MI and eqn. 2.22. Here we consider only the definition of " ψ ontic" and " ψ epistemic" models.To motivate the definitions of ψ ontic and epistemic, one has to consider Einstein’sargument for incompleteness of Quantum Mechanics, which is most famously put downin the EPR paper[8]. However one must note Einstein himself did not write the EPRpaper ( Podolsky did), and Einstein was not satisfied with the outcome, writing in a letterto Schrodinger[28] shortly after the paper was published, "For reasons of language, thiswas written by Podolsky after many discussions. But still it has not come out as wellas I really wanted; on the contrary, the main point was, so to speak, buried by theerudition.". He reproduced his own version of the argument in the same letter[28], alsolater in the paper ’Physics and Reality’ [29], and in his autobiographical notes [19]. Wewill concern ourselves here with Einstein’s preferred argument, which will lead us moresimply to the distinction between ψ ontic and epistemic hidden variable theories. Note that in Einstein’s preferred version, it is not Quantum Mechanics, but the wavefunction,which is proven an incomplete description of system.
The following are assumed:1. Two spatially separated systems have separate real states.(separability)2. All interactions propagate at speed less than or equal to the speed of light.(locality)3. An experimenter’s choice of measurement is independent of his past.(free-will)4. A system’s real state is completely determined by its past.(determinism)Now let us consider two particles in a spin singlet state, | ψ (cid:105) = | (cid:105)| (cid:105)−| (cid:105)| (cid:105)√ . Let theparticles be handed over to Alice and Bob, who travel away from each other, so that anycommunication by means of light signals between them takes a certain amount of time.From assumption 1, let the real states of Alice’s particle be λ and of Bob’s particlebe λ . Now consider Alice making her decision to measure her particle’s spin along acertain direction after she has separated from Bob.11et Alice make a measurement along ˆ a direction. Then Bob’s particle’s wavefunc-tion will collapse to an eigenket in ˆ σ · ˆ a basis. From assumption 2, as Alice is far awayfrom Bob, her actions cannot exert a causal influence on his particle immediately. Thus,Bob’s particle remains in the state λ .From assumption 3, one can consider the case of Alice making a measurement along ˆ a (cid:48) direction, without changing any event in her past. From assumption 4, as Bob’s par-ticle’s past is unchanged, it still has the real state λ . Now however, Bob’s particle’swavefunction has collapsed to an eigenket in ˆ σ · ˆ a (cid:48) basis.Therefore, given the 4 assumptions above, we have two different wavefunctions(eigenketsin ˆ σ · ˆ a or ˆ σ · ˆ a (cid:48) basis) that describe the same real state λ of Bob’s particle. Thus, thewavefunction is an incomplete description of the system. Note however the relationship, λ described by 2 different wavefunctions ⇒ wavefunction is incomplete description of systemis strict. That is, λ described by only one wavefunction (cid:54)⇒ wavefunction is complete description of system.This point will become clear in the next section 2.3.2. To formalise the issue of incompleteness, we have to first define the real state of a sys-tem.
Definition 2.3.2.1.
The real state of the system λ is the state, which if known, gives usthe most complete knowledge of measurement results on that system. Thus, if a systemhas a wavefunction | ψ (cid:105) and a measurement M is performed on it, then the probability f getting the k th outcome given the real state λ , will have the following property: p ( k | λ, M ) = p ( k | λ, x, M ) (2.24) where x is any other variable. Specifically, p ( k | λ, M ) = p ( k | λ, | ψ (cid:105) , M ) (2.25)Thus, if λ is the real state of the system, eqn. 2.3 reduces to p ( k || ψ (cid:105) , M ) = (cid:90) Λ dλp ( k || ψ (cid:105) , M, λ ) p ( λ || ψ (cid:105) , M ) (2.26) = (cid:90) Λ dλp ( k | M, λ ) p ( λ || ψ (cid:105) , M ) (2.27)further, since ontological models assume MI, (2.28) = (cid:90) Λ dλp ( k | M, λ ) p ( λ || ψ (cid:105) ) (2.29)Now we are in a position to define incompleteness. From section 2.3.1, we see thatthe wavefunction is an incomplete description of the system if more than one wavefunc-tion, say | ψ (cid:105) and | ψ (cid:105) , can correspond to the same real state λ of the system. Infact,this tells us something stronger than incompleteness: the wavefunction describes theobserver’s information about the system, and not the system itself. The following dis-cussion explains why.Consider the real state of the system to be defined by λ = ( λ , λ , λ , ....λ N ) , where λ i s are hypothetical variables which together describe the system. Now for a fixed λ ,if more than one wavefunction can be used to describe λ , | ψ (cid:105) cannot have a one to onerelation with any of the λ i s that form the description of λ . Thus the wavefunction doesnot describe the system itself, even incompletely.It instead describes the information different observers have about the system. Ananalogy with classical statistical mechanics is useful here. Consider a particle in a gasdefined by the microstate ( p , q ) in the phase space, where p > and the Hamiltonian H = p + q . Then the particle can be regarded as part of the ensemble whose momenta13 are positive, or as part of the ensemble(microcanonical) whose energy is E = p + q .The two ensembles will be described by two different density functions ρ and ρ (cid:48) , withthe crucial property that they will overlap atleast on ( p , q ) . As different observerswill describe the particle by different ensembles, the density operator describes theobserver’s incomplete information about the system. Similarly, as the wavefunction | ψ (cid:105) represents a pure state ensemble, assigning different wavefunctions to the same systemamounts to describing the same system by different ensembles, based on incompleteinformation about the system.Thus we have the following definitions, Definition 2.3.2.2. If p ( λ || ψ (cid:105) ) p ( λ || ψ (cid:105) ) > (2.30) for some λ ∈ Λ , then the model is ψ epistemic and | ψ (cid:105) describes an observer’s infor-mation about the system. Definition 2.3.2.3. If p ( λ || ψ (cid:105) ) p ( λ || ψ (cid:105) ) = 0 ∀ λ ∈ Λ (2.31) then, the model is ψ ontic and | ψ (cid:105) describes the system, either completely or incom-pletely. Definition 2.3.2.4. If p ( λ || ψ (cid:105) ) = δ ( λ − λ | ψ (cid:105) ) (2.32)(2.33) then | ψ (cid:105) completely describes the system, and the model is called ψ ontic complete . Definition 2.3.2.5. If λ = ( λ , λ , λ , ....λ N ) (2.34) and p ( λ || ψ (cid:105) ) = δ ( λ − λ | ψ (cid:105) ) × p ( λ , λ , λ , ....λ N , || ψ (cid:105) , λ ) (2.35)14 hen | ψ (cid:105) incompletely describes the system, and the model is called ψ ontic incomplete . ψ epistemic models offer an intuitive explanation of the imperfect indistinguishabilityof 2 non-orthogonal quantum states. To see this, we first prove a lemma. Lemma 2.3.3.1.
For ontological models, two orthogonal kets | ψ (cid:105) and | ψ ⊥ (cid:105) in Hilbertspace dimension d N do not have overlapping distributions over Λ . ( even if the modelis ψ epistemic)Consider a measurement basis M consisting of the projector | ψ ⊥ (cid:105)(cid:104) ψ ⊥ | .From eqn. 2.29, we have p ( ψ ⊥ || ψ (cid:105) , M ) = |(cid:104) ψ ⊥ | ψ (cid:105)| = 0 (2.36) = (cid:90) Λ dλp ( ψ ⊥ | M, λ ) p ( λ || ψ (cid:105) ) = 0 (2.37) and p ( ψ ⊥ || ψ ⊥ (cid:105) , M ) = |(cid:104) ψ ⊥ | ψ ⊥ (cid:105)| = 1 (2.38) = (cid:90) Λ dλp ( ψ ⊥ | M, λ ) p ( λ || ψ ⊥ (cid:105) ) = 1 (2.39) Eqn. 2.37 implies the supports of p ( ψ ⊥ | M, λ ) , and p ( λ || ψ (cid:105) ) are disjoint.( Support of aprobability distribution p ( λ | x )( p ( y | λ )) is defined as the set { λ | p ( λ | x )( p ( y | λ )) > } ).But from eqn. 2.39 we also know that the support of p ( λ || ψ ⊥ (cid:105) ) is a subset of sup-port of p ( ψ ⊥ | M, λ ) , as p ( λ || ψ ⊥ (cid:105) ) is a normalised distribution. Hence p ( λ || ψ (cid:105) , M ) and p ( λ || ψ ⊥ (cid:105) , M ) have disjoint supports. So in the case of ontological models, only non-orthogonal kets can share an overlapover Λ space. Now suppose one is handed a system but not told the ket that describes it.Only the information that the ket is either | ψ (cid:105) or | φ (cid:105) where < |(cid:104) φ | ψ (cid:105)| < is provided.To determine which ket it is, one measures it in the basis M = {| φ (cid:105)(cid:104) φ | , | φ ⊥ (cid:105)(cid:104) φ ⊥ |} . If15ne gets the result φ then one says the ket was | φ (cid:105) and if one gets φ ⊥ then one says theket was | ψ (cid:105) . The process is not error-free: | ψ (cid:105) can also give the result φ , with probabil-ity |(cid:104) φ | ψ (cid:105)| .For an epistemic model, there is a finite probability that the hidden variable λ de-scribing the system is from the overlap region between | ψ (cid:105) and | φ (cid:105) in Λ space. In thatcase, given only the hidden variable λ , there is no way of ascertaining which ket, | ψ (cid:105) or | φ (cid:105) , it came from as the result of measurement for ontological models depends only on λ . Now one can ask how much of the error probability such a model can explain.Denoting by Λ γ the support of p ( λ | γ (cid:105) ) , the probability that upon preparing a systemin | ψ (cid:105) one gets a λ in the overlap of Λ φ and Λ ψ is (cid:82) Λ φ dλp ( λ || ψ (cid:105) ) . Definition 2.3.3.1.
