Why the quantum? Insights from classical theories with a statistical restriction Classical statistical theory + fundamental restriction on statistical distributions ⇓ A large part of quantum theory In the sense of reproducing the operational predictions Classical statistical theory + fundamental restriction on statistical distributions ⇓ A large part of quantum theory In the sense of reproducing the operational predictions ψ ψ' i.e. quantum states emerge as statistical distributions (epistemic states) Classical theory Statistical theory for the classical theory Restricted Statistical theory for the classical theory Mechanics Liouville mechanics Restricted Liouville mechanics = Gaussian quantum mechanics Bits Statistical theory of bits Restricted statistical theory of bits ≃ Stabilizer theory for qubits Trits Statistical theory of trits Restricted statistical theory of trits = Stabilizer theory for qutrits These theories include: • Most basic quantum phenomena e.g. noncommutativity, Interference, coherent superposition, collapse, complementary bases, no-cloning, … • Most quantum information-processing tasks e.g. teleportation, key distribution, quantum error correction, improvements in metrology, dense coding, … • A large part of entanglement theory e.g. monogamy, distillation, deterministic and probabilistic single copy entanglement transformation, catalysis, … • A large part of the formalism of quantum theory e.g. Choi-Jamiolkowski isomorphism, Naimark extension, Stinespring dilation, multiple convex decompositions of states, … Categorizing quantum phenomena Those not arising in a restricted statistical classical theory Those arising in a restricted statistical classical theory Wave-particle duality noncommutativity Teleportation entanglement Quantized spectra Key distribution collapse Interference No cloning Coherent superposition Bell inequality violations Quantum eraser Bell-Kochen-Specker theorem Improvements in metrology Computational speed-up Particle statistics Pre and post-selection “paradoxes” Categorizing quantum phenomena Those arising in a restricted statistical classical theory Those not arising in a restricted statistical classical theory Interference Bell inequality violations Noncommutativity Computational speed-up Entanglement Bell-Kochen-Specker theorem Collapse Certain aspects of items on the left Wave-particle duality Others… Teleportation No cloning Key distribution Improvements in metrology Quantum eraser Coherent superposition Pre and post-selection “paradoxes” Others… Quantized spectra? Particle statistics? Others… Categorizing quantum phenomena Those arising in a restricted statistical classical theory Those not arising in a restricted statistical classical theory Interference Bell inequality violations Noncommutativity Computational speed-up Entanglement Bell-Kochen-Specker theorem Collapse Certain aspects of items on the left Wave-particle duality Others… Teleportation Still surprising! No cloning Key distribution Find more! Not so strange after all! Improvements in metrology Focus on these Quantum eraser Coherent superposition Pre and post-selection “paradoxes” Others… Quantized spectra? Particle statistics? Others… A research program Speculative possibility for an axiomatization of quantum theory Principle 1: There is a fundamental restriction on observers capacities to know and control the systems around them Principle 2: ??? (Some change to the classical picture of the world) Classical theory Statistical theory for the classical theory Restricted Statistical theory for the classical theory Mechanics Liouville mechanics Restricted Liouville mechanics = Gaussian quantum mechanics Bits Statistical theory of bits Restricted statistical theory of bits ≃ Stabilizer theory for qubits Trits Statistical theory of trits Restricted statistical theory of trits = Stabilizer theory for qutrits Classical theory Statistical theory for the classical theory Restricted Statistical theory for the classical theory Mechanics Liouville mechanics Restricted Liouville mechanics = Quadrature quantum mechanics Bits Statistical theory of bits Restricted statistical theory of bits ≃ Stabilizer theory for qubits Trits Statistical theory of trits Restricted statistical theory of trits = Stabilizer theory for qutrits Classical complementarity as a statistical restriction with broad applicability Joint work with