Foundations of Quantum Theory Provide an adequate interpretation Explore nonclassical phenomena Determine principles from which quantum theory may be derived What’s the problem? “Orthodox” postulates of quantum theory 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. 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 Inconsistencies of the orthodox interpretation By the collapse postulate (applied to the system) By unitary evolution postulate (applied to isolated system that includes the apparatus) Indeterministic and discontinuous evolution Deterministic and continuous evolution Determinate properties Indeterminate properties The quantum measurement problem If the measurement apparatus is treated externally If the measurement apparatus is treated internally U is a linear operator False starts on the measurement problem • Interpret coherent superposition as disjunction Means either or with probabilities |a|2 and |b|2 respectively This is a denial of the representational completeness of ψ False starts on the measurement problem • 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 ψ False starts on the measurement problem • Appeal to environment-induced decoherence This doesn’t help False starts on the measurement problem • Appeal to differences in the state of the apparatus But for the interaction to be considered a measurement, we require And by linearity The postulated evolution does not correspond to a proper measurement Responses to the measurement problem 1. Deny universality of quantum dynamics • Quantum-classical hybrid models • Collapse models 2. Deny representational completeness of ψ • 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 Effective preparation Measurement Operational Quantum Mechanics Preparation Transformation Measurement Effective Measurement The real formalism of operational quantum theory Operational Quantum Mechanics Preparation Density operator Measurement 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 Density operator Transformation Trace-preserving completely positive linear map (CP map) Measurement Positive operator-valued measure (POVM) Operational Quantum Mechanics Preparation Transformation Effective preparation ρ ! ρ′ = T (ρ) Measurement Operational Quantum Mechanics Preparation Transformation Measurement Effective Measurement Operational postulates of quantum theory 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 Every transformation is associated with a trace-preserving completelypositive linear map ρ ! ρ′ = T (ρ) Every measurement outcome k is associated with a tracenonincreasing completely-positive linear map Tk such that No mention of “physical states” or their evolution How density operators and POVMs arise in the operational approach Operational Quantum Mechanics Preparation Measurement Vector Hermitian operator Operational Quantum Mechanics Preparation Measurement Density operator Hermitian operator Ensembles jψ1 i f pi g ρ f Πk g jψ2 i jψ3 i p( k) = = = i i i p( k|i) p( i) ψi|¦ k |ψipi T r( ¦ k |ψiψi|) pi = T r( ¦ k i pi|ψiψi|) where p( k) = T r( ¦ k ρ) ρ= i pi|ψiψi| “Density operator” hψjρjψi ¸ 0 8jψi 2 H Unit trace Tr(ρ) = 1 Positive Reduced density operators a ρs jψi sa (s) s ( s) p( k) = T r sa[( ¦ k ⊗ Ia) |ψsaψ|] ( s) = T r s[¦ k ( T r a( |ψsaψ|) ] ( s) f Ia g p( k) = T r( ¦ k ρs) where ρs = T r a ( |ψsaψ|) f Πk g ↔ Pure preparation ↔ Mixed preparation Multiplicity of convex decompositions Multiplicity of purifications Operational Quantum Mechanics Preparation Measurement Density operator Hermitian operator Operational Quantum Mechanics Preparation Measurement Density operator Projection valued measure (PVM) Operational Quantum Mechanics Preparation Density operator Measurement Position operator valued measure (POVM) Mixtures of measurements k = (i, j) f pi g (1) f Πj g ρ (2) f Πj g (3) f Πj g p( i, j) = p( j|i) p( i) ( i) = T r( ¦ j ρ) pi ( i) = T r( pi¦ j ρ) Ei,j p( k) = T r( Ek ρ) hψjEk jψi ¸ 0 8jψi 2 H Sum to identity k Ek = I Positive {Ek } “Positive operator valued