The degree of epistemicity Ω( ψ, φ ) between | ψ (cid:105) and | φ (cid:105) in an onto-logical model is defined by the equation (cid:90) Λ φ dλp ( λ || ψ (cid:105) ) = Ω( ψ, φ ) |(cid:104) φ | ψ (cid:105)| (2.40)If Ω( ψ, φ ) = 1 , then the probability of making an error given that the ket wasactually | ψ (cid:105) is the same as the probability that the ket | ψ (cid:105) has a hidden variable λ in theregion Λ ψ ∩ Λ φ . Thus the model completely explains the errors in terms of overlap ofdistributions over Λ . Definition 2.3.3.2. If Ω( ψ, φ ) = 1 for all possible pairs | ψ (cid:105) and | φ (cid:105) , then the ontologi-cal model is called maximally epistemic . In section 2.1, we introduced the density function p ( λ | ρ, M ) , which depends on ρ . Canpreparing the same ρ by different procedures lead to different p ( λ | ρ, M ) ?16onsider the maximally mixed density operator in 2 dimension, ρ = 1 / | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) (2.41) = 1 / | + (cid:105)(cid:104) + | + |−(cid:105)(cid:104)−| ) (2.42)where ˆ σ · ˆ z = ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) and ˆ σ · ˆ x = ( | + (cid:105)(cid:104) + | + |−(cid:105)(cid:104)−| ) .From eqn. 2.41, we see one prepare ρ by having an equal number of | (cid:105) and | (cid:105) ketsin the ensemble, while from eqn. 2.42, we see the same ρ can be prepared by having anequal number of | + (cid:105) and |−(cid:105) kets in the ensemble. However, the second ensemble is different from the first, though they have the same description in terms of ρ . Definition 2.4.0.1.
Hidden variable models where the distribution p ( λ | ρ, M ) dependson detail beyond the density operator ρ are called preparation-contextual . A preparation in these models is specified by not just ρ , but the context of prepara-tion S P , leading to distribution p ( λ | ρ, S P , M ) . For example the context here is whetherone uses 2.41 or 2.42 to prepare one’s ensemble.One can easily extend the argument to pure states. Consider preparing a pure stateensemble of vertically polarized photons by two methods: a) Passing unpolarized lightthrough a polarizer with its transmission axis oriented vertically; and b) Passing un-polarized light through a polarizing prism (like Wollaston prism) with its optical axisoriented such as to give us two separate beams, one horizontally polarized and anothervertically, and selecting only the latter photons into our ensemble. Both methods give usthe same ensemble, but the method of preparation is different. A hidden variable modelwhich assigns different p ( λ || ψ (cid:105) , S P , M ) to the same ensemble depending on the con-text(method of preparation) can in principle exist, but has not been proposed so far [30].There can be contexts in not only defining preparation, but also measurement. Wealready know from the expression p ( k | ρ, M, λ ) that the response function depends onthe measurement basis in general. Consider an N dimensional Hilbert space where N > . Let us prepare a system in state say ρ and calculate the probability of gettinga measurement result corresponding to the projector | φ (cid:105)(cid:104) φ | , which equals tr ( ρ | φ (cid:105)(cid:104) φ | ) .17t the hidden variable level, the probability is p ( φ | ρ, M, λ ) , which must be integratedover Λ to give tr ( ρ | φ (cid:105)(cid:104) φ | ) .However, we have several choices for our basis to perform the measurement corre-sponding to | φ (cid:105)(cid:104) φ | . We can have the basis M = {| φ (cid:105)(cid:104) φ | , | φ ⊥ (cid:105)(cid:104) φ ⊥ | , | φ ⊥ (cid:105)(cid:104) φ ⊥ | , .... | φ ⊥ N − (cid:105)(cid:104) φ ⊥ N − |} (2.43)or M (cid:48) = {| φ (cid:105)(cid:104) φ | , | φ ⊥ (cid:105) (cid:48) (cid:104) φ ⊥ | (cid:48) , | φ ⊥ (cid:105) (cid:48) (cid:104) φ ⊥ | (cid:48) , .... | φ ⊥ N − (cid:105) (cid:48) (cid:104) φ ⊥ N − | (cid:48) } (2.44) Definition 2.4.0.2. If p ( φ | ρ, M, λ ) (cid:54) = p ( φ | ρ, M (cid:48) , λ ) in general for a hidden variablemodel, where M contains the projector | φ (cid:105)(cid:104) φ | , then the model is measurement contex-tual or Kochen-Specker contextual . It means that, given λ , the probability of getting an outcome at the hidden variablelevel corresponding to a projector | φ (cid:105)(cid:104) φ | depends on how the projective measurementis implemented, even though the operational probability tr ( ρ | φ (cid:105)(cid:104) φ | ) is the same. Onecan similarly define contexts for POVM elements as well.18 HAPTER 3Consequences of violating Measurement Independence
In this chapter we study the consequences of violating Measurement Independence tounderstand the assumption better. We begin by considering several theorems proved inthe ontological models framework and see if they still remain valid in the MD case.
For the case of ontological models, we have a number of results that severely con-strain epistemicity. For such models, Maroney[32] proved that maximal epistemicityfor Hilbert space dimension d ≥ is impossible, while the epistemic explanation ofindistinguishability as overlap over ontic states itself has been proven[24] as arbitrarilybad for certain quantum states in d ≥ . There are certain other results as well; thatmaximally epistemic ⇒ Kochen-Specker noncontextual [25], and maximally epistemic ⇔ Reciprocity ∩ Determinism [30]. It is natural to ask whether these and other resultscan be generalized over the broader class of measurement dependent(MD) models too.In the subsections we denote by Λ γ the support of p ( λ || γ (cid:105) ) Let us first consider Maroney’s result.
Theorem 3.1.1.
Maroney’s theorem:
For the class of ontological models, the degree ofepistemicity Ω( ψ, φ ) cannot equal to 1 for arbitrary states in Hilbert space dimensiongreater than or equal to 3. The argument considers three measurements M , M and M , and some states | a (cid:105) , | b (cid:105) , | c (cid:105) , | p (cid:105) , | m (cid:105) in Hilbert space of dimension 3. From M , it is concluded that Λ a ∩ Λ p ∩ Λ m = Λ c ∩ Λ p ∩ Λ m = ∅ (3.1)nd from M that Λ b ∩ Λ p ∩ Λ m = ∅ (3.2)Both these results are combined to yield (Λ a ∪ Λ b ∪ Λ c ) ∩ Λ p ∩ Λ m = ∅ (3.3)which is used to derive the final result. In an MD model however, the distributions overontic space change as measurements are changed, and equations 3.1 and 3.2 cannotbe combined to give 3.3. The same reasoning applies to generalizing this result todimensions greater than 3. Hence, Maroney’s theorem cannot be applied to MD models. To state the result of Barrett et al [24], we first need to define a few notions:
Definition 3.1.2.1.
The classical overlap between two states | ψ (cid:105) and | φ (cid:105) is defined as w C = 1 − / (cid:90) Λ dλ | p ( λ | ψ (cid:105) ) − p ( λ || φ (cid:105) ) | (3.4) Definition 3.1.2.2.
The quantum overlap between two states | ψ (cid:105) and | φ (cid:105) is defined as w Q = 1 − (cid:112) − |(cid:104) ψ | φ (cid:105)| (3.5) Theorem 3.1.2.