Olaf Schreiber Building upon: RS, quant-ph/0401052 [Phys. Rev. A 75, 032110 (2007)] S. van Enk, arxiv:0705.2742 [Found. Phys. 37, 1447 (2007)] D. Gross, quant-ph/0602001 [J. Math. Phys. 47, 122107 (2006)] S. Bartlett, T. Rudolph, RS, unpublished A fact about operational quantum theory: Jointly-measurable observables = a commuting set of observables (relative to matrix commutator) This suggests a restriction on a classical statistical theory: Jointly-knowable variables = a commuting set of variables (relative to Poisson bracket) Continuous degrees of freedom Configuration space: Rn ∋ ( x1 , x2 , . . . , xn) Phase space: - ≡ R2n ∋ ( x1 , p1 , x2 , p2 , . . . , xn, pn) ≡ m Functionals on phase space: F : - → R Xk ( m) = xk Pk ( m) = pk Poisson bracket of functionals: The linear functionals / canonical variables are: Independent of m Discrete degrees of freedom Zd = {0, 1, . . . , d − 1} Configuration space:( Zd) n ∋ ( x1 , x2 , . . . , xn) Phase space: - ≡ ( Zd) 2n ∋ ( x1 , p1 , x2 , p2 , . . . , xn, pn ) ≡ m Functionals on phase space: F : - → Zd Xk ( m) = xk Pk ( m) = pk Poisson bracket of functionals: ( F [m + exi ] − F [m]) ( G[m + epi ] − G[m]) −( F [m + epi ] − F [m]) ( G[m + eqi ] − G[m]) The linear functionals / canonical variables are: Independent of m A canonically conjugate pair e.g. A commuting pair e.g. The principle of classical complementarity: An observer can only have knowledge of the values of a commuting set of canonical variables and is maximally ignorant otherwise. Symplectic geometry Symplectic inner product ω( m, m′) = mT Jm′ ω : - ×- →R where Thus ω( m, m′) = ′ ′ i q i pi − pi q i The linear functionals F= i ( aiXi + bi Pi ) form a dual space {X1 , P1 , . . . , Xn, Pn} is dual to {ex1 , ep1 , . . . , exn , epn } Poisson bracket of functionals = symplectic inner product of vectors Valid epistemic states: These are specified by: Example: A set of known variables V V = {X1 , P2 } ∀F, G ∈ V : [F, G] = 0 A valuation of the known variables v( X1 ) = 2, v( P2 ) = 2 v : V → R( Z d ) Equivalently, An isotropic subspace V ⊆ Ω∗ v ∀F, G ∈ V : ω( F, G) = 0 A valuation vector v ∈ V*⊆ Ω v : ∀F ∈ V, F T v = v( F ) V The ontic states consistent with the epistemic state (V,v) are {m ∈ - | ∀F ∈ V : F ( m) = v( F ) } = {m ∈ - | ∀F ∈ V : F T m = F T v} = {m ∈ - | PV m = v} ≡V⊥+ v (Dirac-delta / Kronecker delta) The associated distribution is 1δ pV,v ( m)) = N V ⊥ + v ( m)) Example V = {X1 , X2 } v( X1 ) = 1, v( X2 ) = 2 V ⊥ + v = {m ∈ - | X1 ( m) = 1, X2 ( m) = 2} = {( 1, s, 2, t) | s, t ∈ R} “Heisenberg picture” and “Schrodinger picture” v Valid reversible transformations: Those that preserve the Poisson bracket / symplectic inner product: The group of symplectic affine transformations (Clifford group) where and Symplectic Affine (Heisenberg-Weyl) Valid reproducible measurements: Any commuting set of canonical variables Restricted Liouville mechanics - 2n = R X P Valid epistemic states for a single degree of freedom X X V v P v X V P X V v P V=∅ P Valid epistemic states for a pair of degrees of freedom Restricted statistical theory of trits - = ( Z3 ) 2n 2 1 0 0 1 2 Valid epistemic states for a single trit Commuting sets: The singleton sets the empty set Canonical variables known known 2 1 0 2 1 0 0 1 2 2 1 0 2 1 0 0 1 2 2 1 0 0 1 2 0 1 2 2 1 0 0 1 2 0 1 2 2 1 0 0 1 2 Nothing known 2 1 0 0 1 2 0 1 2 2 1 0 2 1 0 known 2 1 0 2 1 0 0 1 2 0 1 2 2 1 0 known 0 1 2 0 1 2 Valid epistemic states for a pair of trits a1 , b1 , a2 , b2 ∈ Z3 Canonical variables How to represent this graphically 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 1 variable known known 22 21 20 12 11 10 02 01 00 known 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 00 01 02 10 11 12 20 21 22 2 variables known and known 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 1 variable known known 22 21 20 12 11 10 02 01 00 known 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 00 01 02 10 11 12 20 21 22 2 variables known and known 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 Valid reproducible measurements On a single trit 2 1 0 2 1 0 0 1 2 2 1 0 2 1 0 0 1 2 0 1 2 0 1 2 On a pair of trits 22 21 20 12 11 10 02 01 00 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 etc. 00 01 02 10 11 12 20 21 22 Restricted statistical theory of bits - = ( Z2 ) 2n 1 0 0 1 A single bit a, b ∈ Z2 Canonical variables Epistemic states of