Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)

Foundations of Quantum Mechanics Spekkens, Rob Jul 26, 2010

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Foundations of Quantum Theory Provide an adequate interpretation Explore nonclassical phenomena Determine principles from which quantum theory may be derived What’s the problem? Representational completeness of ψ. The rays of Hilbert space correspond one-to-one with the physical states of the system. Measurement. If the Hermitian operator A with spectral projectors {Pk} is measured, the probability of outcome k is 〈ψ|Pk |ψ〉. These probabilities are objective -- indeterminism. “Orthodox” postulates of quantum theory Evolution of isolated systems. It is unitary, therefore deterministic and continuous. Evolution of systems undergoing measurement. If Hermitian operator A with spectral projectors {Pk} is measured and outcome k is obtained, the physical state of the system changes discontinuously, First problem: the term “measurement” is not defined in terms of the more primitive “physical states of systems”. Isn’t a measurement just another kind of physical interaction? Two strategies: (1) Realist strategy: Eliminate measurement as a primitive concept and describe everything in terms of physical states (2) Operational strategy: Eliminate “the physical state of a system” as a primitive concept and describe everything in terms of operational concepts “It would seem that the theory is exclusively concerned about "results of measurement", and has nothing to say about anything else. What exactly qualifies some physical systems to play the role of "measurer"? ” - John Bell “In a strict sense, quantum theory is a set of rules allowing the computation of probabilities for the outcomes of tests which follow specified preparations.” - Asher Peres The realist strategy Indeterministic and discontinuous evolution By the collapse postulate (applied to the system) By unitary evolution postulate (applied to isolated system that includes the apparatus) Deterministic and continuous evolution Inconsistencies of the orthodox interpretation Determinate properties Indeterminate properties If the measurement apparatus is treated externally The quantum measurement problem If the measurement apparatus is treated internally U is a linear operator  • Interpret coherent superposition as disjunction False starts on the measurement problem Means either or with probabilities |a|2 and |b|2 This is a denial of the representational completeness of ψ respectively • Interpret the reduced density operator as a proper mixture Either contradicts original assignment of entangled state Or is a denial of the representational completeness of False starts on the measurement problem ψ • Appeal to environment-induced decoherence False starts on the measurement problem This doesn’t help • Appeal to differences in the state of the apparatus But for the interaction to be considered a measurement, we require False starts on the measurement problem And by linearity The postulated evolution does not correspond to a proper measurement Responses to the measurement problem 2. Deny representational completeness of ψ 1. Deny universality of quantum dynamics • Quantum-classical hybrid models • Collapse models • ψ-ontic hidden variable models (e.g. Bohmian mechanics) • ψ-epistemic hidden variable models 3. Deny that there is a unique outcome • Everett’s relative state interpretation (many worlds) 4. Deny some aspect of classical logic or classical probability theory • Quantum logic and quantum Bayesianism 5. Deny some other feature of the realist framework? The operational strategy  Operational Quantum Mechanics Preparation Measurement Vector Hermitian operator Operational Quantum Mechanics Preparation Measurement Effective preparation Update map Operational Quantum Mechanics Preparation Transformation Measurement Vector Unitary map Hermitian operator Operational Quantum Mechanics Preparation Transformation Measurement Effective preparation Operational Quantum Mechanics Transformation Measurement Preparation Effective Measurement The real formalism of operational quantum theory Preparation Measurement Operational Quantum Mechanics Density operator