Barrett’s theorem : No maximally epistemic ontological model canreproduce the quantum predictions for a system of dimension d ≥ .Moreover, as the dimension of Hilbert space d → ∞ , the ratio of classical over quantumoverlap will tend to zero for atleast some pairs of quantum states. The argument considers the d + 1 mutually unbiased orthonormal bases of a d ( ≥ -dimensional Hilbert space [36], of which | c (cid:105) is an element of one such basis, and theother bases are {| e γi (cid:105)} , where i, γ ∈ { , , ....d } ( γ ranges over the bases and i over theelements). From PP-incompatibility[37] of {| c (cid:105) , | e αi (cid:105) , | e βj (cid:105)} , a measurement M having20utcomes { f , f , f , f } with the following properties is considered: (cid:90) Λ eαi p ( f | λ ) p ( λ || e αi (cid:105) ) dλ = 0 (3.6) (cid:90) Λ eβj p ( f | λ ) p ( λ || e βj (cid:105) ) dλ = 0 (3.7) (cid:90) Λ c p ( f | λ ) p ( λ || c (cid:105) ) dλ = 0 (3.8) (cid:90) Λ eαi p ( f | λ ) p ( λ || e αi (cid:105) ) dλ = (cid:90) Λ eβj p ( f | λ ) p ( λ || e βj (cid:105) ) dλ = (cid:90) Λ c p ( f | λ ) p ( λ || c (cid:105) ) dλ = 0 (3.9)From M , it is concluded that Λ e αi ∩ Λ e βj ∩ Λ c = ∅ (3.10)It is then further assumed that, | e αi (cid:105) and | e αj (cid:105) being orthogonal, Λ e αi ∩ Λ e αj = ∅ (3.11)The final result is derived using both eqn. 3.10 and 3.11. In a MD model however,eqn. 3.11 would actually correspond to a measurement M (cid:48) (cid:54) = M , where the outcomesare { e αi , e αj ... } . Two orthogonal states can have overlapping supports in such a model,depending on the measurement being performed (see 3.1.3.1 for full discussion). So,Barrett et al’s result also fails to be applicable. Ref. [25] contains the following theorem:
Theorem 3.1.3.
The following are true for ontological models in all dimensions ofHilbert space:(i) Maximally ψ -epistemic ⇒ Kochen-Specker noncontextual ∩ Determinism(ii) Preparation noncontextual ⇒ Maximally ψ -epistemic.the relationship strict for both.
21o check their validity in MD models, we must first generalize the notion of maxi-mal epistemicity appropriately.
Definition 3.1.3.1.
The degree of epistemicity Ω M ( ψ, φ ) between | ψ (cid:105) and | φ (cid:105) , states inHilbert space d N , is defined by (cid:90) Λ φ | M dλp ( λ || ψ (cid:105) , M ) = Ω M ( φ, ψ ) × |(cid:104) ψ | φ (cid:105)| (3.12) when measuring both in measurement basis M = {| φ (cid:105)(cid:104) φ | , | φ ⊥ (cid:105)(cid:104) φ ⊥ | , | φ ⊥ (cid:105)(cid:104) φ ⊥ | , ... | φ ⊥ N − (cid:105)(cid:104) φ ⊥ N − |} and where Λ φ | M is the support of p ( λ || φ (cid:105) , M ) . While generalizing this notion, it is important to consider the appropriate measure-ment M . Two orthogonal quantum states may have finite overlap over ontic space when M does not contain a projector corresponding to either of the states, but nothing can beinferred from this overlap about indistinguishability. For a maximally epistemic model, Ω M ( φ, ψ ) = 1 for arbitrary | ψ (cid:105) and | φ (cid:105) for all M that contain projector of one of them. Validity of relation i)
We prove the following,
Theorem 3.1.4.
For a measurement dependent hidden variable model,Maximally ψ epistemic ⇒ Determinism (3.13)
Maximally ψ epistemic (cid:54)⇒ Kochen-Specker noncontextual (3.14)
Proof : Maximal epistemicity for measurement dependent models means (cid:82) Λ | M p ( | φ (cid:105)| λ, M ) p ( λ || ψ (cid:105) , M ) dλ = (cid:82) Λ φ | M p ( | φ (cid:105)| λ, M ) p ( λ || ψ (cid:105) , M ) dλ , which implies p ( | φ (cid:105)| λ, M ) = 0 almost everywhere on Λ | M \ Λ φ | M , where M consists of | φ (cid:105)(cid:104) φ | as one of its projectors. Thus, the model is deterministic.The argument is true for any other M (cid:48) containing | φ (cid:105)(cid:104) φ | ( it is pointless to discuss p ( | φ (cid:105)| λ, M (cid:48) ) without such context), however, as Λ φ | M (cid:54) = Λ φ | M (cid:48) , the model is notmeasurement non-contextual in general. ψ ontic deterministic, MD modelsare possible. An example is the following: A ψ ontic, deterministic MD model Let the measurement basis M = {| e i (cid:105)(cid:104) e i |} ,where i = 1 , ....n , for a n-dimensional Hilbert space system. Let us define, x = 0 (3.15) x i = |(cid:104) e i | ψ (cid:105)| for i > (3.16)Then the probability distribution of λ is defined to be p ( λ || ψ (cid:105) , M ) = |(cid:104) e i | ψ (cid:105)| for λ ∈ ( i − (cid:88) j =0 x j , i (cid:88) j =0 x j ) (3.17)The important thing to note here is λ is divided into n bins, each of length |(cid:104) e j | ψ (cid:105)| for j =1 , ..n . The probability density of λ in each zone is again |(cid:104) e j | ψ (cid:105)| . λ is normalized of-course: (cid:90) p ( λ ||| ψ (cid:105) , M ) dλ = n (cid:88) j =1 (cid:90) x j x j − p ( λ || ψ (cid:105) , M ) dλ (3.18) = n (cid:88) j =1 |(cid:104) e j | ψ (cid:105)| (3.19) = 1 (3.20)The response function, which explicitly depends on | ψ (cid:105) , is defined as p ( e i || ψ (cid:105) , λ, M ) = Θ( λ − i − (cid:88) j =0 x j ) − Θ( λ − i (cid:88) j =0 x j ) (3.21)where Θ is Heaviside Step function and x j ( | ψ (cid:105) , M ) .23he model reproduces Quantum Mechanics predictions: p ( e i || ψ (cid:105) , M ) = (cid:90) p ( e i || ψ (cid:105) , λ, M ) × p ( λ || ψ (cid:105) , M ) dλ (3.22) = (cid:90) (cid:80) ij =0 x j (cid:80) i − j =0 x j |(cid:104) e i | ψ (cid:105)| dλ (3.23) = |(cid:104) e i | ψ (cid:105)| (3.24) Validity of relation ii)
Now let us check (ii), by considering its contrapositive as done in [25]. Consider a 2-Dmodel not maximally epistemic, so there exist | ψ (cid:105) and | φ (cid:105) such that (cid:90) Λ | M p ( | φ (cid:105)| λ, M ) p ( λ || ψ (cid:105) , M ) dλ > (cid:90) Λ φ | M p ( | φ (cid:105)| λ, M ) p ( λ || ψ (cid:105) , M ) dλ where M = {| φ (cid:105)(cid:104) φ | , | φ ⊥ (cid:105)(cid:104) φ ⊥ |} . Thus there exists a set of finite measure Ω M which isdisjoint from Λ φ | M but overlaps with Λ ψ | M , such that λ ∈ Ω M ⇒ p ( | φ (cid:105)| λ, M ) > .Now consider two preparations ρ = ( | ψ (cid:105)(cid:104) ψ | + | ψ ⊥ (cid:105)(cid:104) ψ ⊥ | ) = I/ and ρ = ( | φ (cid:105)(cid:104) φ | + | φ ⊥ (cid:105)(cid:104) φ ⊥ | ) = I/ , with corresponding distributions p ( λ | ρ , M ) = { p ( λ || ψ (cid:105) , M )+ p ( λ || ψ ⊥ (cid:105) , M ) } and p ( λ | ρ , M ) = { p ( λ || φ (cid:105) , M )+ p ( λ | φ ⊥ (cid:105) , M ) } , we see that Λ ρ ∩ Ω M is a finite set while Λ ρ ∩ Ω M = ∅ .This is because, as in the specific case of ontological models, Ω M is disjoint fromboth Λ φ | M and Λ φ ⊥ | M , however it shares an overlap with Λ ψ | M . Hence the relation (ii)holds for MD models. The relation is strict again, as maximally epistemic but prepara-tion contextual models are possible. As an example, consider a modified MD KochenSpecker model : Modified KS Model I
