N-qubit Proofs of the Kochen-Specker Theorem

Mordecai Waegell P.K. Aravind

Worcester Polytechnic Institute

Worcester, MA, USA

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.

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

The 600-cell

Vertices = 120 (60 rays) Edges = 720 Faces = 1200 Cells = 600 tetrahedra Critical Parity Proofs ≈ 108

605-754 System

The 120-cell

Vertices = 600 (300 rays) Edges = 1200 Faces = 720 Cells = 120 dodecahedra

3009-6754 System

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 607-1054 set contains numerous critical proofs of the KS theorem. We enumerated roughly 1.5 x 108 distinct parity proofs, and estimate that there are at least 109 in total.

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

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.

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

The N-qubit KS Theorem and Kernels

ZZZZ ZZXX XXII XIZX IXXZ

ZZXXZZ XXZZXX ZZZZII XIXIZX IXIXXZ

ZZZ ZXX XZX XXZ

Kernels and Observable-based KS Proofs

Critical Observable-based KS Proofs

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^(2s) – 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.

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 2O critical parity proofs, where O is the total number of observables.

Observable-based BKS Proof

Projector-based BKS Proof

Observable-based BKS Proof

Projector-based BKS Proof

Proof O-I Symbol R-B Symbol # of Parity Proofs

4-qubit Star 122-154413 1616322

5444 - 1168122101484614 212 = 4,096

4-qubit Windmill 132-5423 844402

5 - 2188644 213 = 8,192

4-qubit Clock 132-2463 1625244

4 - 28166164 213 = 8,192

4-qubit Whorl 202-14123 825484

4 - 1886444 220 = 1,048,576

5-qubit Star 122-154413 1626324

5484 - 1168122101484614

212 = 4,096

5-qubit Wheel 152-3553 4826208

4 - 316301054 215 = 32,768

6-qubit Arch 112-152433 1646168

512164 - 116412610 2812634 211 = 2,048

6-qubit Arrow

132-2544 3246328

5 - 4161612128 213 = 8,192

6-qubit Pinwheel 162-5443 40851616

4 - 198126104 216 = 65,536

Family of Star Proofs for all odd N

Family of Star Proofs for all Even N

Family of Wheel Proofs for all Odd N

Family of Whorl Proofs for all Even N

General Family of Kite Proofs for all N

A class of minimal N(odd)-qubit IDs

Z I Z I Z Z

X X X

Y Y X

Z I I I Z I Z I I Z

I I Z I Z

I I I Z Z

X X X X X

Y Y Y Y X

N=3 N=5

Z I I Z I Z

I Z

I I Z Z

X X X X

Y Y Y X

N

A class of minimal N(even)-qubit IDs

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

N=4 N=6

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

N

The shortest Kite for N qubits

ZZZZZII XXZXXZZ

YIXZZXX

IYIXIZX

IIXIXXZ

IIZZZZZZZZZ ZIZZZXXIXXI

IZXXIZZZIII

XIXIXXIXZXX

IXIXIIXIIZZ

YYIIXIIXXIX

N=7 M=5

N=11 M=6

ZZZZZZZZIIIIIIII XXXXIIIIZZZZIIII

YIIIXXXIXIIIZZII

IYIIYIIIIXXXXIZI

IIIIIYIXYYIIIIXZ

IIYIIIIYIIYIYXIX

IIIYIIYIIIIYIYYY

N=16 M=7

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.

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.

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.

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.

Thank You

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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): 011012.