Position operator valued measure (POVM) Operational Quantum Mechanics Preparation Measurement Effective preparation Update map Trace-decreasing completely positive linear map where Operational Quantum Mechanics Preparation Transformation Measurement Trace-preserving completely positive linear map (CP map) Positive operator-valued measure (POVM)Density operator Operational Quantum Mechanics Preparation Transformation Measurement Effective preparation ρ ! ρ′ = T (ρ) Operational Quantum Mechanics Transformation Measurement Preparation Effective Measurement Every preparation P is associated with a density operator ρ Every measurement M is associated with a positive operator-valued measure {Ek}.  The probability of M yielding outcome k given a preparation P is Operational postulates of quantum theory Every measurement outcome k is associated with a trace- nonincreasing completely-positive linear map Tk such that Every transformation is associated with a trace-preserving completely- positive linear map No mention of “physical states” or their evolution ρ ! ρ′ = T (ρ) How density operators and POVMs arise in the operational approach Operational Quantum Mechanics Preparation Measurement Vector Hermitian operator Preparation Measurement Operational Quantum Mechanics Density operator Hermitian operator jψ1i Ensembles f pig jψ2i jψ3i f Πkgρ Positive Unit trace “Density operator” p(k) = T r( ¦ kρ)p(k) = ∑ i p(k|i)p( i) = ∑ i 〈ψi|¦ k|ψi〉pi = ∑ i T r( ¦ k|ψi〉〈ψi|)pi = T r( ¦ k ∑ i pi|ψi〉〈ψi|) hψjρjψi ¸ 0 8jψi 2 H Tr(ρ) = 1 ρ = ∑ i pi|ψi〉〈ψi|where Reduced density operators jψi sa f Π(s)k g s a f Iag ρs p(k) = T rsa[( ¦ ( s)k ⊗ Ia) |ψ〉sa〈ψ|] = T rs[¦ ( s)k ( T ra( |ψ〉sa〈ψ|) ] p(k) = T r( ¦ ( s)k ρs) where ρs = T ra( |ψ〉sa〈ψ|) ↔ Pure preparation ↔ Mixed preparation Multiplicity of convex decompositions Multiplicity of purifications Preparation Measurement Operational Quantum Mechanics Density operator Hermitian operator Preparation Measurement Operational Quantum Mechanics Density operator Projection valued measure (PVM) Preparation Measurement Operational Quantum Mechanics Density operator Position operator valued measure (POVM)  f pig f Π(1)j g f Π(2)j g (3) ρ k = (i, j) Mixtures of measurements f Πj gp( i, j) = p( j|i)p( i) = T r( ¦ ( i)j ρ)pi = T r(pi¦ ( i)j ρ) Ei,j Positive hψjEk jψi ¸ 0 8jψi 2 H Sum to identity ∑ k Ek = I p(k) = T r(Ekρ) “Positive operator valued measure (POVM)”{Ek} f 1 2 , 1 2 g f j0i h0j, j1i h1jg k = (i, j) f j+i h+j, j¡ i h¡ jg f 1 2 j0i h0j , 1 2 j1i h1j , 1 2 j+i h+j , 1 2 j¡ i h¡ jg 1 4 j0i h0j + 14 j1i h1j + 14 j+i h+j + 14 j¡ i h¡ j = 12I Recall j 2 f 1, 2, 3, 4, 5, 6, 7, 8, 9, ...g k 2 f 1, 2, 3, ...g {Ek} {Fj} Coarse-graining p(k) = ∑ j∈Sk p( j) T r(Ekρ) = ∑ j∈Sk T r(Fjρ) ∀ρ = T r[( ∑ j∈Sk Fj)ρ] ∀ρ Ek = ∑ j∈Sk Fj Note: the Ek need not be rank 1 f 1 2 , 1 2 g f j0i h0j, j1i h1jg k = (i, j) f j+i h+j, j¡ i h¡ jg Example f 1 2 j0i h0j , 1 2 j1i h1j , 1 2 j+i h+j , 1 2 j¡ i h¡ jg f 1 2 j0i h0j + 1 2 j1i h1j , 1 2 j+i h+j + 1 2 j¡ i h¡ jg= f 1 2 I , 1 2 Ig f j0i h0j, j1i h1jg f q, 1 ¡ qg f 1 2 I , 1 2 Ig Another example f qj0i h0j , qj1i h1j , (1 ¡ q)1 2 I , (1 ¡ q)1 2 Ig = f 1 + q 2 j0i h0j + 1 ¡ q 2 j1i h1j , 1 ¡ q 2 j0i h0j + 1 + q 2 j1i h1jg f qj0i h0j + (1 ¡ q)1 2 I , qj1i h1j + (1 ¡ q)1 2 Ig Noisy S·z f 1 2 j0i h0j , 1 2 j1i h1j , 1 2 j+i h+j , 1 2 j¡ i h¡ jg f 1 2 j0i h0j , 1 2 j1i h1j , 1 2 j+i h+j , 1 2 j¡ i h¡ jg f E0, E1g Noisy S· n f 1 2 j0i h0j , 1 2 j1i h1j , 1 2 j+i h+j , 1 2 j¡ i h¡ jg f F0, F1g Noisy S· n⊥ Note: General conditions for joint measurability of POVMs are not known ρs f Π(sa)k g Measurement by coupling to an ancilla p(k) = T rsa[¦ ( sa)k (ρs ⊗ τa) ] = T rs[T ra( ¦ ( sa)k τa) ρs] E (s) k ρs E (s) = Tr (Π (sa) τ ) Example = { √ 2−1( |0〉|0〉 + |1〉|1〉) ,√ 2−1( |0〉|0〉 − 1〉|1〉) ,√ 2−1( |0〉|1〉 + |1〉|0〉) ,√ 2−1( |0〉|1〉 − |1〉|0〉)} f jΦ1i , jΦ2i , jΦ3i , jΦ4i g k a k a = hθjajΦk i sahΦkjsajθi a hθjajΦ1(2)i sa = p 2 −1 [cos(θ/2)j0i s § sin(θ/2)j1i s] = p 2 −1j § θi s hθjajΦ3(4)i sa = p 2 −1 [sin(θ/2)j0i s § cos(θ/2)j1i s] = p 2 −1jπ ¨ θi s θ = π/4 Naimark’s theorem: Every POVM can be implemented by coupling to an ancilla and implementing a projective measurement f Ekg = f 1 2 jθi hθj , 1 2 j ¡ θi h¡ θj , 1 2 jπ ¡ θi hπ ¡ θj , 1 2 jπ + θi hπ + θjg Preparation Measurement Operational Quantum Mechanics Density operator Position operator valued measure (POVM) Towards an operational axiomatization of quantum theory Convex theories Quantum theory Category Theory Framework Classical theory Convex theories with maximal dual cone C*  algebraic theories Possibilistic Theories Classical Statistical Theories with epistemic restriction Preparation Measurement A framework for convex operational theories sP =   Pr( 1|M,P ) Pr( 2|M,P ) Pr( 1|M ′, P ) Pr( 2|M ′, P ) Pr( 3|M ′, P ) . . .   