Let the hidden variable be λ (cid:48) = ( λ, ˆ λ ) , where ˆ λ is a vector onthe surface of Bloch sphere, and λ is a discrete variable taking values λ i and λ i ⊥ . λ and24 λ are correlated. The model, where M = {|| i (cid:105)(cid:104) i | , | i ⊥ (cid:105)(cid:104) i ⊥ |} is defined as : p ( λ i ( i ⊥ ) || ψ (cid:105) , M ) = 1 / (3.25) p (ˆ λ || ψ (cid:105) , M, λ i ( i ⊥ ) ) = 2 /π × Θ(ˆ i (ˆ i ⊥ ) · ˆ λ ) × Θ( ˆ ψ · ˆ λ ) × ˆ ψ · ˆ λ (3.26) p ( λ (cid:48) || ψ (cid:105) , M ) = p (ˆ λ || ψ (cid:105) , M, λ k ) × p ( λ k || ψ (cid:105) , M ) (3.27) p ( | l (cid:105)(cid:104) l || λ (cid:48) , M ) = p ( | l (cid:105)(cid:104) l || λ k , M ) = δ lk (3.28)It can be checked that the model reproduces Quantum Mechanics predictions. p ( | i (cid:105)(cid:104) i ||| ψ (cid:105) , M ) = (3.29) (cid:90) Σ k p ( | i (cid:105)(cid:104) i || λ k , M ) × p (ˆ λ || ψ (cid:105) , M, λ k ) × p ( λ k || ψ (cid:105) , M ) d ˆ λ (3.30) = (cid:90) Σ k δ ik × /π × Θ(ˆ k · ˆ λ ) × Θ( ˆ ψ · ˆ λ ) × ˆ ψ · ˆ λ × d ˆ λ (3.31) = (cid:90) /π × Θ(ˆ i · ˆ λ ) × Θ( ˆ ψ · ˆ λ ) × ˆ ψ · ˆ λ × d ˆ λ (3.32) = |(cid:104) ψ | i (cid:105)| (3.33)The model is maximally epistemic as follows: Consider | ψ (cid:105) and | φ (cid:105) , measured in M = {| ψ (cid:105)(cid:104) ψ | , | ψ ⊥ (cid:105)(cid:104) ψ ⊥ |} . Then p ( λ ψ ⊥ , ˆ λ || ψ (cid:105) , M ) = 0 , and p ( λ ψ , ˆ λ || ψ (cid:105) , M ) > on ahemisphere with ˆ ψ at its center. So, the overlap over ontic space with | φ (cid:105) is (cid:90) p ( λ ψ , ˆ λ || φ (cid:105) , M ) d ˆ λ (3.34) = (cid:90) /π × Θ( ˆ ψ · ˆ λ ) × Θ( ˆ φ · ˆ λ ) × ˆ φ · ˆ λ × d ˆ λ (3.35) = |(cid:104) ψ | φ (cid:105)| (3.36)To check preparation contextuality, consider ρ = | (cid:105)(cid:104) | + | (cid:105)(cid:104) | and ρ = ( | π/ , (cid:105)(cid:104) π/ , | + | π/ , π (cid:105)(cid:104) π/ , π | ) , where | θ, φ (cid:105) = cos( θ/ | (cid:105) + e iφ sin( θ/ | (cid:105) .Though ρ = ρ , it can easily be checked that for M = {| (cid:105)(cid:104) | , | (cid:105)(cid:104) |} , Λ ρ | M ⊂ Λ ρ | M . Before stating Ballentine’s result, we define:25 efinition 3.1.4.1.
The core ξ k | M of a response function p ( k | λ, M ) is defined as the set { λ | p ( k | λ, M ) = 1 } . Definition 3.1.4.2.
An ontological model satisfies reciprocity , or is reciprocal, if Λ ψ = ξ ψ | M for all M that contain | ψ (cid:105)(cid:104) ψ | as a projector. Now Ballentine’s result is;
Theorem 3.1.5.
The following relation holds for ontological models in all dimensionsof Hilbert space:
M aximally epistemic ⇔ Determinism ∩ Reciprocity (3.37)We first generalise the notion of reciprocity for MD models
Definition 3.1.4.3.
A MD hidden variabel model is reciprocal if the following holds Λ ψ | M = ξ ψ | M (3.38) for all M that contain | ψ (cid:105)(cid:104) ψ | as a projector, and where Λ ψ | M is the support of thedistribution p ( λ || ψ (cid:105) , M ) . From the arguments leading to eqn. 3.13 we can immediately confirm the forwardimplication of 3.37. The converse also holds true, as all contributions to (cid:82) Λ p ( φ | λ, M ) p ( λ || ψ (cid:105) , M ) dλ must come from Λ φ | M . Hence the relation is true in MDmodels. et al’s results[38] We first define a few notions used in their paper,26 efinition 3.1.5.1.
Upon performing a measurement M = {| φ (cid:105)(cid:104) φ | , | φ (cid:105)(cid:104) φ | , .... | φ N (cid:105)(cid:104) φ N |} on state | ψ (cid:105) , the randomness in an ontological model valid in d N , in occurrence of i th result( corresponding to | φ i (cid:105)(cid:104) φ i | is defined as : I O ( ψ, φ i ) = (cid:90) Λ r | M dλp ( φ i | λ, M ) × p ( λ || ψ (cid:105) ) (3.39) where Λ r | M = Λ ψ ∩ ( S φ i | M \ C φ i | M ) (3.40) and S φ i | M and C φ i | M are defined as: λ ∈ C φ i | M ⇔ p ( φ i | λ, M ) = 1 (3.41) λ ∈ S φ i | M ⇔ p ( φ i | λ, M ) > (3.42) Definition 3.1.5.2.
Upon performing a measurement M = {| φ (cid:105)(cid:104) φ | , | φ (cid:105)(cid:104) φ | , .... | φ N (cid:105)(cid:104) φ N |} on state | ψ (cid:105) , the randomness in Quantum Mechanics in occurrence of i th result( cor-responding to | φ i (cid:105)(cid:104) φ i | is defined as : I Q ( ψ, φ ) = |(cid:104) ψ | φ (cid:105)| (3.43)The paper contains the following theorem: Theorem 3.1.6.
The order of randomness of ontological reciprocal models is equalto that of Quantum Mechanics for arbitrary states and measurements in Hilbert spacedimension d ≥ , assuming a basis independent measure of degree of epistemicity[32]. We define randomness for MD hidden variable models in section 3.3.3, but hereonly note as sufficient to prove their result as not valid for MD case, that an assumptionof theirs in deriving their result is to assume the validity of Maroney’s theorem. As wehave seen, this cannot be maintained in MD models, and thus their argument fails.In section 3.3.3, we show how a MD model can be reciprocal and have zero ran-domness. 27ence, we see that not all theorems related to epistemicity are valid once the as-sumption of measurement independence is relaxed. In particular, none of the theoremsthat rule out maximal epistemicity in d ≥ can be extended to cover MD models, a factthat we exploit in section 3.3. Brans replaces eqn.2.11 (cid:104) ˆ σ · ˆ a ⊗ ˆ σ · ˆ b (cid:105) = (cid:90) dλA (ˆ a, λ ) B (ˆ b, λ ) ρ ( λ ) with (cid:104) ˆ σ · ˆ a ⊗ ˆ σ · ˆ b (cid:105) = (cid:90) dλA (ˆ a, λ ) B (ˆ b, λ ) ρ ( λ | ˆ a, ˆ b ) (3.44)so that λ and ˆ a , ˆ b are now correlated.A simple formulation of the Brans model is as follows. Consider λ replaced by ( λ (cid:48) i , λ (cid:48) j , ˆ A , ˆ B ) where λ (cid:48) i , λ (cid:48) j are the parts of hidden variable describing the 2 particles, and ˆ A , ˆ B are the parts of hidden variable that determine the measurement choices. Then, ρ ( λ (cid:48) i , λ (cid:48) j , ˆ A, ˆ B | ˆ a, ˆ b ) = δ ( ˆ A − ˆ a ) × δ ( ˆ B − ˆ b ) |(cid:104) ψ singlet | ( | i (cid:105) ˆ A ⊗ | j (cid:105) ˆ B ) | (3.45)where i, j ∈ { + , −} and | k (cid:105) ˆ A ( ˆ B ) denotes an eigenstate of ˆ σ · ˆ A ( ˆ B ) A ( λ, ˆ a ) = A ( λ (cid:48) i ) = i × (3.46) B ( λ, ˆ b ) = B ( λ (cid:48) j ) = j × (3.47)The model reproduces Quantum Mechanical correlations:28 ˆ σ · ˆ a ⊗ ˆ σ · ˆ b (cid:105) = (cid:90) dλA (ˆ a, λ ) B (ˆ b, λ ) ρ ( λ | ˆ a, ˆ b ) (3.48) = (cid:90) (cid:88) ij A ( λ (cid:48) i ) B ( λ (cid:48) j ) ρ ( λ (cid:48) i , λ (cid:48) j , ˆ A, ˆ B | ˆ a, ˆ b ) d ˆ Ad ˆ B (3.49) = (cid:90) (cid:88) ij i.j.δ ( ˆ A − ˆ a ) .