Constructions of Mutually Unbiased Bases

Klappenecker, Andreas; Rötteler, Martin

doi:10.1007/978-3-540-24633-6_10

Cited by 297 publications

(341 citation statements)

References 21 publications

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“…The space H = C 4 is also known to admit a full set of MUBs since dim H = 4 is the second power of the prime number 2; for example [22],…”

Section: When Dim H ≥mentioning

confidence: 99%

Information Geometry of Randomized Quantum State Tomography

Fujiwara

Yamagata

2018

Entropy

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Suppose that a d-dimensional Hilbert space H C d admits a full set of mutually unbiased bases |1 (a) , . . . , |d (a) , where a = 1, . . . , d + 1. A randomized quantum state tomography is a scheme for estimating an unknown quantum state on H through iterative applications of measurementswhere the numbers of applications of these measurements are random variables. We show that the space of the resulting probability distributions enjoys a mutually orthogonal dualistic foliation structure, which provides us with a simple geometrical insight into the maximum likelihood method for the quantum state tomography.

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“…The space H = C 4 is also known to admit a full set of MUBs since dim H = 4 is the second power of the prime number 2; for example [22],…”

Section: When Dim H ≥mentioning

confidence: 99%

Information Geometry of Randomized Quantum State Tomography

Fujiwara

Yamagata

2018

Entropy

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“…That would be a total of (d + 1)d = d 2 + d projectors, but because not all of the expectation values of these projectors are independent, this set can be related to a rank-one POVM with d 2 elements, which is the minimum number of elements of a I-complete POVM [8]. It is also known [9] that for any value of d there does exist at least 3 MUBs (and it is conjectured [10,11] that for some values of d no more than that), which can be related to a rank-one POVM with 3d − 2 elements. This is the smallest number not ruled out by Theorem II, and so one might hope that this POVM could be PSIR-complete.…”

Section: Introductionmentioning

confidence: 99%

Pure-state informationally complete and “really” complete measurements

Finkelstein

2004

Phys. Rev. A

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I construct a POVM which has 2d rank-one elements and which is informationally complete for generic pure states in d dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (quant-ph/0404137). I show that if a rank-one POVM is required to be informationally complete for all pure states in d dimensions, it must have at least 3d − 2 elements. I also show that, in a POVM which is informationally complete for all pure states in d dimensions, for any vector there must be at least 2d − 1 POVM elements which do not annihilate that vector.

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“…Experimental results demonstrate the practicability of those schemes [3][4][5]. Different construction methods for MUBs are known [2,[6][7][8][9][10]]. For a d-level system, i. e., a system described by a d × d density matrix, one would need d + 1 mutually unbiased bases, since any measurement statistically reveals d − 1 parameters.…”

mentioning

confidence: 99%

“…, 256. In general, the construction of MUBs is quite involved and uses methods from abstract algebra and the mathematical theory of finite fields and rings [10], which are far apart from most methods that are commonly used in physics. We overcome this problem in such a way, that our construction (although not its proof) is very easy to follow and to apply by anyone with just basic knowledge of linear algebra.…”

mentioning

confidence: 99%

Construction of mutually unbiased bases with cyclic symmetry for qubit systems

Seyfarth

Ranade

2011

Phys. Rev. A

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For the complete estimation of arbitrary unknown quantum states by measurements, the use of mutually unbiased bases has been well-established in theory and experiment for the past 20 years. However, most constructions of these bases make heavy use of abstract algebra and the mathematical theory of finite rings and fields, and no simple and generally accessible construction is available. This is particularly true in the case of a system composed of several qubits, which is arguably the most important case in quantum information science and quantum computation. In this letter, we close this gap by providing a simple and straightforward method for the construction of mutually unbiased bases in the case of a qubit register. We show that our construction is also accessible to experiments, since only Hadamard and controlled phase gates are needed, which are available in most practical realizations of a quantum computer. Moreover, our scheme possesses the optimal scaling possible, i. e., the number of gates scales only linearly in the number of qubits.In quantum physics, the estimation of the state of a system is of high practical value [1]. It is well-known that for the complete estimation of a state, known as state tomography, a single measurement is not sufficient, even if performed many times to get the statistics of such measurement. It is necessary to measure a state in various different bases. The best choice of such bases for an arbitrary system is so-called mutually unbiased bases (MUBs), which offer the highest information outcome, as already stated by Wootters and Fields [2]: mutually unbiased bases provide an optimal means of determining an ensemble's state. Experimental results demonstrate the practicability of those schemes [3][4][5]. Different construction methods for MUBs are known [2, 6-10]. For a d-level system, i. e., a system described by a d × d density matrix, one would need d + 1 mutually unbiased bases, since any measurement statistically reveals d − 1 parameters. Unfortunately, it is not even known whether d + 1 such bases exist in every d-level system. Mutually unbiased bases are related to different topics in mathematics and physics, e. g., quantum cryptography, foundations of physics [11], orthogonal Latin squares or hidden-variable models [12,13] and even Feynman's path integral [14].In this letter, we want to focus on a system which is of particular interest in quantum information processing, namely, a quantum register built of qubits. We propose a complete set of mutually unbiased bases for quantum registers of size 1, 2, 4, 8, . . . , 256. In general, the construction of MUBs is quite involved and uses methods from abstract algebra and the mathematical theory of finite fields and rings [10], which are far apart from most methods that are commonly used in physics. We overcome this problem in such a way, that our construction (although not its proof) is very easy to follow and to apply by anyone with just basic knowledge of linear algebra. Moreover, our construction is applicable to experi...

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Constructions of Mutually Unbiased Bases

Cited by 297 publications

References 21 publications

Information Geometry of Randomized Quantum State Tomography

Information Geometry of Randomized Quantum State Tomography

Pure-state informationally complete and “really” complete measurements

Construction of mutually unbiased bases with cyclic symmetry for qubit systems

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