rM,k =   0 0 0 1 0 . . .   Preparation Measurement A framework for convex operational theories sP =   Pr( pass|M1, P ) Pr( pass|M2, P ) . . . Pr( pass|MK, P )   Suppose there are K fiducial measurements (pass-fail mmts from which one can infer the statistics for all mmts) What can we say about f? “operational state” Operational states form a convex set (w,1-w) Convex linear Also true for fiducial mmts, so Closed under convex combination Convex linearity implies linearity If f is convex linear on opt’l states Then f is linear on opt’l states Proof: Note that: Thus: Convex linearity implies linearity If f is convex linear on opt’l states Then f is linear on opt’l states Therefore Preparation Measurement A convex operational theory sP ∈ S “operational effects”“operational states” S = Convex set R = Interval of positive cone S and R characterize the operational theory! Operational classical theory S = a simplex R = the unit hypercube s can be any probability distributions r can be any vector of conditional probabilities Operational quantum theory Recall: The Hermitian operators on H of dimension d form a real Euclidean vector space of dimension d2 S = the convex set of positive trace-one operators The inner product is (A,B) = Tr(AB) R = the set of all positive operators less than identity hψjEk jψi ¸ 0 8jψi 2 H∑ k Ek = I An axiomatization must derive S and R See e.g. L. Hardy, quant-ph/0101012, and J. Barrett, quant-ph/0508211 Is the operational interpretation satisfactory? f 1 2 jθi hθj , 1 2 j ¡ θi h¡ θj , 1 2 jπ ¡ θi hπ ¡ θj , 1 2 jπ + θi hπ + θjg θ = π/4 Naimark’s theorem: Every POVM can be implemented by coupling to an ancilla and implementing a projective measurement P2 P4 P6P5 P1 M1 M3 M10P7 M2 M6 M4 Two approaches to axiomatization Operational approach Ontological approach M8P8 M9 Preparations Measurements Axioms are constraints on experimental statistics p(k|M,P) Axioms are constraints on the ontology and its dynamics Back to realist approaches (this time allowing for hidden variables) Preparation P An ontological model of an operational theory Measurement M ψ|ψ1〉 |ψ3〉 |ψ2〉 Deterministic hidden variable model for pure states and projective measurements It is assumed that the outcomes are deterministic given λ Example: the Kochen-Specker model for a 2d system The KS model cannot be generalized to mixed states, POVMs or higher dimensions Suppose A and B share       += 11002 1ψ If A measures }1,0{ B’s state becomes            with probability 1/2 with probability 1/2 0 1 If A measures },{ −+ B’s state becomes            with probability 1/2 with probability 1/2 + − A B Example: Statistically restricted classical theories “Steering” Consider Einstein’s version of the EPR argument ])44[]33[]22[]11([ 4 1),( +++=′ λλµ If A measures {1,2} vs. {3,4} Her knowledge of B is updated to If A measures  {1,3} vs. {2,4} with prob. 1/2 with prob. 1/2 with prob. 1/2 with prob. 1/2 Her knowledge of B is updated to λ λ λ λ λ Alice’s initial knowledge of B In a statistically restricted classical theory the convex set of operational states exhibits - Convexly extremal states can be classically mixed - non-simplicial shape / ambiguous mixtures - Convexly extremal states can be correlated Noncommutativity Entanglement Collapse Wave-particle duality Teleportation No cloning Key distribution Improvements in metrology Quantum eraser Coherent superposition Pre and post-selection “paradoxes” Others… Bell inequality violations Contextuality Computational speed-up Certain aspects of items on the left Others… Categorizing quantum phenomena Those not arising in a restricted statistical classical theory Those arising in a restricted statistical classical theory Type 1 Nonclassicality Type 2 Nonclassicality Bell’s theorem John S. Bell (1928-1990) SA pair of two-outcome measurements S T T ES S ES T ET S ET T There are two possible measurements, S and T, with two outcomes each: green or red Suppose which of S or T occurs at each wing is chosen at random Scenario 1 1.  