δ ( ˆ B − ˆ b ) . |(cid:104) ψ singlet | ( | i (cid:105) ˆ A ⊗ | j (cid:105) ˆ B ) | d ˆ Ad ˆ B (3.50) = (cid:88) ij i.j. |(cid:104) ψ singlet | ( | i (cid:105) ˆ A ⊗ | j (cid:105) ˆ B ) | (3.51) = |(cid:104) ψ singlet | ( | + (cid:105) ˆ A ⊗ | + (cid:105) ˆ B ) | − |(cid:104) ψ singlet | ( | + (cid:105) ˆ A ⊗ |−(cid:105) ˆ B ) | − |(cid:104) ψ singlet | ( |−(cid:105) ˆ A ⊗ | + (cid:105) ˆ B ) | + |(cid:104) ψ singlet | ( |−(cid:105) ˆ A ⊗ |−(cid:105) ˆ B ) | (3.52)The model satisfies Bell’s locality condition and determinism, from eqns. 3.46 and3.47 respectively. But it does not satisfy MI from 3.45. Let us find out how the mea-surement choices ˆ a and ˆ b are correlated with the hidden variables λ (cid:48) i and λ (cid:48) j describingthe two particles. From eqn. 3.45, ρ ( λ (cid:48) i , λ (cid:48) j | ˆ a, ˆ b ) = (cid:90) d ˆ Ad ˆ Bρ ( λ (cid:48) i , λ (cid:48) j , ˆ A, ˆ B | ˆ a, ˆ b ) (3.53) = (cid:90) d ˆ Ad ˆ Bδ ( ˆ A − ˆ a ) × δ ( ˆ B − ˆ b ) |(cid:104) ψ singlet | ( | i (cid:105) ˆ A ⊗ | j (cid:105) ˆ B ) | (3.54) = |(cid:104) ψ singlet | ( | i (cid:105) ˆ a ⊗ | j (cid:105) ˆ b ) | (3.55)Now let’s find how each individual particle is correlated with the measurement29hoices, ρ ( λ (cid:48) i | ˆ a, ˆ b ) = (cid:88) j ρ ( λ (cid:48) i , λ (cid:48) j | ˆ a, ˆ b ) (3.56) = (cid:88) j |(cid:104) ψ singlet | ( | i (cid:105) ˆ a ⊗ | j (cid:105) ˆ b ) | (3.57) = tr ( | ψ singlet (cid:105)(cid:104) ψ singlet || i (cid:105) ˆ a (cid:104) i | ˆ a ⊗ I ) (3.58)and similarly for ρ ( λ (cid:48) j | ˆ a, ˆ b ) . We thus find that the particles are correlated only withthe local measurement choices. That is, Alice’s particle’s hidden variable state is cor-related with her choice and has not correlation with Bob’s choice of measurement, andvice versa.Now we generalise this model to cover arbitrary perparations and measurements.The model was first generalised by MJW Hall for arbitrary preparations and projectivemeasurements[34] having d × d outcomes, where d and d are integers. d N - The gener-alised Brans Model For an N dimensional system in Hilbert space, measuring ρ with POVM elements M = { E , E , E , .....E N ......E X } where (cid:80) Xi =1 E i = ˆ I , the Generalized Brans model is asfollows : λ ∈ { λ , λ , ...., λ X } (3.59) p ( λ j | ρ, M ) = tr ( ρE j ) (3.60) p ( k | λ j , M ) = δ kj (3.61)Note that a POVM in general can have any number of measurement results, even30nfinite[35], so X is not restricted by N . The model satisfies Born rule: p ( k | ρ, M ) = (cid:88) j p ( k | λ j , M ) p ( λ j | ρ, M ) (3.62) = (cid:88) j δ kj tr ( ρE j ) (3.63) = tr ( ρE k ) (3.64) We introduced the notion of degree of epistemicity Ω( ψ, φ ) between two kets | ψ (cid:105) and | φ (cid:105) for ontological models in section 2.3.3. We also generalised the notion to measure-ment dependent models in 3.1.3.1, which we apply to the model. Theorem 3.3.1.
The generalised Brans model is maximally epistemic in arbitrary di-mensions of Hilbert space.
Proof : Consider two pure states | ψ (cid:105) and | φ (cid:105) in d N , satisfying |(cid:104) φ | ψ (cid:105)| (cid:54) = 0 . To distin-guish between | ψ (cid:105) and | φ (cid:105) , we measure | ψ (cid:105) in an orthonormal basis M = {| φ (cid:105)(cid:104) φ | , | φ ⊥ (cid:105)(cid:104) φ ⊥ | , | φ ⊥ (cid:105)(cid:104) φ ⊥ | , .... | φ ⊥ N − (cid:105)(cid:104) φ ⊥ N − |} . Now, from eqn. 3.60 p ( λ || φ (cid:105) , M ) = |(cid:104) φ | φ (cid:105)| = 1 (3.65) p ( λ || ψ (cid:105) , M ) = |(cid:104) ψ | φ (cid:105)| (3.66) From eqn. 3.12 replacing (cid:82) dλ by (cid:80) i p ( λ || ψ (cid:105) , M ) = |(cid:104) ψ | φ (cid:105)| × Ω M ( ψ, φ ) (3.67) ⇒ Ω M ( ψ, φ ) = 1 (3.68) One may note maximal epistemicity is not possible for ontological models for d N > .One may consider the model in context of a POVM measurement that does not make31he error of misidentification. Consider a POVM M with elements E = √
21 + √ × | (cid:105)(cid:104) | (3.69) E = √
21 + √ × |−(cid:105)(cid:104)−| (3.70) E = I − E − E (3.71)designed to distinguish between | (cid:105) and | + (cid:105) . From eqn. 3.60 , we have p ( λ || (cid:105) , M ) = p ( λ || + (cid:105) , M ) > (3.72) p ( λ || (cid:105) , M ) = p ( λ || + (cid:105) , M ) = 0 (3.73) p ( λ || (cid:105) , M ) = p ( λ || + (cid:105) , M ) > (3.74)The measurement fails to distinguish between | (cid:105) and | + (cid:105) when we get result E . Themodel explains it by saying that each time the measurement fails, the hidden variablestate was in the overlap of the two kets in λ . The question whether a model satisfies Preparation Independence (PI) is important asthe PBR theorem [23] rules out all epistemic models without using the assumption ofMI. However the result was proven for ontological models and the definition of PI wasrestricted. Here we generalise this notion for MD case first.
Definition 3.3.2.1.
A hidden variable(not necessarily ontological) model satisfies
Prepa-ration Independence if the following is true for a product state | ψ (cid:105)⊗| ψ (cid:105)⊗| ψ (cid:105) ... | ψ N (cid:105) , p ( λ || ψ (cid:105) ⊗ | ψ (cid:105) ⊗ | ψ (cid:105) ... | ψ N (cid:105) , M ) (3.75) = p ( λ , λ , λ ...λ N || ψ (cid:105) ⊗ | ψ (cid:105) ⊗ | ψ (cid:105) ... | ψ N (cid:105) , M )= p ( λ || ψ (cid:105) , M ) × p ( λ || ψ (cid:105) , M ) × p ( λ || ψ (cid:105) , M ) .... × p ( λ N || ψ N (cid:105) , M ) (3.76) where p ( λ i || ψ i (cid:105) , M ) denotes the distribution of | ψ i (cid:105) over Λ when a measurement M s being performed on a N dimensional tensor product state of which | ψ i (cid:105) is the i th part. Now let us test whether the generalised Brans model satisfies Preparation Indepen-dence(PI). Consider | ψ (cid:105) = | ψ (cid:105) ⊗ | ψ (cid:105) ⊗ | ψ (cid:105) ... | ψ N (cid:105) , where say | ψ i (cid:105) is a qubit, so thatthe entire state is a n system. Assume λ = ( λ , λ , λ ...λ n ) , where each λ k takes 2values λ k and λ k , which gives us n distinct values of λ , from λ to λ n . Let us mea-sure it in an entangled orthonormal basis M = {| φ (cid:105)(cid:104) φ | , | φ (cid:105)(cid:104) φ | , .... | φ n (cid:105)(cid:104) φ n |} . If λ i = ( λ a .....λ ck ...λ dn ) , then p ( λ i || ψ (cid:105) , M ) = p ( λ a .....λ ck ...λ dn || ψ (cid:105) , M ) = |(cid:104) φ i | ψ (cid:105)| (3.77)and (3.78) p ( λ ck || ψ (cid:105) k , M ) = (cid:88) λ j ,j (cid:54) = k p ( λ .....λ ck ...λ n || ψ (cid:105) , M ) (3.79) = (cid:88) m |(cid:104) φ m | ψ (cid:105)| ( n − terms) (3.80)Clearly, p ( λ i || ψ (cid:105) , M ) (cid:54) = p ( λ a || ψ (cid:105) k , M ) × ....p ( λ ck || ψ (cid:105) k , M ) × .....p ( λ dn || ψ (cid:105) k , M ) (3.81)and hence PI is not satisfied. In a recent result, a quantification was given for the amount of randomness contained inontological models[38]. The same question can be raised for MD case, and we gener-alise their notion to MD case first.
Definition 3.3.3.1.
