N-qubit Proofs of the Kochen-Specker Theorem

Transcription

1 N-qubit Proofs of the Kochen-Specker Theorem Mordecai Waegell P.K. Aravind Worcester Polytechnic Institute Worcester, MA, USA

2 Introduction Proofs of the 2-qubit KS Theorem based on regular polytopes in 4 dimensions, and using only real-valued quantum states. Nonclassical Structures within the N-qubit Pauli group, and the connection between entanglement, contextuality, and nonlocality. Proofs and Tests of the KS Theorem and the GHZ theorem based on N-qubit Pauli Observables.

3 The Peres-Mermin Square and The 24-cell Vertices = 24 (12 rays) Edges = 96 Faces = 96 Cells = 24 tetrahedra

4 The 600-cell System Vertices = 120 (60 rays) Edges = 720 Faces = 1200 Cells = 600 tetrahedra Critical Parity Proofs 10 8

5 The 120-cell System Vertices = 600 (300 rays) Edges = 1200 Faces = 720 Cells = 120 dodecahedra

6 The 2-qubit Pauli Group The Complete 2-qubit Pauli group contains 15 observables that form 15 mutually commuting sets of 3 observables. Taking the joint eigenbasis of each of these 15 maximally commuting sets gives rise to 60 rays in the Hilbert Space of 2 qubits, which form 105 orthogonal bases. This set contains numerous critical proofs of the KS theorem. We enumerated roughly 1.5 x 10 8 distinct parity proofs, and estimate that there are at least 10 9 in total.

7 Nonclassical Structures Critical Identity Products (IDs) Critical Kernels Critical Observable-based KS Proofs Proofs of Bell s Theorem without inequalities Saturated Sets of Projectors and Parity Proofs

8 Identity Products (IDs) An ID is a set of mutually commuting N-qubit Pauli observables whose overall product is ± I, the identity in the space of all N qubits. An ID is critical iff no subset of observables and/or qubits can be removed such that the remaining set is still an ID. If an ID is critical, then the states in its joint eigenbasis are entangled over all N qubits. A critical ID need not be a maximal commuting set, and if it has less then N+1 observables, then the entangled projectors of its joint eigenbasis have ranks higher than one.

9 Critical IDs, Entanglement, and Graph States ZZ XX YY ZZZ ZXX XZX XXZ ZZZ YYZ XIX IXX ZZZZ YYXX XIZX IXXZ ZZZI YYIZ XIYY IXXX ZZZZ ZZXX XXII XIZX IXXZ ZZZZ YYZZ XIXI IXIX IIXX ZZZZ YIYI XIZX IYYZ IXIX ZZZI YYIZ XIXX IXXX IIZZ ZZZI YYZZ XZXX IZIX IXXZ ZZZI YYIZ XZXZ IZZX IXXX ZZZI ZYYZ ZXYY YYZZ XYIY

10 The N-qubit KS Theorem and Kernels ZZZ ZXX XZX XXZ ZZZZ ZZXX XXII XIZX IXXZ ZZXXZZ XXZZXX ZZZZII XIXIZX IXIXXZ

11 Kernels and Observable-based KS Proofs

12 Critical Observable-based KS Proofs

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16 Saturated Sets of Projectors We obtain a saturated set by taking the rays from the joint eigenbases of every ID in an Observable-based KS proof. In addition to the eigenbases, these rays form a number of other orthogonal bases, based on how the IDs of the proof are interconnected. Each pair of IDs form H = 2^(2 s ) 2 Hybrid bases in addition to their eigenbases, where s is the number of observables shared by both IDs. The complete set of bases formed by the rays from any pair of IDs is saturated, and therefore the entire set is also saturated.

17 Parity Proofs of the KS Theorem The saturated set of rays obtained from an Observable-based KS proof is not critical, but contains a variety of critical parity proofs as subsets. There is a special class of Observable-based KS proofs in which no two IDs share more than one common observable and no observable appears in more than two IDs. For these proofs, the saturated set contains exactly 2 O critical parity proofs, where O is the total number of observables.