Whenever the same measurement is made on A and B, the outcomes always agree 2.  Whenever different measurements are made on A and B, the outcomes always disagree S and S or T and T S and T or T and S SS TT There are two possible measurements, S and T, with two outcomes each: green or red Suppose which of S or T occurs at each wing is chosen at random Scenario 2 1.  Whenever the same measurement is made on A and B, the outcomes always disagree 2.  Whenever different measurements are made on A and B, the outcomes always agree S and S or T and T S and T or T and S ST S T There are two possible “measurements”, S and T, with two outcomes each: green or red Suppose which of S or T occurs at each wing is chosen at random Scenario 3 1. Whenever the measurement T is made on both A and B, the outcomes always disagree 2.  Otherwise, the outcomes always agree T and T S and S or S and T or T and S ST S T The game can be won at most 75% of the time by local strategies Using quantum theory, it can be won ≃85% of the time Q: How could you cheat and win the game all the time? A: Communication of the choice of measurement in one wing to the system in the opposite wing But there’s a problem… Mmt is chosen Outcome is registered Mmt is chosen Outcome is registered t x Tension with the theory of relativity Experiment can distinguish: 1)  the quantum predictions 2) the predictions of any locally causal theory Quantum theory is corroborated! Would access to randomness help to generate the correlations? If the detector inefficiencies are sufficiently high, can particles obeying local causality simulate the correlations on the detected pairs? Is there a problem if the choice of measurement is made before the particles are sent to the detectors? No.  It will only decrease the degree of correlation Yes.  This is the detector loophole. Yes.  This is the locality loophole. Is the proof robust to experimental imperfections?  (e.g. the detector sometimes registers the wrong outcome) Yes.  The Bell inequality may still be violated. When seeking a realist explanation of these experiments, the mystery is the tension between: 1) No superluminal signalling (independence of statistics at one wing on choice of measurement at the other) 2) The necessity of superluminal influences (dependence of particular outcomes at one wing on choice of measurement at the other) The quantum correlations p( success) = 14 [ p( agree|SS) + p( agree|ST ) + p( agree|TS) + p( disagree|TT ) ] Realist theories that are locally causal predict p( success) · 0.75 A Bell Inequality Quantum theory predicts that one can achieve p( success) ≃ 0.85 ST S T S T S T T T S S p(success) = 1 2 + 1 2 p 2 ' 0.85 The Bell-inequality violation in quantum theory jψiAB = 1p 2 (j0iAj0iB + j1iAj1iB) ST S T T T S S = cos2(π/8) = 1 2 + 1 2 p 2 jh+n̂jAh+m̂jB jψiAB j2 = jh+m̂j + n̂i j2 = cos2(θ/2) Ah+n̂jψiAB = [cos(θ/2)Ah0j + sin(θ/2)Ah1j] 1p 2 (j0iAj0iB + j1iAj1iB) = cos(θ/2)j0iB + sin(θ/2)j1iB = j + n̂iB p( agree|SS) = p( agree|ST ) = p( agree|TS) = p( disagree|TT ) jψiAB = 1p 2 (j0iAj0iB + j1iAj1iB) p(success) = 1 2 + 1 2 p 2 ' 0.85 The Bell-inequality violation in quantum theory No signalling in quantum theory {EAk } {F B j } ρAB p( j) = ∑ k p(k, j) = ∑ k T rAB[ (EAk ⊗ FBj ) ρAB] = T rAB[ ( IA ⊗ FBj ) ρAB] Independent of choice of measurement at A Note that                       for A and B space-like separated[EAk , F B j ] = 0 Nonlocality in more depth “The [beables] in any space-time region 1 are determined by those in any space region V, at some time t, which fully closes the backward light cone of 1. Because the region V is limited, localized, we will say the theory exhibits local determinism. -- J.S. Bell t x 1 V tx A B C “A theory will be said to be locally causal if the probabilities for the values of local beables in a space-time region A are unaltered by specification of values of local