In a hidden variable model valid in d N , if we measure in an or-thonormal basis M = {| φ (cid:105)(cid:104) φ | , | φ (cid:105)(cid:104) φ | , .... | φ N (cid:105)(cid:104) φ N |} on state | ψ (cid:105) , the randomness in occurence of i th result( corresponding to | φ i (cid:105)(cid:104) φ i | ) is defined as : ( ψ, φ i ) = (cid:90) Λ r | M dλp ( φ i | λ, M ) × p ( λ || ψ (cid:105) , M ) (3.82) where Λ r | M = Λ ψ | M ∩ ( S φ i | M \ C φ i | M ) (3.83) and S φ i | M and C φ i | M are defined as: λ ∈ C φ i | M ⇔ p ( φ i | λ, M ) = 1 (3.84) λ ∈ S φ i | M ⇔ p ( φ i | λ, M ) > (3.85)Now let us check its randomness. As this model is completely deterministic, fromeqn. 3.83 we find Λ r | M = ∅ . Hence there is no randomness in this model. However,the Generalized Brans model is also Reciprocal, which follows directly from eqn. 3.37which holds for MD models. This proves that reciprocal models in MD case have norestrictions on randomness unlike ontological models. A quantum channel is a communication channel through which information can betransferred. In ref. [39], Montina showed how ψ epistemic ontological models can beused to derive finite communication(FC) protocols for classical simulation of quantumchannels.The protocol Montina describes is the following. Let the quantum channel consistof Alice choosing a state | ψ (cid:105) and sending it to Bob, who then chooses to perform ameasurement M on it. Bob is unaware of | ψ (cid:105) and Alice is unaware of M . A classicalsimulation of this process using ontological models consists of the following. Alicechooses the state | ψ (cid:105) and generates a variable λ according to the probability distribu-tion p ( λ || ψ (cid:105) ) . She communicates the value of λ to Bob, who now simulates the mea-34urement M by the probability distribution p ( k | λ, M ) which gives the probability ofobtaining the k th outcome. The simulation is exact if, p ( k || ψ (cid:105) , M ) = (cid:90) Λ dλp ( k | λ, M ) p ( λ || ψ (cid:105) ) (3.86)This is ofcourse an ontological model. Since λ is a continuous variable in generalhowever, Alice needs to communicate an infinite amount of information to Bob. Onecan reduce this communication cost by the following procedure. Instead of Alice di-rectly communicating the value of λ , she communicates an amount of information thatallows Bob to generate λ according the distribution p ( λ || ψ (cid:105) ) . The minimum amount ofcommunication required per round for this is equal to the mutual information I ( λ : | ψ (cid:105) ) between λ and | ψ (cid:105) [39]. If N simulations are performed in parallel, then in the limit oflarge N , the asymptotic communication cost is strictly equal to I ( λ : | ψ (cid:105) ) [39], wherefor two continuous variables x and y , I ( x : y ) = h ( x ) + h ( y ) − h ( x, y ) (3.87)and (3.88) h ( x ) = − (cid:90) dxp ( x ) log e ( p ( x )) (3.89)We will not derive these results but make use of them here.For a ψ ontic model however, I ( λ : | ψ (cid:105) ) is infinite as p ( λ || ψ (cid:105) ) contains a delta func-tion. Hence, for the simulation to have only finite amount of communication betweenAlice and Bob, ψ epistemic ontological models are the only choice.The questions we ask here are: Can MD models too simulate quantum channels?and, can simulation by MD models offer any advantage over simulation by ontologicalmodels? We first give a MD hidden variable model for qubits and then give a protocolto use it to simulate quantum channels. Later we discuss how measurement dependentmodels can be more advantageous than ontological models for such simulations.35 .4.1 Modified Kochen Specker model II Let the qubit | a (cid:105) be denoted by ˆ a on the Bloch sphere, and the measurement M = {| b (cid:105)(cid:104) b | , | b ⊥ (cid:105)(cid:104) b ⊥ |} be denoted by ˆ b on the Bloch sphere. Then, the model is defined by p (ˆ λ | ˆ a, ˆ b ) = Θ(ˆ λ · ˆ a ) | ˆ λ · ˆ b | π (3.90) p (ˆ k | ˆ λ ) = Θ(ˆ λ · ˆ k ) (3.91)where Θ is the Heaviside Step function, and ˆ k ∈ { ˆ b, − ˆ b } . The density function for ˆ λ isnormalized, (cid:90) p (ˆ λ | ˆ a, ˆ b ) d ˆ λ = 1 (3.92)and the model reproduces Quantum Mechanics predictions, p ( ± ˆ b | ˆ a, ˆ b ) = (cid:90) Θ(ˆ λ · ˆ a ) | ˆ λ · ˆ b | π Θ( ± ˆ λ · ˆ b ) d ˆ λ (3.93) = 1 ± ˆ a · ˆ b (3.94)(3.95)It can be checked that the model is maximally epistemic. Let Alice prepare her qubit along ˆ a . As she does not know what measurement ˆ b Bobwill choose, she cannot generate the complete probability distribution 3.90. She insteadgenerates a uniform distribution over the hemisphere with ˆ a at its center, Θ(ˆ λ · ˆ a ) / π .She sends ˆ λ according to this distribution to Bob.At his end Bob does not accept all the ˆ λ s Alice is sending. He first chooses hismeasurement ˆ b and then attaches a weight of | ˆ λ · ˆ b | to the uniform distribution sent byAlice. He picks up more ˆ λ s from the regions | ˆ λ · ˆ b | is high and less from the regions36here | ˆ λ · ˆ b | is low so as to generate the final distribution Θ(ˆ λ · ˆ a ) | ˆ λ · ˆ b | /π . Effectively,Bob picks up ˆ λ s with probability distribution | ˆ λ · ˆ b | /π over the hemisphere defined by Θ(ˆ λ · ˆ a ) .In this protocol, Alice does not generate the final distribution of ˆ λ . She only hasto communicate such that Bob can generate a uniform distribution over the hemispherewith ˆ a at its center. The communication required is I (ˆ a : ˆ λ ) . Let us calculate the same. Here, p (ˆ λ | ˆ a, ˆ b ) = Θ(ˆ λ · ˆ a ) | ˆ λ · ˆ b | π (3.96) p (ˆ b | ˆ λ ) = Θ(ˆ λ · ˆ b ) (3.97)Assuming p (ˆ a ) = π , we have, h (ˆ a ) = − (cid:90) π log e ( 14 π ) d ˆ a (3.98) = log e (4 π ) (3.99)As ˆ λ depends on both ˆ a and ˆ b , assuming p (ˆ b ) = π we first find p (ˆ λ | ˆ a ) = (cid:90) p (ˆ λ | ˆ a, ˆ b ) p (ˆ b ) d ˆ b (3.100) = 12 π Θ(ˆ λ · ˆ a ) (3.101)and, p (ˆ λ ) = (cid:90) p (ˆ λ | ˆ a ) p (ˆ a ) d ˆ a (3.102) = (cid:90) π Θ(ˆ λ · ˆ a ) × π d ˆ a (3.103) = 14 π (3.104)37herefore, h (ˆ λ ) = h (ˆ a ) = log e (4 π ) . Now, h (ˆ λ, ˆ a ) = − (cid:90) p (ˆ λ | ˆ a ) p (ˆ a ) log e ( p (ˆ λ | ˆ a ) p (ˆ a )) d ˆ λd ˆ a (3.105) = − π (cid:90) Θ(ˆ λ · ˆ a ) log e ( Θ(ˆ λ · ˆ a )8 π ) d ˆ λd ˆ a (3.106) = 18 π { log e (8 π ) (cid:90) Θ(ˆ λ · ˆ a ) d ˆ λd ˆ a − (cid:90) Θ(ˆ λ · ˆ a ) log e (Θ(ˆ λ · ˆ a )) d ˆ λd ˆ a } (3.107) = log e (2) nats = 1 bit (3.108)Now Alice sends information for each ˆ λ . But Bob does not use all the ˆ λ s Alice issending. Hence we will have a correction factor for this. Alice sends " π " amount of ˆ λ sto Bob, as she generates a uniform distribution over a hemisphere. Bob finally selects π amount of ˆ λ , as the final distribution is Θ(ˆ λ · ˆ a ) | ˆ λ · ˆ b | /π . Thus, he selects only halfof the ˆ λ s Alice is sending. As Alice sends information for all ˆ λ s nevertheless, the com-munication cost involved in this protocol is twice that of calculated, per round.We thus see that it is indeed possible to use MD models for classical simulation ofquantum channels. As we saw in section 3.3.1, MD models are not constrained in their maximal epistemic-ity in higher dimensions of Hilbert space unlike ontological models, where maximalepistemicity is impossible for d N > . Thus, one can in principle develop protocolsfor classical simulation of quantum channels invloving qutrits or higher dimensionalsystems, by using MD maximally epistemic hidden variable models.38 .5 A Measurement Dependent model that cross corre-lates particles and measurement choices in EPR sce-nario In section 3.2.1 we saw that in the Brans model, the hidden variable state of Alice hasno correlation with the measurement choice of Bob, and vice versa. It is of interest todevelop a model where the particles are correlated with both measurement choices, asthis has foundational implications, discussed in section 3.5.2. First we present a modelthat achieves this.
