Structure of the sets of mutually unbiased bases with cyclic symmetry

Seyfarth, Ulrich; Sánchez-Soto, Luis L.; Leuchs, Gerd

doi:10.1088/1751-8113/47/45/455303

Cited by 7 publications

(5 citation statements)

References 81 publications

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“…In addition, . R, BR, and A have to be symmetric and R has to be invertible [57]. It turns out that when R = 1 1 m and A = 0 m , the resulting complete sets exhibit an entanglement structure with three completely factorizable classes, which, following the original work [57], will be called field-based sets, as the generators represent a finite field.…”

Section: Mutually Unbiased Bases: Basic Backgroundmentioning

confidence: 99%

“…R, BR, and A have to be symmetric and R has to be invertible [57]. It turns out that when R = 1 1 m and A = 0 m , the resulting complete sets exhibit an entanglement structure with three completely factorizable classes, which, following the original work [57], will be called field-based sets, as the generators represent a finite field. When R is not a polynomial in B and A = 0 m , the generators form an additive group, where for only two of their classes the Pauli operators commute on each qubit separately: they are denoted as group-based sets, Finally, whenever R is not a polynomial in B, and A is not the product of any polynomial in B with R added to a diagonal matrix, the resulting cyclic set of MUBs has only a single class left, where the Pauli operators commute on all qubits separately.…”

Section: Mutually Unbiased Bases: Basic Backgroundmentioning

confidence: 99%

“…of the characteristic polynomial of B has to be d + 1. In addition, R, BR, and A have to be symmetric and R has to be invertible [57].…”

Section: Mutually Unbiased Bases: Basic Backgroundmentioning

confidence: 99%

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Practical implementation of mutually unbiased bases using quantum circuits

2015

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The number of measurements necessary to perform the quantum state reconstruction of a system of qubits grows exponentially with the number of constituents, creating a major obstacle for the design of scalable tomographic schemes. We work out a simple and efficient method based on cyclic generation of mutually unbiased bases. The basic generator requires only Hadamard and controlled-phase gates, which are available in most practical realizations of these systems. We show how complete sets of mutually unbiased bases with different entanglement structures can be realized for three and four qubits. We also analyze the quantum circuits implementing the various entanglement classes.

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Section: Mutually Unbiased Bases: Basic Backgroundmentioning

confidence: 99%

Section: Mutually Unbiased Bases: Basic Backgroundmentioning

confidence: 99%

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Practical implementation of mutually unbiased bases using quantum circuits

2015

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“…In [5], [6], the authors have constructed m+1 number of orthonormal bases for R m with coherence 1/ √ m, where m is a power of two. Some of the well known structured IUNTFs are mutually unbiased bases (MUBs) ( [12], [13], [14], [15], [16]). Two orthonormal bases B and B ′ of an m−dimensional complex inner-product space are called mutually unbiased if and only…”

Section: Introductionmentioning

confidence: 99%

Construction of Structured Incoherent Unit Norm Tight Frames

Sasmal¹,

Jampana²,

Sastry³

2017

Preprint

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The exact recovery property of Basis pursuit (BP) and Orthogonal Matching Pursuit (OMP) has a relation with the coherence of the underlying frame. A frame with low coherence provides better guarantees for exact recovery. In particular, Incoherent Unit Norm Tight Frames (IUNTFs) play a significant role in sparse representations. IUNTFs with special structure, in particular those given by a union of several orthonormal bases, are known to satisfy better theoretical guarantees for recovering sparse signals. In the present work, we propose to construct structured IUNTFs consisting of large number of orthonormal bases. For a given r, k, m with k being less than or equal to the smallest prime power factor of m and r < k, we construct a CS matrix of size mk × (mk × m r ) with coherence at most r k , which consists of m r number of orthonormal bases and with density 1 m . We also present numerical results of recovery performance of union of orthonormal bases as against their Gaussian counterparts.

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“…They can arise as eigenvectors of a MUB cycling operators, operators that cycle through all the d + 1 bases in a MUB. MUB-balanced states are known to exist if d = 2 n where unitary MUB-cyclers exist [17,20], and if d = (prime) n = 3 modulo 4 where anti-unitary MUB-cyclers exist [19,21]. These states have some intriguing and useful properties [19,21,22].…”

Section: Introductionmentioning

confidence: 99%

States that are far from being stabilizer states

Andersson¹,

Bengtsson²,

Blanchfield³

et al. 2015

J. Phys. A: Math. Theor.

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Abstract:Stabilizer states are eigenvectors of maximal commuting sets of operators in a finite Heisenberg group. States that are far from being stabilizer states include magic states in quantum computation, MUB-balanced states, and SIC vectors. In prime dimensions the latter two fall under the umbrella of Minimum Uncertainty States (MUS) in the sense of Wootters and Sussman. We study the correlation between two ways in which the notion of "far from being a stabilizer state" can be quantified, and give detailed results for low dimensions. In dimension 7 we identify the MUB-balanced states as being antipodal to the SIC vectors within the set of MUS, in a sense that we make definite. In dimension 4 we show that the states that come closest to being MUS with respect to all the six stabilizer MUBs are the fiducial vectors for Alltop MUBs.

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Structure of the sets of mutually unbiased bases with cyclic symmetry

Cited by 7 publications

References 81 publications

Practical implementation of mutually unbiased bases using quantum circuits

Practical implementation of mutually unbiased bases using quantum circuits

Construction of Structured Incoherent Unit Norm Tight Frames

States that are far from being stabilizer states

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