7 research outputs found
Finite Geometry Behind the Harvey-Chryssanthacopoulos Four-Qubit Magic Rectangle
A "magic rectangle" of eleven observables of four qubits, employed by Harvey
and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a
16-dimensional Hilbert space, is given a neat finite-geometrical
reinterpretation in terms of the structure of the symplectic polar space of the real four-qubit Pauli group. Each of the four sets of observables of
cardinality five represents an elliptic quadric in the three-dimensional
projective space of order two (PG) it spans, whereas the remaining set
of cardinality four corresponds to an affine plane of order two. The four
ambient PGs of the quadrics intersect pairwise in a line, the resulting
six lines meeting in a point. Projecting the whole configuration from this
distinguished point (observable) one gets another, complementary "magic
rectangle" of the same qualitative structure.Comment: 5 pages, 1 figure; Version 2 - slightly expanded, accepted in Quantum
Information & Computatio
Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry
Employing five commuting sets of five-qubit observables, we propose specific
160-661 and 160-21 state proofs of the Bell-Kochen-Specker theorem that are
also proofs of Bell's theorem. A histogram of the 'Hilbert-Schmidt' distances
between the corresponding maximal bases shows in both cases a noise-like
behaviour. The five commuting sets are also ascribed a finite-geometrical
meaning in terms of the structure of symplectic polar space W(9,2).Comment: 10 pages, 2 figure
On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits
The Bell-Kochen-Specker theorem (BKS) theorem rules out realistic {\it
non-contextual} theories by resorting to impossible assignments of rays among a
selected set of maximal orthogonal bases. We investigate the geometrical
structure of small BKS-proofs involving real rays and
-dimensional bases of -qubits (). Specifically, we look at the
parity proof 18-9 with two qubits (A. Cabello, 1996), the parity proof 36-11
with three qubits (M. Kernaghan & A. Peres, 1995 \cite{Kernaghan1965}) and a
newly discovered non-parity proof 80-21 with four qubits (that improves work of
P. K Aravind's group in 2008). The rays in question arise as real eigenstates
shared by some maximal commuting sets (bases) of operators in the -qubit
Pauli group. One finds characteristic signatures of the distances between the
bases, which carry various symmetries in their graphs.Comment: version to appear in European Physical Journal Plu
'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon
Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13
pages] have given a number of distinct sets of three-qubit observables, each
furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that
two of these sets/configurations, namely the and ones, can uniquely be extended into geometric hyperplanes of the
split Cayley hexagon of order two, namely into those of types and in the
classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773-797].
Moreover, employing an automorphism of order seven of the hexagon, six more
replicas of either of the two configurations are obtained
Quantum magic rectangles: Characterization and application to certified randomness expansion
We study a generalization of the Mermin-Peres magic square game to arbitrary
rectangular dimensions. After exhibiting some general properties, these
rectangular games are fully characterized in terms of their optimal win
probabilities for quantum strategies. We find that for rectangular
games of dimensions there are quantum strategies that win with
certainty, while for dimensions quantum strategies do not
outperform classical strategies. The final case of dimensions is
richer, and we give upper and lower bounds that both outperform the classical
strategies. Finally, we apply our findings to quantum certified randomness
expansion to find the noise tolerance and rates for all magic rectangle games.
To do this, we use our previous results to obtain the winning probability of
games with a distinguished input for which the devices give a deterministic
outcome, and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput.
46, 1304 (2017)].Comment: 23 pages, 3 figures; published version with minor correction
Nonlocal games and their device-independent quantum applications
Device-independence is a property of certain protocols that allows one to ensure their proper execution given only classical interaction with devices and assuming the correctness of the laws of physics. This scenario describes the most general form of cryptographic security, in which no trust is placed in the hardware involved; indeed, one may even take it to have been prepared by an adversary.
Many quantum tasks have been shown to admit device-independent protocols by augmentation with "nonlocal games". These are games in which noncommunicating parties jointly attempt to fulfil some conditions imposed by a referee. We introduce examples of such games and examine the optimal strategies of players who are allowed access to different possible shared resources, such as entangled quantum states. We then study their role in self-testing, private random number generation, and secure delegated quantum computation. Hardware imperfections are naturally incorporated in the device-independent scenario as adversarial, and we thus also perform noise robustness analysis where feasible.
We first study a generalization of the Mermin–Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these "magic rectangle" games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for m×n magic rectangle games with dimensions m,n≥3, there are quantum strategies that win with certainty, while for dimensions 1×n quantum strategies do not outperform classical strategies. The final case of dimensions 2×n is richer, and we give upper and lower bounds that both outperform the classical strategies. As an initial usage scenario, we apply our findings to quantum certified randomness expansion to find noise tolerances and rates for all magic rectangle games. To do this, we use our previous results to obtain the winning probabilities of games with a distinguished input for which the devices give a deterministic outcome and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput. 46, 1304 (2017)].
Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. We use our 3×n magic rectangle games to obtain a self-test for n Bell states where one side needs only to measure single-qubit Pauli observables. The protocol requires small input sizes [constant for Alice and O(log n) bits for Bob] and is robust with robustness O(n⁵/²√ε), where ε is the closeness of the ideal (perfect) correlations to those observed. To achieve the desired self-test, we introduce a one-side-local quantum strategy for the magic square game that wins with certainty, we generalize this strategy to the family of 3×n magic rectangle games, and we supplement these nonlocal games with extra check rounds (of single and pairs of observables).
Finally, we introduce a device-independent two-prover scheme in which a classical verifier can use a simple untrusted quantum measurement device (the client device) to securely delegate a quantum computation to an untrusted quantum server. To do this, we construct a parallel self-testing protocol to perform device-independent remote state preparation of n qubits and compose this with the unconditionally secure universal verifiable blind quantum computation (VBQC) scheme of J. F. Fitzsimons and E. Kashefi [Phys. Rev. A 96, 012303 (2017)]. Our self-test achieves a multitude of desirable properties for the application we consider, giving rise to practical and fully device-independent VBQC. It certifies parallel measurements of all cardinal and intercardinal directions in the XY-plane as well as the computational basis, uses few input questions (of size logarithmic in n for the client and a constant number communicated to the server), and requires only single-qubit measurements to be performed by the client device