maximal knowledge known 1 0 known 1 0 0 1 1 0 1 0 0 1 1 0 0 1 known 0 1 1 0 0 1 0 1 Epistemic states of non-maximal knowledge Nothing known 1 0 0 1 A pair of bits a1 , b1 , a2 , b2 ∈ Z2 Canonical variables 11 10 01 00 00 01 10 11 1 variable known known known 11 10 01 00 11 10 01 00 00 01 10 11 00 01 10 11 2 variables known and known 11 10 01 00 00 01 10 11 1 variable known known known 11 10 01 00 11 10 01 00 00 01 10 11 00 01 10 11 2 variables known and 11 10 01 00 00 01 10 11 known 1 0 1 0 0 1 1 0 0 1 0 1 Equivalence of these restricted statistical theories to “subtheories” of quantum theory Look to a representation of quantum theory on phase space – the Wigner representation Restricted Liouville mechanics = Quadrature Quantum Mechanics - 2n = R X P Quadrature quantum mechanics ^ : L2 ( Rn) → L2 ( Rn) Hermitian operators: F Commutator: The quadrature operators are: Quadrature states are eigenstates of a commuting set of quadrature operators Specified by an isotropic subspace V and a valuation vector v∈V (Quadrature transformations and measurements take quadrature states to quadrature states) Wigner representation of quantum mechanics Weyl operator Quantum state ρ Characteristic function Wigner function For quadrature state associated with V, v 1δ WV,v ( m) = N V ⊥ + v ( m) Equivalence of states implies equivalence of measurements and transformations Therefore Theorem: Restricted statistical Liouville mechanics is empirically equivalent to quadrature quantum mechanics Restricted statistical theory of trits = Stabilizer theory for qutrits - = ( Z3 ) 2n 2 1 0 0 1 2 C C3 Equivalence of states implies equivalence of measurements and transformations Therefore Theorem: The restricted statistical theory of trits is empirically equivalent to the Stabilizer theory for qutrits Restricted statistical theory of bits ≃ Stabilizer theory for qubits - = ( Z2 1 0 0 1 2n ) Analogously to what we did for trits, one can: Define stabilizer theory for qubits Define Gross’ discrete Wigner function for qubits Find: Wigner function can be negative for qubit stabilizer states The restricted statistical theory of bits is not equivalent but very close to the Stabilizer theory for qubits Knowledge balance vs. classical complementarity Contrast: The principle of classical complementarity: An observer can only have knowledge of the values of a commuting set of canonical variables and otherwise is maximally ignorant. The knowledge-balance principle: The only distributions that can be prepared are those that correspond to knowing at most half the information From: RS, quant-ph/0401052 [Phys. Rev. A 75, 032110 (2007)] The same epistemic states are found to be valid, but the logic is different… Example: 11 10 01 00 is forbidden 00 01 10 11 Knowledge-balance principle: It is forbidden by an assumption of locality and the existence of nontrivial measurements: Find X1=1 Principle of epistemic complementarity: It is forbidden because it corresponds to X2 = 0 and X1 + P2 = 0 but [X2, X1 + P2] ≠ 0 What about applying knowledge-balance to trits? (See S. van Enk, arxiv:0705.2742) Valid epistemic states for a pair of systems are different! and known 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 Allowed by knowledge-balance, but corresponding to nothing in QM! Knowledge-Balance Principle 2003-2008 Long live Symplectic Structure! Beyond classical complementarity: could a different statistical restriction get us closer to quantum theory? NO for discrete degrees of freedom Supplementing the unitary representation of the Clifford group with a single non-Clifford unitary yields all unitaries YES for continuous degrees of freedom In addition to rotations and displacements in phase space, one can add squeezing – one gets all the quadratic Hamiltonians (Bartlett, Rudolph, Spekkens, unpublished) The classical uncertainty principle: The only Liouville distributions that can be prepared are those satisfying and that have maximal entropy for a given set of second-order moments. The theory is empirically equivalent to Gaussian quantum mechanics Why the restricted statistical theory of bits Is not equivalent to qubit stabilizer theory Even number of correlations Odd number of correlations Qubit stabilizer theory is nonlocal and contextual (e.g. GHZ) Restricted statistical theory of bits is local and noncontextual According to Knowledge-Balance Valid epistemic