beables in a space-time region B, when what happens in the backward light cone of A is already sufficiently specified, for example by a full specification of local beables in a space-time region C.” -- J. S. Bell p(XAjXB, λC) = p(XAjλC) Local causality At x a B b Locality causality implies p(Aja, b, B, λ) = p(Aja, λ) p(Bja, b, A, λ) = p(Bjb, λ) and implies factorizability p(A,Bja, b, λ) = p(Aja, λ)p(Bjb, λ) λ a, b – settings A, B -- outcomes p(AjB, a, b, λ) = p(Aja, λ) Recall Bayes’ rule p(A,B) = p(AjB)p(B) p(A,Bja, b, λ) = p(AjB, a, b, λ)p(Bja, b, λ) p(A,BjC) = p(AjB,C)p(BjC) therefore By local causality p(Bja, b, λ) = p(Bjb, λ) Factorizability from local causality Thus p(A,Bja, b, λ) = p(Aja, λ)p(Bjb, λ) aa’ b 1 4 [ p( agree|ab) + p( agree|ab′) + p( agree|a′b) + p( disagree|a′b′) ] · 3/4 b’+ - + - + - + - C(a, b) = ( + 1)p( agree|ab) + (−1)p( disagree|ab) ] |C(a, b) + C(a′, b) + C(a, b′) − C(a′, b′) | · 2 Define The Clauser-Horn-Shimony-Holt (CHSH) inequality These (equivalent) inequalities can be derived from local causality See e.g. J.S. Bell, Speakable and Unspeakable, Chap. 16, App. 2 Applications of nonlocality Magic is a natural force that can be used to override the usual laws of nature. -- Harry Potter entry in wikipedia Bell–inequality violations are natural phenomena that can be used to override the usual (classical-like) laws of nature Quantum Spellcraft Based on Bell-inequality violation Reduction in communication complexity Buhrman, Cleve, van Dam, SIAM J.Comput. 30 1829 (2001) Brassard, Found. Phys. 33, 1593 (2003) Device-independent secure key distribution Barrett, Hardy, Kent, PRL 95, 010503 (2005) Acin, Gisin, Masanes, PRL. 97, 120405 (2006) Enhancing zero-error channel capacity Cubitt, Leung, Matthews, Winter, arXiv:0911.5300 Monogamy of Bell-inequality violating correlations Alice Bob Adversary Why isn’t the world more nonlocal? The traditional notion of noncontextuality in quantum theory A given vector may appear in many different measurements |ψ1〉 |ψ3〉 |ψ2〉 |ψ1〉 |ψ′3〉 |ψ′2〉 The traditional notion of noncontextuality: Every vector is associated with the same regardless of how it is measured (i.e. the context) Traditional notion of noncontextuality |ψ1〉 |ψ3〉 |ψ2〉 |ψ1〉 |ψ′3〉 |ψ′2〉 1 0 0 1 0 0 The traditional notion of noncontextuality: For every λ, every basis of vectors receives a 0-1 valuation, wherein exactly one element is assigned the value 1 (corresponding to the outcome that would occur for λ), and every vector is assigned the same value regardless of the basis it is considered a part (i.e. the context). |ψ1〉 |ψ3〉 |ψ2〉 |ψ1〉 |ψ′3〉 |ψ′2〉 0 0 1 0 1 0 The traditional notion of noncontextuality: For every λ, every basis of vectors receives a 0-1 valuation, wherein exactly one element is assigned the value 1 (corresponding to the outcome that would occur for λ), and every vector is assigned the same value regardless of the basis it is considered a part (i.e. the context). Ernst Specker (with son) and Simon Kochen Bell-Kochen-Specker theorem: A noncontextual hidden variable model of quantum theory for Hilbert spaces of dimension 3 or greater is impossible. John S. Bell Example: The CEGA algebraic 18 ray proof in 4d: Cabello, Estebaranz, Garcia-Alcaine, Phys. Lett. A 212, 183 (1996) 0,0,0,1 0,0,1,0 1,1,0,0 1,-1,0,0 0,0,0,1 0,1,0,0 1,0,1,0 1,0,-1,0 1,-1,1,-1 1,-1,-1,1 1,1,0,0 0,0,1,1 1,-1,1,-1 1,1,1,1 1,0,-1,0 0,1,0,-1 0,0,1,0 0,1,0,0 1,0,0,1 1,0,0,-1 1,-1,-1,1 1,1,1,1 1,0,0,-1 0,1,-1,0 1,1,-1,1 1,1,1,-1 1,-1,0,0 0,0,1,1 1,1,-1,1 -1,1,1,1 1,0,1,0 0,1,0,-1 1,1,1,-1 -1,1,1,1 1,0,0,1 0,1,-1,0 In each of the 9 columns, one ray is assigned 1, the other three 0 Therefore, 9 rays must be assigned 1 But each ray appears twice and so there must be an even number of rays assigned 1 CONTRADICTION! Each of the 18 rays appears twice in the following list Example: The CEGA algebraic 18 ray proof in 4d: Cabello, Estebaranz, Garcia-Alcaine, Phys. Lett. A 212, 183 (1996) 0,0,0,1 0,0,1,0 1,1,0,0 1,-1,0,0 0,0,0,1 0,1,0,0 1,0,1,0 1,0,-1,0 1,-1,1,-1 1,-1,-1,1 1,1,0,0 0,0,1,1 1,-1,1,-1 1,1,1,1 1,0,-1,0 0,1,0,-1 0,0,1,0 0,1,0,0 1,0,0,1 1,0,0,-1 1,-1,-1,1 1,1,1,1 1,0,0,-1 0,1,-1,0 1,1,-1,1 1,1,1,-1 1,-1,0,0 0,0,1,1 1,1,-1,1 -1,1,1,1 1,0,1,0 0,1,0,-1 1,1,1,-1 -1,1,1,1 1,0,0,1 0,1,-1,0 In each of the 9 columns, one ray is assigned 1, the other three 0 Therefore, 9 rays must be assigned 1 But each ray appears twice and so there must be an even number of rays assigned 1 CONTRADICTION! Each of the 18 rays appears twice in the following list  Example: Kochen and Specker’s original algebraic 117 ray proof in 3d Example: Clifton’s state-specific 8 ray proof in 3d CONTRADICTION! The traditional notion of noncontextuality: For every λ, every projector  P is assigned a value 0 or 1 regardless of how it is measured (i.e. the context) Coarse-graining of a measurement implies a coarse- graining of the value (because it is just post-processing) Every measurment has some outcome for all P Example: Bell’s proof in 3d based on Gleason’s theorem CONTRADICTION But there is no ρ such that ω(P)=0 or 1 for all P (Any given ρ can only achieve a 0-1 valuation on a single basis) the value assigned to A should be independent of whether it is measured together with B or together with C (i.e. the context) The traditional notion of noncontextuality: Measure A = measure projectors onto eigenspaces of A, Measure A with B = measure projectors onto joint eigenspaces of A and B, then coarse-grain over B outcome is independent of context is independent of contextTherefore For Hermitian operators A, B, C satisfying Measure A with C = measure projectors onto joint eigenspaces of A and C, Then coarse-grain over C outcome Functional relationships among commuting Hermitian operators must be respected by their values Proof: the possible sets of eigenvalues one can simultaneously assign to L, M, N,… are specified by their joint eigenstates. By acting the first equation on each of the joint eigenstates, we get the second. If then Example: Mermin’s magic square proof in 4d I I I I I ¡ I v(X1) v(X2) v(X1X2) = 1 v(Y1) v(Y2) v(Y1Y2) = 1 v(X1Y2) v(Y1X2) v(Z1Z2) = 1 v(X1) v(Y2) v(X1Y2) = 1 v(Y1) v(X2) v(Y1X2) = 1 v(X1X2) v(Y1Y2) v(Z1Z2) = −1 X1 X2 Y2 Y1 X1X2 Y1Y2 Z1Z2X1Y2 Y1X2 Product of LHSs = +1 Product of RHSs = -1 CONTRADICTION X1 X2 X1X2 = I Y1 Y2 Y1Y2 = I X1Y2 Y1X2 Z1Z2 = I X1 Y2 X1Y2 = I Y1 X2 Y1X2 = I X1X2 Y1Y2 Z1Z2 = −I Aside: Local determinism is an instance of traditional noncontextuality where the context is remote is either measured withSAa - I B IA - SBb IA - SBb′or with v(SAa )Therefore                 is the same for the two contexts the value assigned to A should be independent of whether it is measured together with B or together with C (i.e. the context) Recall traditional noncontextuality: For Hermitian operators A, B, C satisfying Every proof of the impossibility of a locally deterministic model is a proof of the impossibility of a traditional noncontextual model This is local determinism jψiAB = 1 2 4∑ i=1 jiiAjiiB Aside: Traditional noncontextuality can sometimes be justified by local causality Perfect correlation when same mmt is made on both wings + local causality  Traditional noncontextual hidden variable model for mmts on one wing CONTRADICTION! The generalized notion of noncontextuality Problems with the traditional definition of noncontextuality: - applies only to sharp measurements - applies only to deterministic hidden variable models - applies only to models of quantum theory A better notion of noncontextuality would determine - whether any given theory admits a noncontextual model - whether any given experimental data can be explained by a noncontextual model Preparation P A realist model of an operational theory Measurement M A realist model of an operational theory is noncontextual if Operational equivalence of two experimental procedures Equivalent representations in the realist model Generalized definition of noncontextuality: P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Operational equivalence classes P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Difference of Equivalence class P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Difference of context P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Example from quantum theory Different density op’s P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Example from quantum theory P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Example from