measure (POVM)” 1 1 f , g 2 2 k = (i, j) f j0i h0j, j1i h1jg f j+i h+j, j¡ i h¡ jg 1 1 1 1 f j0i h0j , j1i h1j , j+i h+j , j¡ i h¡ jg 2 2 2 2 Recall 1 4 j0i h0j + 14 j1i h1j + 14 j+i h+j + 14 j¡ i h¡ j = 12 I Coarse-graining k 2 f 1, 2, 3, ...g {Ek } j 2 f 1, 2, 3, 4, 5, 6, 7, 8, 9, ...g {Fj } p( k) = T r( Ek ρ) = j∈Sk j∈Sk = T r[( p( j) T r( Fj ρ) ∀ρ j∈Sk Ek = j∈Sk Fj Fj ) ρ] ∀ρ Note: the Ek need not be rank 1 Example 1 1 f , g 2 2 k = (i, j) f j0i h0j, j1i h1jg f j+i h+j, j¡ i h¡ jg 1 1 1 1 f j0i h0j , j1i h1j , j+i h+j , j¡ i h¡ jg 2 2 2 2 1 1 1 1 1 1 = f I , Ig f j0i h0j + j1i h1j , j+i h+j + j¡ i h¡ jg 2 2 2 2 2 2 Another example f q, 1 ¡ qg f j0i h0j, j1i h1jg 1 1 f I , Ig 2 2 1 1 f qj0i h0j , qj1i h1j , (1 ¡ q) I , (1 ¡ q) Ig 2 2 1 1 f qj0i h0j + (1 ¡ q) I , qj1i h1j + (1 ¡ q) Ig 2 2 Noisy S·z =f 1+q 1¡ q 1¡ q 1+q j0i h0j + j1i h1j , j0i h0j + j1i h1jg 2 2 2 2 1 1 1 1 f j0i h0j , j1i h1j , j+i h+j , j¡ i h¡ jg 2 2 2 2 1 1 1 1 f j0i h0j , j1i h1j , j+i h+j , j¡ i h¡ jg 2 2 2 2 f E 0 , E1 g Noisy S· n 1 1 1 1 f j0i h0j , j1i h1j , j+i h+j , j¡ i h¡ jg 2 2 2 2 f F 0 , F1 g Noisy S· n⊥ Note: General conditions for joint measurability of POVMs are not known Measurement by coupling to an ancilla (sa) f Πk ρs ( sa) p( k) = T r sa[¦ k ( ρs ⊗ τa) ] ( sa) = T r s[T r a( ¦ k τa) ρs] (s) Ek g Example f jΦ1 i , jΦ2 i , jΦ3 i , jΦ4 i g √ −1 = { 2 ( |0|0 + |1|1) , √ −1 2 ( |0|0 − 1|1) , √ −1 2 ( |0|1 + |1|0) , √ −1 2 ( |0|1 − |1|0) } ρs (s) (sa) Ek = Tra (Πk τa ) = hθj a jΦk i sa hΦk j sa jθi a hθj a jΦ1(2) i sa = hθj a jΦ3(4) i sa = p p 2 −1 2 [cos(θ/2)j0i s § sin(θ/2)j1i s ] = −1 [sin(θ/2)j0i s § cos(θ/2)j1i s ] = p p 2 −1 2 j § θi s −1 jπ ¨ θi s 1 1 1 1 f Ek g = f jθi hθj , j ¡ θi h¡ θj , jπ ¡ θi hπ ¡ θj , jπ + θi hπ + θjg θ = π/4 2 2 2 2 Naimark’s theorem: Every POVM can be implemented by coupling to an ancilla and implementing a projective measurement Operational Quantum Mechanics Preparation Density operator Measurement Position operator valued measure (POVM) Towards an operational axiomatization of quantum theory Quantum theory Classical theory Category Theory Framework Possibilistic Theories Classical Statistical Theories with epistemic restriction Convex theories C* algebraic theories Convex theories with maximal dual cone A framework for convex operational theories Preparation sP = P r( 1|M, P ) P r( 2|M, P ) P r( 1|M ′, P ) P r( 2|M ′, P ) P r( 3|M ′, P ) .. Measurement rM,k = 0 0 0 1 0 .. A framework for convex operational theories Preparation Measurement Suppose there are K fiducial measurements (pass-fail mmts from which one can infer the statistics for all mmts) sP = P r( pass|M1 , P ) P r( pass|M2 , P ) ... P r( pass|MK , P ) “operational state” What can we say about f? Operational states form a convex set (w,1-w) Also true for fiducial mmts, so Closed under convex combination Convex linear 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 A convex operational theory Preparation sP ∈ S Measurement “operational states” “operational effects” S = Convex set R = Interval of positive cone S and R characterize the operational theory! Operational classical theory s can be any probability distributions S = a simplex r can be any vector of conditional probabilities R = the unit hypercube Operational quantum theory Recall: The Hermitian operators on H of dimension d form a real Euclidean vector space of dimension d2 The inner product is (A, B) = Tr(AB) S = the convex set of positive trace-one operators hψjE k jψi ¸ 0 8jψi 2 H k Ek = I R = the set of all positive operators less than identity 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? 