states for a single system 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 Valid epistemic states for a pair of systems Plus permutations of rows and columns The same epistemic states are found to be valid, but the logic is different… Example: 11 10 01 00 is forbidden 00 01 10 11 Knowledge-balance principle: It is forbidden by an assumption of locality and the existence of nontrivial measurements: Find X1=1 Are the two theories always equivalent? Principle of epistemic complementarity: It is forbidden because it corresponds to X2 = 0 and X1 + P2 = 0 but [X2, X1 + P2] ≠ 0 What about applying knowledge-balance to trits? (See S. van Enk, arxiv:0705.2742) Valid epistemic states for a single system are the same Valid epistemic states for a pair of systems are slightly different! and and known 22 21 20 12 11 10 02 01 00 known 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 Ruled out by locality 00 01 02 10 11 12 20 21 22 Allowed by locality, but corresponding to nothing in QM! Valid epistemic states for a pair of degrees of freedom 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 How to represent this graphically 2 1 0 00 01 02 10 11 12 20 21 22 0 1 2 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 Uncorrelated pure epistemic states X2=0 X1=0 P1=0 X1+P1=0 X1-P1=0 P2=0 X2+P2=0 X2-P2=0 Correlated pure epistemic states X1+X2=0 P1-P2=0 X1+X2=0 X1+P1-P2=0 X1+X2=0 X1-P1+P2=0 X1-X2=0 P1+P2=0 X1-X2=0 X1-P1-P2=0 X1-X2=0 P1+X2+P2=0 X1-P2=0 P1-X2=0 X1-P2=0 P1-X2+P2=0 X1-P2=0 P1-X2-P2=0 P1-X2=0 X1+P1-P2=0 X1+P1-X2=0 X1+X2-P2=0 P1-P2=0 X1-P1+X2=0 P1-X2=0 X1-P1-P2=0 X1+P1-X2=0 X1-P1-X2-P2=0 X1-X2-P2=0 X1-P1+X2=0 X1+P2=0 P1+X2=0 X1+P2=0 P1+X2+P2=0 X1+P2=0 P1+X2-P2=0 P1+X2=0 X1+X2+P2=0 X1+P1-P2=0 P1-X2+P2=0 X1+P1+P2=0 P1-P2=0 P1+X2=0 X1+P1+P2=0 P1+P2=0 X1-X2+P2=0 X1-P1-P2=0 P1-X2-P2=0 Valid reversible transformations 1 trit example: 2 trit example: Valid reproducible measurements On a single trit 2 1 0 2 1 0 0 1 2 2 1 0 2 1 0 0 1 2 0 1 2 0 1 2 On a pair of trits 22 21 20 12 11 10 02 01 00 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 etc. 00 01 02 10 11 12 20 21 22 Uncorrelated pure epistemic states X2=0 X1=0 P1=0 X1+P1=0 P2=0 X2+P2=0 Correlated pure epistemic states 1 0 0 1 11 10 01 00 00 01 10 11 1 0 1 0 0 1 1 0 0 1 0 1
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Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)
Why the quantum? Insights from classical theories with a statistical restriction Spekkens, Robert Jul 23, 2010
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Title | Why the quantum? Insights from classical theories with a statistical restriction |
Creator |
Spekkens, Robert |
Contributor | University of British Columbia. Department of Physics and Astronomy Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics Pacific Institute for the Mathematical Sciences |
Date Issued | 2010-07-23 |
Description | A significant part of quantum theory can be obtained by postulating a single conceptual innovation relative to classical theories, namely, that agents face a fundamental restriction on what they can know about the physical state of any system. This talk will consider a particular sort of statistical restriction wherein only classical variables with vanishing Poisson bracket can be known simultaneously. When this principle is applied to a classical statistical theory of three-level systems (trits), the resulting theory is found to be operationally equivalent to the stabilizer formalism for qutrits. Applied to a classical theory of harmonic oscillators, it yields quantum mechanics restricted to quadrature eigenstates and observables. Finally, applied to a classical statistical theory of bits, it yields a theory that is almost equivalent to (but interestingly different from) the stabilizer formalism for qubits. I will discuss the significance of these results for the project of deriving the formalism of quantum theory from physical principles. |
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Language | eng |
Date Available | 2016-11-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
IsShownAt | 10.14288/1.0103165 |
URI | http://hdl.handle.net/2429/30074 |
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Peer Review Status | Unreviewed |
Scholarly Level | Faculty |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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