quantum theory P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Preparation noncontextual model P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Preparation contextual model P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Definition of preparation noncontextual model: λλ λ λ λ (a) Some states of a qubit (c) A preparation contextual model of these (Kochen-Specker, 1967) (a) (b) (c) (b) A preparation noncontextual model of these (RWS, PRA 75, 032110, 2007) P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Difference of context P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 |ψ1〉 |ψ3〉 |ψ2〉 |ψ1〉 |ψ′3〉 |ψ′2 〉 Example from quantum theory P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Example from quantum theory P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Example from quantum theory P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Example from quantum theory E = qj π 4 i hπ 4 j + (1 ¡ q)1 2 I f E, I ¡ Eg f E, I ¡ Eg E = 1 2 j0i h0j + 1 2 j+i h+j P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Measurement noncontextual model P2 P4 P3 P6P5 P1 M1 M5 M3 M10M8 P9 P8 P7 M2 M6 M4 M7 M9 Measurement contextual model universal noncontextuality = noncontextuality for preparations and measurements Generalized noncontextuality in quantum theory Defining noncontextuality in quantum theory P P’ Preparation Noncontextuality in QT Defining noncontextuality in quantum theory M M’ Measurement Noncontextuality in QT Preparation-based proof of contextuality (i.e. of the impossibility of a noncontextual realist model of quantum theory) Important features of realist models Representing one-shot distinguishability: Representing convex combination: Proof based on finite construction in 2d Proof based on finite construction in 2d σa σb σA σc σCσB  By preparation noncontextuality  CONTRADICTION But then the RHS of decomposition (4) is From decompositions (1)-(3) Example: A “reverse” Gleason theorem for all dimensions for some effect Ελ Suppose preparation noncontextuality CONTRADICTION If one knew λ, one could retrodict with certainty which state was prepared from an orthogonal basis, for any basis. There is no effect such that finding it would allow one to achieve such a retrodiction. is convex-linear in ρ By preparation noncontextuality But PNC for σx cannot be justified by local causality PNC for I/2 can be justified by local causality Aside: justifying preparation noncontextuality by local causality Any bipartite Bell-type proof of nonlocality proof of preparation contextuality→ (proof due to Jon Barrett) Also, Measurement contextuality New definition versus traditional definition |ψ1〉 |ψ3〉 |ψ2〉 |ψ1〉 |ψ′3〉 |ψ′2〉 How to formulate the traditional notion of noncontextuality: This is equivalent to assuming: |ψ1〉 |ψ3〉 |ψ2〉 |ψ1〉 |ψ′3〉 |ψ′2〉 M M’ coarse-grain |ψ2〉 and |ψ3〉measure coarse-grain |ψ’2〉 and |ψ’3〉measure traditional notion of noncontextuality = M But recall that the most general representation was Therefore: outcome determinism for sharp measurements revised notion of noncontextuality for sharp measurements and So, the new definition of noncontextuality is not simply a generalization of the traditional notion For sharp measurements, it is a revision of the traditional notion Local determinism: We ask: Does the outcome depend on space-like separated events (in addition to local settings and λ)? Local causality: We ask: Does the probability of the outcome depend on space-like separated events (in addition to local settings and λ)? Traditional notion of measurement noncontextuality: We ask: Does the outcome depend on the measurement context (in addition to the observable and λ)? The revised notion of measurement noncontextuality: We ask: Does the probability of the outcome depend on the measurement context (in addition to the observable and λ)? Noncontextuality and determinism are separate issues No-go theorems for previous notion are not necessarily no-go theorems for the new notion! In face of contradiction, could give up ODSM traditional notion of noncontextuality = outcome determinism for sharp measurements revised notion of noncontextuality for sharp measurements and preparation noncontextuality outcome determinism for