1 1 1 1 f jθi hθj , j ¡ θi h¡ θj , jπ ¡ θi hπ ¡ θj , jπ + θi hπ + θjg 2 2 2 2 θ = π/4 Naimark’s theorem: Every POVM can be implemented by coupling to an ancilla and implementing a projective measurement Two approaches to axiomatization Operational approach P1 P2 M1 P4 Ontological approach M2 M4 M3 P6 P5 M6 P7 M10 P8 Preparations M8 M9 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) An ontological model of an operational theory Preparation P Measurement M Deterministic hidden variable model for pure states and projective measurements ψ |ψ1〉 |ψ2〉 |ψ3〉 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 Example: Statistically restricted classical theories Consider Einstein’s version of the EPR argument Suppose A and B share ψ = 1 2 If A measures B 0 0 + 1 1 {0 , 1} B’s state becomes If A measures A 0 1 with probability 1/2 with probability 1/2 + − with probability 1/2 with probability 1/2 {+ , − } B’s state becomes “Steering” 1 µ (λ ′, λ ) = ([11] + [22] + [33] + [44]) 4 λ Alice’s initial knowledge of B If A measures {1,2} vs. {3,4} λ with prob. 1/2 λ with prob. 1/2 λ with prob. 1/2 Her knowledge of B is updated to If A measures {1,3} vs. {2,4} Her knowledge of B is updated to λ with prob. 1/2 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 Categorizing quantum phenomena Those arising in a restricted statistical classical theory Noncommutativity Entanglement Collapse Wave-particle duality Teleportation No cloning Key distribution Improvements in metrology Quantum eraser Coherent superposition Pre and post-selection “paradoxes” Others… Type 1 Nonclassicality Those not arising in a restricted statistical classical theory Bell inequality violations Contextuality Computational speed-up Certain aspects of items on the left Others… Type 2 Nonclassicality Bell’s theorem John S. Bell (1928-1990) A pair of two-outcome measurements S S T T S S E S T E T S E T T E 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 S and S or T and T 2. Whenever different measurements are made on A and B, the outcomes always disagree S and T or T and S S S T 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 2 1. Whenever the same measurement is made on A and B, the outcomes always disagree S and S or T and T 2. Whenever different measurements are made on A and B, the outcomes always agree S and T or T and S S S T 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 S S T 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… Tension with the theory of relativity t x Outcome is registered Outcome is registered Mmt is chosen Mmt is chosen 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? No. It will only decrease the degree of correlation Is the proof robust to experimental imperfections? (e.g. the detector sometimes registers the wrong outcome) Yes. The Bell inequality may still be violated. If the detector inefficiencies are sufficiently high, can particles obeying local causality simulate the correlations on the detected pairs? Yes. This is the detector loophole. Is there a problem if the choice of measurement is made before the particles are sent to the detectors? Yes. This is the locality loophole. 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|T S) + p( disagree|T T ) ] 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 The Bell-inequality violation in quantum theory T jψi AB S S 1 p = (j0i A j0i B + j1i A j1i B ) T 2 1 1 p(success) = + p 2 2 2 ' 0.85 T S S T S S T T The Bell-inequality violation in quantum theory T jψi AB S S 1 p = (j0i A j0i B + j1i A j1i B ) T 2 1 1 p(success) = + p 2 2 2 ' 0.85 S T A h+n̂jψi AB S T 1 = [cos(θ/2)A h0j + sin(θ/2)A h1j] p (j0i A j0i B + j1i A j1i B ) 2 = cos(θ/2)j0i B + sin(θ/2)j1i B = j + n̂i B jh+n̂j A h+m̂j B jψi AB j 2 = jh+m̂j + n̂i j 2 = cos2 (θ/2) p( agree|SS) = p( agree|ST ) = p( agree|T S) = p( disagree|T T ) 1 1 2 = cos (π/8) = + p 2 2 2 No signalling in quantum theory {EkA} {FjB } ρAB p( j) = p( k, j) = k T r AB [ ( EkA ⊗ FjB ) ρAB ] k = T r AB [ ( I A ⊗ FjB ) ρAB ] Independent of choice of measurement at A Note that [EkA , FjB ] = 0 for A and B space-like separated 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 t x 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 Local causality p(XA jXB , λC ) = p(XA jλC ) t x A B a b λ a, b – settings A, B -- outcomes 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) Factorizability from local causality Recall Bayes’ rule