The model is a modified version of the local, deterministic and MD model for singletstate correlations given by Hall in [18], and reformulated in [34]. Here we make it sep-arable too.Let the ontic states of the two particles entangled in singlet state be denoted by ˆ λ and ˆ λ ; both are vectors on a unit sphere. Let the corresponding experimenters makemeasurements along ˆ a and ˆ b directions. Then, the measurement results are given by, A (ˆ λ , ˆ a ) = Sign (ˆ λ · ˆ a ) (3.109) B (ˆ λ , ˆ b ) = Sign (ˆ λ · ˆ b ) (3.110)where Sign ( x ) is the sign function. So, the model is deterministic and local.The probability distribution of the ontic states is given by, p (ˆ λ , ˆ λ | ˆ a, ˆ b, | ψ (cid:105) singlet ) = 14 π − (ˆ a · ˆ b ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } − (1 − φ ˆ a ˆ b π ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } δ (ˆ λ + ˆ λ ) (3.111)where φ ˆ a ˆ b is the angle between ˆ a and ˆ b . 39he marginals are, p (ˆ λ | ˆ a, ˆ b, | ψ (cid:105) singlet ) = (cid:90) d ˆ λ p (ˆ λ , ˆ λ | ˆ a, ˆ b, | ψ (cid:105) singlet ) (3.112) = 14 π a · ˆ b ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } − φ ˆ a ˆ b π ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } (3.113)And similar for p (ˆ λ | ˆ a, ˆ b, | ψ (cid:105) singlet ) .From eqn. 3.113 we see that ˆ λ is correlated with measurement choices of both experimenters.That the model correctly reproduces singlet state correlations can be proven. Theprobability Alice gets value x and Bob y on measuring spins along ˆ σ · ˆ a and ˆ σ · ˆ b respectively is, p ( x, y || ψ (cid:105) singlet , M = ˆ σ · ˆ a ⊗ ˆ σ · ˆ b ) = (cid:90) d ˆ λ d ˆ λ δ x,A (ˆ λ , ˆ a ) δ y,B (ˆ λ , ˆ b ) p (ˆ λ , ˆ λ | ˆ a, ˆ b, | ψ (cid:105) singlet ) (3.114) = (cid:90) d ˆ λ d ˆ λ δ x,A (ˆ λ , ˆ a ) δ y,B (ˆ λ , ˆ b ) π − (ˆ a · ˆ b ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } − (1 − φ ˆ a ˆ b π ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } δ (ˆ λ + ˆ λ ) (3.115) = (cid:90) d ˆ λ δ x,A (ˆ λ , ˆ a ) δ y,B ( − ˆ λ , ˆ b ) π a · ˆ b ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } − φ ˆ a ˆ b π ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } (3.116)where δ is the Kronecker delta function. Eqn. 3.116 is the same integral as in [34], thusensuring correctness. In reference [16], the authors claim to prove a theorem that any locally causal [22], sep-arable model that reproduces quantum mechanical predictions must be ψ epistemic inEPR scenario. Their argument consists of the following assumption: Consider a framewhere Alice makes the first measurement. Let her choose her measurement direction40igure 3.1: Condition of local-causalityeither along ˆ a or ˆ a (cid:48) . If the model is locally causal, then they argue that the measure-ment choices made by Alice, at spacelike separation from Bob, should not affect thedistribution of Bob’s particle’s hidden variable state λ B . p ( λ B | ˆ a ) = p ( λ B | ˆ a (cid:48) ) (3.117)The assumption is incorrect to the best of our understanding. Instead of being aconsequence of local-causality, it is a consequence of MI. As defined in their own paperin agreement with Bell [22], Definition 3.5.2.1.
Consider an event x occuring at spacetime region A and an event y occuring at spacetime region B, where A and B are space-like separated. Then local-causality is the condition that p ( x | λ C , y ) = p ( x | y ) (3.118) where λ C contains a complete specification of events in space time region C that screensoff B from the intersection of backward light cones of A and B. (refer Fig. 3.1) No such complete description is provided in eqn. 3.117. But the equation makessense if one assumes MI, as the measurement choices then have correlations only withevents in their future light cones.Now let us consider the MD model introduced in the last section 3.5.1, which leads tothe following theorem:
Theorem 3.5.1.
A measurement dependent model which is local causal need not be ψ epistemic. roof : Consider that, in a frame where Alice makes the first measurement, shechooses her measurement direction either along ˆ a or ˆ a (cid:48) . Accordingly, the probabilitydistribution of ˆ λ of Modified Hall model, p (ˆ λ | ˆ a, ˆ b, | ψ (cid:105) singlet ) = 14 π a · ˆ b ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } − φ ˆ a ˆ b π ) Sign { (ˆ λ · ˆ a )(ˆ λ · ˆ b ) } (3.119) p (ˆ λ | ˆ a (cid:48) , ˆ b, | ψ (cid:105) singlet ) = 14 π a (cid:48) · ˆ b ) Sign { (ˆ λ · ˆ a (cid:48) )(ˆ λ · ˆ b ) } − φ ˆ a (cid:48) ˆ b π ) Sign { (ˆ λ · ˆ a (cid:48) )(ˆ λ · ˆ b ) } (3.120) differs, but this does not imply a failure of local-causality. Eqns. 3.109 and 3.110explicitly show the local nature of the model.Hence, the argument that they should be same as a consequence of local-causality,cannot be held for MD models, and epistemicity cannot be derived as in [16]. ψ ontic ontological model be converted to ψ epistemic by introducing Measurement Dependence? In sections 3.1.3 and 3.4.1, we modified an already maximally epistemic ontologicalmodel[31] to maximally epistemic MD model in 2 ways. The question remains if onecan convert a ψ ontic ontological model to ψ epistemic by introducing measurementdependence. Here we prove it in the affirmative by modifying the ψ ontic Bell-Merminmodel[40] to a maximally ψ epistemic MD model. Here, the hidden variable λ (cid:48) = ( λ, ˆ λ ) , where ˆ λ is a vector on the surface of Blochsphere, and λ is a discrete variable taking values λ i and λ j . λ and ˆ λ are correlated. Themodel, where M = {| i (cid:105)(cid:104) i | , | j (cid:105)(cid:104) j |} is an orthonormal basis, is defined as : p ( λ k = i ( j ) || ψ (cid:105) , M ) = 1 / (3.121) p (ˆ λ || ψ (cid:105) , M, λ k ) = 1 / (2 π ) × Θ(ˆ k · ( ˆ ψ + ˆ λ )) (3.122) p ( λ (cid:48) || ψ (cid:105) , M ) = p (ˆ λ || ψ (cid:105) , M, λ k ) × p ( λ k || ψ (cid:105) , M ) (3.123) p ( | l (cid:105)(cid:104) l || λ (cid:48) , M ) = p ( | l (cid:105)(cid:104) l || λ k , M ) = δ lk (3.124)42here Θ is the step function. The model reproduces Quantum Mechanics predictions: p ( | i (cid:105)(cid:104) i ||| ψ (cid:105) , M ) = (cid:90) (cid:88) k p ( | i (cid:105)(cid:104) i || λ k , M ) × p (ˆ λ || ψ (cid:105) , M, λ k ) × p ( λ k || ψ (cid:105) , M ) d ˆ λ (3.125) = (cid:90) (cid:88) k δ ik × / (4 π ) × Θ(ˆ k · ( ˆ ψ + ˆ λ )) d ˆ λ (3.126) = (cid:90) / (4 π ) × Θ(ˆ i · ( ˆ ψ + ˆ λ )) d ˆ λ (3.127) = |(cid:104) ψ | i (cid:105)| (3.128)It can be checked that the model is maximally ψ epistemic, as follows. Considertwo states | i (cid:105) and | ψ (cid:105) , measured in the basis M = {| i (cid:105)(cid:104) i | , | j (cid:105)(cid:104) j |} .Then for | i (cid:105) , p ( λ j , ˆ λ || i (cid:105) , M ) = 1 / (4 π ) × Θ(ˆ j · (ˆ i + ˆ λ )) (3.129) = 1 / (4 π ) × Θ( − j · ˆ λ )) (3.130) = 0 (almost everywhere) (3.131)and (3.132) p ( λ i , ˆ λ || i (cid:105) , M ) = 1 / (4 π ) × Θ(ˆ i · (ˆ i + ˆ λ )) (3.133) = 1 / (4 π ) × Θ(1 + ˆ j · ˆ λ )) (3.134) = 1 / (4 π ) (almost everywhere) (3.135)while for | ψ (cid:105) p ( λ i , ˆ λ || ψ (cid:105) , M ) = 1 / (4 π ) × Θ(ˆ i · ( ˆ ψ + ˆ λ )) (3.136)therefore the overlap is (cid:90) d ˆ λ (1 / π ) × Θ(ˆ i · ( ˆ ψ + ˆ λ )) (3.137) = |(cid:104) ψ | i (cid:105)| (3.138)Thus the Bell-Mermin model, which is ψ ontic, can be modified to MD case to be ψ epistemic. 43 HAPTER 4Some observations on Preparation Independence
PI was introduced in 3.3.2. Here we collect a few observations on this assumption.
