Mutually unbiased binary observable sets on<i>N</i>qubits

Lawrence, Jay; Brukner, Časlav; Zeilinger, Anton

doi:10.1103/physreva.65.032320

Cited by 160 publications

(226 citation statements)

References 18 publications

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“…For B n = {B b } b∈{0,1} t a mubs, w ∈ {0, 1} n , and b ∈ {0, 1} t , we denote by |v (b) w the w-th state in basis B b ∈ B n . Lawrence, Brukner, and Zeilinger [11] introduced an alternative construction for maximal mubss based on algebra in the Pauli group. Their construction plays an important role in the security analysis of our qkrs.…”

Section: Mutually Unbiased Basesmentioning

confidence: 99%

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A Quantum Cipher with Near Optimal Key-Recycling

Damgård¹,

Pedersen²,

Salvail³

2005

Advances in Cryptology – CRYPTO 2005

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Abstract. Assuming an insecure quantum channel and an authenticated classical channel, we propose an unconditionally secure scheme for encrypting classical messages under a shared key, where attempts to eavesdrop the ciphertext can be detected. If no eavesdropping is detected, we can securely re-use the entire key for encrypting new messages. If eavesdropping is detected, we must discard a number of key bits corresponding to the length of the message, but can re-use almost all of the rest. We show this is essentially optimal. Thus, provided the adversary does not interfere (too much) with the quantum channel, we can securely send an arbitrary number of message bits, independently of the length of the initial key. Moreover, the key-recycling mechanism only requires one-bit feedback. While ordinary quantum key distribution with a classical one time pad could be used instead to obtain a similar functionality, this would need more rounds of interaction and more communication.

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Section: Mutually Unbiased Basesmentioning

confidence: 99%

“…In [11], it is first shown how to partition the set of 4 n − 1 non-trivial Pauli operators {O i } 4 n −1 i=1 into 2 n + 1 subsets, each containing 2 n − 1 commuting members. Second, each such partitioning is shown to define a maximal mubs.…”

Section: Mutually Unbiased Basesmentioning

confidence: 99%

A Quantum Cipher with Near Optimal Key-Recycling

Damgård¹,

Pedersen²,

Salvail³

2005

Advances in Cryptology – CRYPTO 2005

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“…3); five lines of a spread represent nothing but the five maximum subsets of three mutually commuting operators each, whose associated bases are mutually unbiased. 10,21 It is a straightforward exercise to associate the points of the quadrangle with the operators/observables C i , Eq. (8), in such a way to recover Table 2, after substituting the "−"/"+" sign for any two points of the quadrangle which are/are not on a common line.…”

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confidence: 99%

Projective ring line encompassing two-qubits

2008

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The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF (2) is found to fully accommodate the algebra of 15 operators -generalized Pauli matrices -characterizing two-qubit systems. The relevant sub-configuration consists of 15 points each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified with the points in such a one-to-one manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical notions of neighbor/distant answering, respectively, to the operational ones of commuting/non-commuting. This remarkable configuration can be viewed in two principally different ways accounting, respectively, for the basic 9+6 and 10+5 factorizations of the algebra of the observables. First, as a disjoint union of the projective line over GF (2) × GF (2) (the "Mermin" part) and two lines over GF (4) passing through the two selected points, the latter omitted. Second, as the generalized quadrangle of order two, with its ovoids and/or spreads standing for (maximum) sets of five mutually non-commuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open up rather unexpected vistas for an algebraic geometrical modelling of finite-dimensional quantum systems and give their numerous applications a wholly new perspective. Projective lines defined over finite associative rings with unity/identity 1−7 have recently been recognized to be an important novel tool for getting a deeper insight into the underlying algebraic geometrical structure of finite dimensional quantum systems. 8−10 Focusing almost uniquely on the two-qubit case, i.e., the set of 15 operators/generalized four-by-four Pauli spin matrices, of particular importance turned out to be the lines defined over the direct product of the simplest Galois fields, GF (2) × GF (2) × . . . × GF (2). Here, the line defined over GF (2) × GF (2) plays a prominent role in grasping qualitatively the basic structure of so-called Mermin squares, 9,10 i. e., three-by-three arrays in certain remarkable 9 + 6 split-ups of the algebra of operators, whereas the line over GF (2) × GF (2) × GF (2) reflects some of the basic features of a specific 8 + 7 ("cubeand-kernel") factorization of the set. 10 Motivated by these partial findings, we started our quest for such a ring line that would provide us with a complete picture of the algebra of all the 15 operators/matrices. After examining a large number of lines defined over commutative rings, 6,7 we gradually realized that a proper candidate is likely to be found in the non-commutative domain and 1

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“…They also play a relevant role in a proper understanding of complementarity [28][29][30][31], in cryptographic protocols [32,33], and in quantum error correction codes [34,35]. Recently, they have also found uses in quantum game theory, in particular to provide a convenient tool for solving the so-called mean king problem [36][37][38][39][40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

Discrete phase-space structure of n-qubit mutually unbiased bases

et al. 2009

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We work out the phase-space structure for a system of n qubits. We replace the field of real numbers that label the axes of the continuous phase space by the finite field GF(2 n ) and investigate the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We provide a simple classification of such curves and study in detail the four-and eight-dimensional cases, analyzing also the effect of local transformations. In this way, we provide a comprehensive phase-space approach to the construction of mutually unbiased bases for n qubits.

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Mutually unbiased binary observable sets onNqubits

Cited by 160 publications

References 18 publications

A Quantum Cipher with Near Optimal Key-Recycling

A Quantum Cipher with Near Optimal Key-Recycling

Projective ring line encompassing two-qubits

Discrete phase-space structure of n-qubit mutually unbiased bases

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