p(A, B) = p(AjB)p(B) p(A, BjC) = p(AjB, C)p(BjC) therefore p(A, Bja, b, λ) = p(AjB, a, b, λ)p(Bja, b, λ) By local causality p(AjB, a, b, λ) = p(Aja, λ) p(Bja, b, λ) = p(Bjb, λ) Thus p(A, Bja, b, λ) = p(Aja, λ)p(Bjb, λ) a + - a’ + - 1 [ p( agree|ab) + p( agree|ab′) + 4 Define b + - b’ + - p( agree|a′b) + p( disagree|a′b′) ] · 3/4 C( a, b) = ( + 1) p( agree|ab) + ( −1) p( disagree|ab) ] |C( a, b) + C( a′ , b) + C( a, b′ ) − C( a′, b′) | · 2 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 Traditional notion of noncontextuality A given vector may appear in many different measurements |ψ1〉 |ψ2〉 |ψ3〉 |ψ1〉 |ψ′2〉 |ψ′3〉 The traditional notion of noncontextuality: Every vector is associated with the same regardless of how it is measured (i.e. the context) 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〉 1 |ψ2〉 0 |ψ3〉 |ψ1〉 0 1 |ψ′ 2〉 |ψ′3〉 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〉 0 |ψ2〉 1 |ψ3〉 |ψ1〉 0 0 |ψ′ 2〉 |ψ′3〉 0 1 John S. Bell 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. Example: The CEGA algebraic 18 ray proof in 4d: Cabello, Estebaranz, Garcia-Alcaine, Phys. Lett. A 212, 183 (1996) Each of the 18 rays appears twice in the following list 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! Example: The CEGA algebraic 18 ray proof in 4d: Cabello, Estebaranz, Garcia-Alcaine, Phys. Lett. A 212, 183 (1996) Each of the 18 rays appears twice in the following list 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! 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) for all P Every measurment has some outcome Coarse-graining of a measurement implies a coarsegraining of the value (because it is just post-processing) Example: Bell’s proof in 3d based on Gleason’s theorem 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) CONTRADICTION The traditional notion of noncontextuality: For Hermitian operators A, B, C satisfying the value assigned to A should be independent of whether it is measured together with B or together with C (i.e. the context) 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 Measure A with C = measure projectors onto joint eigenspaces of A and C, Then coarse-grain over C outcome is independent of context Therefore is independent of context Functional relationships among commuting Hermitian operators must be respected by their values If then 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. Example: Mermin’s magic square proof in 4d X1 X2 X1 X2 I Y2 Y1 Y1 Y2 I X1 Y2 Y1 X 2 Z1 Z2 I X1 X2 X1 X2 = I Y1 Y2 Y 1 Y2 = I X1 Y2 Y1 X2 Z1 Z2 = I X 1 Y2 X 1 Y2 = I Y1 X2 Y1 X2 = I X1 X2 Y1 Y2 Z1 Z2 = −I I I ¡ I v( X1 ) v( X2 ) v( X1 X2 ) = 1 v( Y1 ) v( Y2 ) v( Y1 Y2 ) = 1 v( X1 Y2 ) v( Y1 X2 ) v( Z1 Z2 ) = 1 Product of LHSs = +1 Product of RHSs = -1 v( X1 ) v( Y2 ) v( X1 Y2 ) = 1 v( Y1 ) v( X2 ) v( Y1 X2 ) = 1 v( X1 X2 ) v( Y1 Y2 ) v( Z1 Z2 ) = −1 CONTRADICTION Aside: Local determinism is an instance of traditional noncontextuality where the context is remote SaA - I B is either measured with I A - SbB or with I A - SbB′ Recall traditional noncontextuality: For Hermitian operators A, B, C satisfying the value assigned to A should be independent of whether it is measured together with B or together with C (i.e. the context) Therefore v(SaA ) is the same for the two contexts This is local determinism Every proof of the impossibility of a locally deterministic model is a proof of the impossibility of a traditional noncontextual model Aside: Traditional noncontextuality can sometimes be justified by local causality 4 jψi AB 1 = jii A jii B 2 i=1 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 A realist model of an operational theory Preparation P Measurement M Generalized definition of noncontextuality: A realist model of an operational theory is noncontextual if Operational equivalence of two experimental procedures Equivalent representations in the realist model M2 Operational equivalence classes M1 M4 M3 M5 M8 P1 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M9 M10 M2 M1 Difference of Equivalence