sharp measurements However, one can prove that |ψ1〉 |ψ3〉 |ψ2〉 Proof preparation noncontextuality outcome determinism for sharp measurements preparation noncontextuality measurement noncontextuality and Therefore: outcome determinism for sharp measurements measurement noncontextuality and We’ve established that preparation noncontextuality outcome determinism for sharp measurements preparation noncontextuality measurement noncontextuality and Therefore: Traditional notion of noncontextuality … and there are many new proofs no-go theorems for the traditional notion of noncontextuality can be salvaged as no-go theorems for the generalized notion We’ve established that Measurement-based proof of contextuality (i.e. of the impossibility of a noncontextual realist model of quantum theory) Proof of contextuality for unsharp measurements in 2d By definition By outcome determinism for sharp measurements By the assumption of measurement noncontextuality CONTRADICTION Example: A variant of Busch’s generalization of Gleason to 2d Busch, Phys. Rev. Lett. 91, 120403 (2003) CONTRADICTION But there is no ρ such that Tr(ρ P)=0 or 1 for all P (Any given ρ can only achieve a 0-1 valuation on a single basis) By outcome determinism for sharp measurements ξP (λ) = 0 or 1   for all projectors P Measurement noncontextuality is convex linear in E for some density operator ρλ ξE(λ) considered as a function of E satisfies the conditions of the generalized Gleason’s theorem The mystery of contextuality There is a tension between 1) the dependence of representation on certain details of the experimental procedure and 2) the independence of outcome statistics on those details of the experimental procedure Noncontextuality and the characterization of classicality From: qis.ucalgary.ca/quantech/wiggalery.html Continuous Wigner function for a harmonic oscillator Common slogan: A quantum state is nonclassical if it has a negative Wigner representation Negativity is not necessary for nonclassicality: the nonclassicality could reveal itself in the negativity of the representation of the measurement rather than the state Negativity is not sufficient for nonclassicality: When considering possibilities for a classical explanation, we need to look at representations other than that of Wigner Classicality as non-negativity Better to ask whether a quantum experiment admits of a classical explanation Examples: • Wigner representation • discrete Wigner representation • Q representation of quantum optics • P representation of quantum optics • Hardy-type formulation of QM using fiducial measurements • Hardy-type formulation of QM using fiducial preparations • … Quasi-probability representations of QM: (e.g. Wootters, quant-ph/0306135) States Measurements See Ferrie and Emerson, J. Phys. A 41 352001 (2008) This provides a classical explanation if and only if for all ρ for all {Ek} A quantum experiment is nonclassical if it fails to admit a quasi-probability representation that is nonnegative for all states and measurements Classicality from nonnegativity, take II: Quasi-probability representations of QM: States Measurements Quasi-probability representations of QM: Nonnegative quasi-probability representation of QM Noncontextual ontological model of QM= Equivalent notions of classicality This provides a classical explanation if and only if for all ρ for all {Ek} States Measurements Noncontextuality inequalities and applications of contextuality Based on noncontextuality-inequality violation Parity-oblivious multiplexing RS, Buzacott, Keehn, Toner, Pryde, PRL 102, 010401 (2009) Quantum Spellcraft Secure key distribution? Horodecki4, Pawlowski, Bourennane, arXiv:1002.2410 Computational advantages? Raussendorf, arXiv:0907.5449 Anders and Browne, Phys. Rev. Lett. 102, 050502 (2009) Why isn’t the world more contextual? The game of parity-oblivious multiplexing b=xy Alice and Bob win if x0, x1 y b Victor Alice Bob The catch: no information about parity (x0 ⊕ x1) can be conveyed! Theorem: For all theories admitting a preparation noncontextual model p(b=xy) ≤ 3/4 A “noncontextuality inequality” RWS, Buzacott, Keehn, Toner, Pryde, PRL 102, 010402 (2009)

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