Numerical evidence for the maximum number of mutually unbiased bases in dimension six

Butterley, Paul; Hall, W. B.

doi:10.1016/j.physleta.2007.04.059

Cited by 58 publications

(79 citation statements)

References 20 publications

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“…As noted repeatedly, this existence problem is open, even in the most intensely studied case of N = 6. 27,87,164,[167][168][169] Since the Galois-Fourier construction of the Heisenberg-Weyl group, which works so well for prime power dimensions, cannot be applied for N = 6, 10, 12, 14, . .…”

Section: Heisenberg-weyl Group Approach For N =mentioning

confidence: 99%

On Mutually Unbiased Bases

Durt

Englert

Bengtsson

et al. 2010

Int. J. Quantum Inform.

680

796

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Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p m , with p prime and m a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12, . . . ).In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, . . . , N − 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two-q-bit Bell states) that are related to the mutually unbiased bases.There is a link to the mathematics of finite affine planes. We also exploit the one-toone correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist -one can find no more than three pairwise unbiased bases -although there is currently no clear-cut demonstration of the case.

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Section: Heisenberg-weyl Group Approach For N =mentioning

confidence: 99%

On Mutually Unbiased Bases

Durt

Englert

Bengtsson

et al. 2010

Int. J. Quantum Inform.

680

796

View full text Add to dashboard Cite

show abstract

“…This criterion, stronger than the one used in [5] is entirely arbitrary, and smaller values could be used at the expense of computational time. The numerical data presented below will retrospectively justify the chosen value of the threshold for zeros of F.…”

Section: Mu Constellations As Global Minimamentioning

confidence: 99%

Maximal sets of mutually unbiased quantum states in dimension 6

Brierley

Weigert

2008

Phys. Rev. A

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We study sets of pure states in a Hilbert space of dimension d which are mutually unbiased (MU), that is, the moduli of their scalar products are equal to zero, one, or 1/ √ d. These sets will be called a MU constellation, and if four MU bases were to exist for d = 6, they would give rise to 35 different MU constellations. Using a numerical minimisation procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension six.

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“…d k , the factors being (powers of) different primes, it is currently unknown whether complete sets of MU bases exist [16]. Interestingly, composite dimensions are rare for small values of d but predominate for large d. While it is possible to construct three MU bases for any d ≥ 2, numerical evidence for d=6 (the smallest composite integer) suggests that no four MU bases exist [17] and that many of their subsets are missing as well [18].Let us now turn to continuous variablesp andq, with [q,p] = i , acting on the Hilbert space L 2 (R) of squareintegrable functions on the real line. The (generalized) eigenstates of position and momentum |q , q ∈ R, and |p , p ∈ R, respectively, are known to satisfy…”

mentioning

confidence: 99%

Mutually unbiased bases for continuous variables

Weigert

Wilkinson

2008

Phys. Rev. A

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The concept of mutually unbiased bases is studied for N pairs of continuous variables. To find mutually unbiased bases reduces, for specific states related to the Heisenberg-Weyl group, to a problem of symplectic geometry. Given a single pair of continuous variables, three mutually unbiased bases are identified while five such bases are exhibited for two pairs of continuous variables. For N = 2, the golden ratio occurs in the definition of these mutually unbiased bases suggesting the relevance of number theory not only in the finite-dimensional setting. 03.65.Ta Mutually unbiased (MU) bases of Hilbert spaces with finite dimension d (as defined by Eq. (1) below) are a useful tool. If you want to experimentally determine the state of a quantum system, given only a limited supply of identical copies, the optimal strategy is to perform measurements with respect to MU bases [1]. To pass a secret message to a second party, you could use quantum cryptography to establish a shared key, a procedure which relies on MU bases in the space. Sending a physical system carrying a spin through a noisy environment, the effect of the interactions on the state of the spin might be modelled by a specific quantum channel, conveniently described in terms of MU bases [5]. Finally, if you happen to be captured by a mean king, you might be able to meet his challenge by knowing about entangled states and MU bases [6,7]. Many of the ideas which underlie physical concepts defined for discrete variables, that is, in a Hilbert space of finite dimension, survive the transition from spin operators to position and momentum operators. Quantum key distribution [8] and quantum teleportation [9], for example, possess counterparts for continuous variables [10] which act on an infinite-dimensional Hilbert space. It is thus natural to inquire into MU bases for continuous variables which, in fact, naturally occur in Feynman's path integral formulation of quantum mechanics [11]. The properties of MU bases in an infinite-dimensional space might also provide new insights into the existence of complete sets of MU bases in spaces of finite dimension not equal to the power of a prime.Let us recall the definition of MU bases in C d and some of their properties. Two orthonormal basessince each state of one basis gives rise to the same probability distribution when measured with respect to the other basis. The value of the overlap κ is not arbitrary but one derives from (1) that κ ≡ 1/ √ d by using the completeness of the basis B b , say.Schwinger [12] describes how to construct two MU bases from any orthonormal basis of C d . They are found to be the eigenbases of two operatorsÛ andV each shifting cyclically the elements of the other basis. These operators satisfy commutation relations of HeisenbergWeyl type,ÛV = e 2πi/dVÛ , describing finite translations in a discrete phase space [13]. This approach has been generalized in [14], where it is shown that if one finds n unitaries each cyclically shifting the eigenbases of all other unitaries then these n bases ...

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Numerical evidence for the maximum number of mutually unbiased bases in dimension six

Cited by 58 publications

References 20 publications

On Mutually Unbiased Bases

On Mutually Unbiased Bases

Maximal sets of mutually unbiased quantum states in dimension 6

Mutually unbiased bases for continuous variables

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