class M3 M5 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M8 P1 M4 M9 M10 M2 M1 M4 M3 M5 Difference of context M8 P1 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M9 M10 M2 Example from quantum theory M1 M4 M3 Different density op’s M5 M8 P1 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M9 M10 M2 Example from quantum theory M1 M4 M3 M5 M8 P1 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M9 M10 M2 Example from quantum theory M1 M4 M3 M5 M8 P1 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M9 M10 M2 Preparation noncontextual model M1 M4 M3 M5 M8 P1 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M9 M10 M2 Preparation contextual model M1 M4 M3 M5 M8 P1 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M9 M10 M2 Definition of preparation noncontextual model: M1 M4 M3 M5 M8 P1 P2 P4 P3 P6 P5 P7 P9 P8 M7 M6 M9 M10 (a) (c) (b) λ (a) Some states of a qubit λ (b) A preparation noncontextual model of these (RWS, PRA 75, 032110, 2007) λ λ λ (c) A preparation contextual model of these (Kochen-Specker, 1967) M2 M1 M4 M3 M5 M8 P1 M7 M6 M9 M10 P2 P4 P3 P6 P5 P7 P9 P8 Difference of context M2 M1 M4 M3 M5 M9 M8 P1 M7 M6 M10 P2 P4 P3 |ψ1〉 P6 P5 P7 |ψ1〉 |ψ2〉 P9 |ψ3〉 |ψ′2 〉 P8 |ψ′3〉 Example from quantum theory M2 M1 M4 M3 M5 M8 P1 M7 M6 M9 M10 P2 P4 P3 P6 P5 P7 P9 P8 Example from quantum theory M2 M1 M4 M3 M5 M8 P1 M7 M6 M9 M10 P2 P4 P3 P6 P5 P7 P9 P8 Example from quantum theory M2 M1 M4 M3 M5 M8 P1 M7 M6 M9 M10 P2 P4 P3 P6 P5 P7 f E, I ¡ Eg P9 P8 π π 1 E = qj i h j + (1 ¡ q) I 4 4 2 f E, I ¡ Eg 1 1 E = j0i h0j + j+i h+j 2 2 Example from quantum theory M2 M1 M4 M3 M5 M8 P1 M7 M6 M9 M10 P2 P4 P3 P6 P5 P7 P9 P8 Measurement noncontextual model M2 M1 M4 M3 M5 M8 P1 M7 M6 M9 M10 P2 P4 P3 P6 P5 P7 P9 P8 Measurement contextual model universal noncontextuality = noncontextuality for preparations and measurements Generalized noncontextuality in quantum theory Defining noncontextuality in quantum theory Preparation Noncontextuality in QT P P’ Defining noncontextuality in quantum theory Measurement Noncontextuality in QT M M’ 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 σC σc σb σA By preparation noncontextuality From decompositions (1)-(3) But then the RHS of decomposition (4) is CONTRADICTION Example: A “reverse” Gleason theorem for all dimensions Suppose preparation noncontextuality is convex-linear in ρ for some effect Ελ 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. CONTRADICTION Aside: justifying preparation noncontextuality by local causality By preparation noncontextuality PNC for I/2 can be justified by local causality But PNC for σx cannot be justified by local causality Also, Any bipartite Bell-type proof of nonlocality (proof due to Jon Barrett) → proof of preparation contextuality Measurement contextuality New definition versus traditional definition How to formulate the traditional notion of noncontextuality: |ψ1〉 |ψ2〉 |ψ3〉 |ψ1〉 |ψ′2〉 |ψ′3〉 This is equivalent to assuming: M |ψ1〉 |ψ2〉 |ψ3〉 measure coarse-grain |ψ2 and |ψ3 M’ |ψ1〉 |ψ′2〉 coarse-grain |ψ’2 and |ψ’3 measure |ψ′3〉 But recall that the most general representation was M Therefore: traditional notion of = noncontextuality revised notion of noncontextuality for sharp measurements and outcome determinism for sharp measurements 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 traditional notion of = noncontextuality revised notion of noncontextuality for sharp measurements and outcome determinism for sharp measurements No-go theorems for previous notion are not necessarily no-go theorems for the new notion! In face of contradiction, could give up ODSM However, one can prove that preparation noncontextuality Proof |ψ1〉 |ψ2〉 |ψ3〉 outcome determinism for sharp measurements We’ve established that preparation noncontextuality outcome determinism for sharp measurements Therefore: measurement noncontextuality measurement noncontextuality and preparation noncontextuality and outcome determinism for sharp measurements We’ve established that preparation