Preparation Independence(PI) was used in [23] as their central assumption. In their pa-per a product state and entangled measurement basis is used. Here we show that for thecase of product state and product state basis, any ontological model satisfying PI willsatisfy quantum predictions. Thus, PI is a natural assumption for this case.
Theorem 4.1.1.
An ontological model which correctly reproduces quantum predictionsfor individual states will reproduce correct predictions for product state measurementson product states if the model satisfies PI.
Proof : If we have a product state ρ = | ψ (cid:105)(cid:104) ψ | ⊗ | φ (cid:105)(cid:104) φ | , where both | ψ (cid:105) and | φ (cid:105) are inn-dimensional Hilbert space, on which projective measurement is performed, where themeasurement basis is also product state, M = ( | e i (cid:105)(cid:104) e i | ⊗ | f j (cid:105)(cid:104) f j | ) i,j ∈{ , , ...n } (4.1) = ( | e i (cid:105)(cid:104) e i | ) i ∈{ , , ...n } ⊗ ( | f j (cid:105)(cid:104) f j | ) j ∈{ , , ...n } (4.2) = | e (cid:105)(cid:104) e || e (cid:105)(cid:104) e | .... | e n (cid:105)(cid:104) e n | ⊗ | f (cid:105)(cid:104) f || f (cid:105)(cid:104) f | .... | f n (cid:105)(cid:104) f n | (4.3) = M ⊗ M (4.4) atisfying (cid:104) e i | e j (cid:105) = 0 and (cid:104) f i | f j (cid:105) = 0 , then probability of getting the k th outcome outof n possibilities, corresponding to k = j + ( i − n , where each k corresponds to aunique i and j: p ( k | ρ, M ) = tr AB ( | ψ (cid:105)(cid:104) ψ | ⊗ | φ (cid:105)(cid:104) φ | × | e i (cid:105)(cid:104) e i | ⊗ | f j (cid:105)(cid:104) f j | ) (4.5) = tr A ( | ψ (cid:105)(cid:104) ψ | e i (cid:105)(cid:104) e i | ) × tr B ( | φ (cid:105)(cid:104) φ | f j (cid:105)(cid:104) f j | ) (4.6) = (cid:90) dλ A p ( i | λ A , M ) p ( λ A || ψ (cid:105)(cid:104) ψ | ) (cid:90) dλ B p ( j | λ B , M ) p ( λ B || φ (cid:105)(cid:104) φ | ) (4.7) = (cid:90) (cid:90) dλ A dλ B × p ( i | λ A , M ) p ( j | λ B , M ) × p ( λ A || ψ (cid:105)(cid:104) ψ | ) p ( λ B || φ (cid:105)(cid:104) φ | ) (4.8) = (cid:90) (cid:90) dλ A dλ B × p ( k | λ A , λ B , M ⊗ M ) × p ( λ A || ψ (cid:105)(cid:104) ψ | ) p ( λ B || φ (cid:105)(cid:104) φ | ) (4.9)It may be noted that one cannot rule out from this theorem models that do not satisfyPI and still reproduce quantum predictions. Hall gave a different proof of PBR theorem [23] based on weakened assumptions[41].Instead of Preparation Independence, he formulated "compatibility" and "local compat-ibility". Here we take a look and see why they are "weaker".The PBR theorem first assumes separability for product states.
Definition 4.2.0.1.
A hidden variable model is separable if p ( λ | ( ρ = ρ ⊗ ρ ⊗ .... ⊗ ρ n ) , M ) > ⇒ λ = ( λ , λ , ...., λ n ) where λ i is the ontic state of i th system. They further assume that the hidden variable state of each individual system is in-dependent of all others: p ( λ , λ , ...., λ n | ρ ⊗ ρ ⊗ .... ⊗ ρ n , M ) = n (cid:89) i =1 p ( λ i | ρ i , M ) (4.10)45 .2.1 Compatibility Denoting by the condition λ ∼ { ρ = ( | ψ (cid:105)(cid:104) ψ | ⊗ ˆ I ⊗ ˆ I.... ⊗ ˆ I ) , M } only if p ( λ | ρ, M ) > , we proceed to consider the case of 2 qubits in product state as in PBR theorem. Definition 4.2.1.1.
A hidden variable model is compatible if a ) λ ∼ { ρ = ( | ψ (cid:105)(cid:104) ψ | ⊗ ˆ I ) , M } b ) λ ∼ { ρ = ( | φ (cid:105)(cid:104) φ | ⊗ ˆ I ) , M } imply the following i ) λ ∼ { ρ = ( | ψ (cid:105)(cid:104) ψ | ⊗ | φ (cid:105)(cid:104) φ | ) , M } , ii ) λ ∼ { ρ = ( | φ (cid:105)(cid:104) φ | ⊗ | ψ (cid:105)(cid:104) ψ | ) , M } , iii ) λ ∼ { ρ = ( | ψ (cid:105)(cid:104) ψ | ⊗ | ψ (cid:105)(cid:104) ψ | ) , M } , iv ) λ ∼ { ρ = ( | φ (cid:105)(cid:104) φ | ⊗ | φ (cid:105)(cid:104) φ | ) , M } where iii ) follows from a ) and iv ) follows from b ) , and i ) and ii ) follow from a ) and b ) jointly. Now let there be a preparation procedure which prepares either of ρ { , , , } = {| ψ (cid:105)(cid:104) ψ |⊗| φ (cid:105)(cid:104) φ | , | ψ (cid:105)(cid:104) ψ | ⊗ | ψ (cid:105)(cid:104) ψ | , | φ (cid:105)(cid:104) φ | ⊗ | φ (cid:105)(cid:104) φ | , | φ (cid:105)(cid:104) φ | ⊗ | ψ (cid:105)(cid:104) ψ |} .In the measurement basis M considered in PBR, p ( k | M, λ ) = 0 if λ ∼ { ρ k , M } .Now assume epistemicity. Let ∃ λ ∼ { ρ k , M } for k = 1 , , , over a set S. However λ ∈ S ⇒ p ( k | λ, M ) = 0 . But we know (cid:80) k p ( k | λ, M ) = 1 . So there’s a contradiction,and hence S is a null set.Denote by the S m the set { λ | λ ∼ { ρ = ( | ψ (cid:105)(cid:104) ψ | ⊗ ˆ I ) , M } ∩ λ ∼ { ρ = ( | φ (cid:105)(cid:104) φ | ⊗ ˆ I ) , M }} From Compatibility we know, S m ⊆ S. Hence S m is also a null set. Hencewe derive the same conclusion of PBR, without even assuming separability for productstates. Thus Compatibility is weaker than Preparation Independence.46 .2.2 Local Compatibility Unlike Compatibility, Local Compatibility assumes separability. Hence it is strongerthan Compatibility.
Definition 4.2.2.1.
A hidden variable model is locally compatible if λ i ∼ ( ρ i , M ) ∀ i ⇒ λ = ( λ , λ , ...., λ n ) ∼ ( ρ ⊗ ρ ⊗ .... ⊗ ρ n , M ) . Considering again the case of 2 qubits,If λ ∼ ( | ψ (cid:105)(cid:104) ψ | , M ) , λ ∼ ( | φ (cid:105)(cid:104) φ | , M ) then, it follows from Local Compatibility, ∃ λ = ( λ , λ ) | λ ∼ ( | ψ (cid:105)(cid:104) ψ | ⊗ | φ (cid:105)(cid:104) φ | , M ) .Now consider epistemicity :Let ∃ λ (cid:48) | λ (cid:48) ∼ {| φ (cid:105)(cid:104) φ | , M } , λ (cid:48) ∼ {| ψ (cid:105)(cid:104) ψ | , M } .Then from Local Compatibility, we have ∃ λ = ( λ (cid:48) , λ (cid:48) ) | λ ∼ ( ρ , , , , M ) . But forsuch λ we have already proven a contradiction in previous section 4.2.1. Thus, eitheror both of our assumptions, Local Compatibility and epistemicity, must be incorrect.Assuming Local Compatibility as correct, epistemicity is ruled out.It is clear that Local Compatibility is a weaker form of PI, as the former assumesseparability but makes no assumption about individual hidden variables being uncorre-lated. Compatibility is weaker than both. 47 HAPTER 5Conclusion5.1 List of new results