18 Observable-based BKS Proof Projector-based BKS Proof

19 Observable-based BKS Proof Projector-based BKS Proof

20 Proof O-I Symbol R-B Symbol # of Parity Proofs 4-qubit Star = 4,096 4-qubit Windmill = 8,192 4-qubit Clock = 8,192 4-qubit Whorl = 1,048,576 5-qubit Star = 4,096 5-qubit Wheel = 32,768 6-qubit Arch = 2,048 6-qubit Arrow = 8,192 6-qubit Pinwheel = 65,536

21 Family of Star Proofs for all odd N

22 Family of Star Proofs for all Even N

23 Family of Wheel Proofs for all Odd N

24 Family of Whorl Proofs for all Even N

25 General Family of Kite Proofs for all N

26 A class of minimal N(odd)-qubit IDs N=3 N=5 N Z I Z Z I I I Z Z I I Z I Z Z I Z I I Z I Z X X X I I Z I Z I Z Y Y X I I I Z Z I I Z Z X X X X X X X X X Y Y Y Y X Y Y Y X

27 A class of minimal N(even)-qubit IDs N=4 N=6 N Z Z Z Z Y Y Z Z X I X I I X I X I I X X Z Z Z Z Z Z Y Y Z Z Z Z X I X I I I I X I X I I I I X I X I I I I X I X I I I I X X Z Z Z Z Z Y Y Z Z Z X I X I I I X I X I I I X I X I I I X X

28 The shortest Kite for N qubits N=7 M=5 ZZZZZII XXZXXZZ YIXZZXX IYIXIZX IIXIXXZ N=11 M=6 IIZZZZZZZZZ ZIZZZXXIXXI IZXXIZZZIII XIXIXXIXZXX IXIXIIXIIZZ YYIIXIIXXIX N=16 M=7 ZZZZZZZZIIIIIIII XXXXIIIIZZZZIIII YIIIXXXIXIIIZZII IYIIYIIIIXXXXIZI IIIIIYIXYYIIIIXZ IIYIIIIYIIYIYXIX IIIYIIYIIIIYIYYY

29 Experimental Tests of Quantum Contextuality Each unique structure that proves the BKS theorem also gives rise to a unique experimental test of quantum contextuality. For an Observable-based BKS proof A with symbol O-I, we define for which the expectation value is given by Because each observable occurs an even number of times throughout the above sum, in any Noncontextual Hidden Variable Theory (NCHVT), However, according to quantum mechanics (and experiment) which violates the above inequality, demonstrating quantum contextuality.

30 Experimental Tests of Quantum Contextuality For a Projector-based BKS proof A with symbol R-B we can similarly define with for each of the R rays in A. Because each ray occurs an even number of times throughout the above sum, in any Noncontextual Hidden Variable Theory (NCHVT), However, according to quantum mechanics (and experiment) which violates the above inequality, demonstrating quantum contextuality.

31 Entanglement and Contextuality; qubit-based Quantum Mechanics Contextuality can be observed in systems without entanglement, as with a single spin-1 particle, but irreducible proofs of contextuality with qubits require that all of the qubits be entangled. If we derive discrete quantum mechanics by assuming that all d- level quantum systems are composed of collections of qubits - in a fashion that generalizes the well-known addition of angular momenta to arbitrary quantum systems - then contextuality is truly a consequence of qubit entanglement and cannot exist without it.

32 Conclusions We have developed the theory of nonclassical structures within the N-qubit Pauli group and used it to derive numerous new proofs of quantum contextuality and nonlocality. We have shown how the various manifestations of nonclassical physics involving qubits are interconnected and inseparable. There are many potential applications for the nonclassical structures we have developed in quantum information processing and the foundations of quantum mechanics, and we plan future work to explore some of these.

33 Thank You Looking for a Postdoc?

34 State Preparation for 2-qubit states encoded in the polarization and orbital-angular-momentum degrees of freedom of a single photon.

35 Cascade Measurements for the Peres-Mermin Square D Ambrosio, Vincenzo, Isabelle Herbauts, Elias Amselem, Eleonora Nagali, Mohamed Bourennane, Fabio Sciarrino, and Adán Cabello. "Experimental implementation of a Kochen- Specker set of quantum tests." Physical Review X 3, no. 1 (2013):