noncontextuality outcome determinism for sharp measurements Therefore: measurement noncontextuality and preparation noncontextuality Traditional notion of noncontextuality no-go theorems for the traditional notion of noncontextuality can be salvaged as no-go theorems for the generalized notion … and there are many new proofs 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) Measurement noncontextuality is convex linear in E ξE(λ) considered as a function of E satisfies the conditions of the generalized Gleason’s theorem for some density operator ρλ By outcome determinism for sharp measurements ξP(λ) = 0 or 1 for all projectors P 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) CONTRADICTION 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 Classicality as non-negativity Continuous Wigner function for a harmonic oscillator Common slogan: A quantum state is nonclassical if it has a negative Wigner representation Better to ask whether a quantum experiment admits of a classical explanation 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 From: qis.ucalgary.ca/quantech/wiggalery.html Quasi-probability representations of QM: States Measurements Examples: • Wigner representation • discrete Wigner representation (e.g. Wootters, quant-ph/0306135) • 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 •… See Ferrie and Emerson, J. Phys. A 41 352001 (2008) Quasi-probability representations of QM: States Measurements This provides a classical explanation if and only if for all ρ for all {Ek} Classicality from nonnegativity, take II: A quantum experiment is nonclassical if it fails to admit a quasi-probability representation that is nonnegative for all states and measurements Quasi-probability representations of QM: States Measurements This provides a classical explanation if and only if for all ρ Nonnegative quasi-probability representation of QM for all {Ek} = Noncontextual ontological model of QM Equivalent notions of classicality Noncontextuality inequalities and applications of contextuality Quantum Spellcraft Based on noncontextuality-inequality violation Parity-oblivious multiplexing RS, Buzacott, Keehn, Toner, Pryde, PRL 102, 010401 (2009) Computational advantages? Raussendorf, arXiv:0907.5449 Anders and Browne, Phys. Rev. Lett. 102, 050502 (2009) Secure key distribution? Horodecki4, Pawlowski, Bourennane, arXiv:1002.2410 Why isn’t the world more contextual? The game of parity-oblivious multiplexing Victor y x0, x1 b Bob Alice Alice and Bob win if b=xy 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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Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)
Foundations of Quantum Mechanics Spekkens, Rob 2010-07-26
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Title | Foundations of Quantum Mechanics |
Creator |
Spekkens, Rob |
Contributor |
Summer School on Quantum Information (10th : 2010 : Vancouver, B.C.) University of British Columbia. Department of Physics and Astronomy Pacific Institute for the Mathematical Sciences |
Date Issued | 2010-07-26 |
Description | The field of quantum foundations seeks to answer questions such as: What do the elements of the mathematical formalism of quantum theory represent? From what physical principles can the formalism be derived? What are the precise ways in which a quantum world differs from a classical world and other possible worlds? These lectures will cover some important foundational topics which touch upon these questions, in particular, operational and realist interpretations of the formalism, the quantum measurement problem, nonlocality and contextuality. |
Subject |
foundations quantum mechanics |
Genre |
Presentation |
Type |
Text Moving Image |
Language | eng |
Date Available | 2016-11-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0040936 |
URI | http://hdl.handle.net/2429/29961 |
Affiliation |
Non UBC |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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