Entropy and Hadamard matrices

Gadiyar, H. Gopalkrishna; Maini, K M Sangeeta; Padma, R.; Sharatchandra, H. S.

doi:10.1088/0305-4470/36/7/103

Cited by 9 publications

(7 citation statements)

References 2 publications

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“…This result is stated in [5]. It is mentioned to have been deduced from a similar problem involving minimizing entropy, which was solved numerically in [7]. Now consider an arbitrary 4 by 4 bistochastic matrix:…”

Section: It Is Orthostochastic If and Only If The Vectorsmentioning

confidence: 78%

On orthostochastic, unistochastic and qustochastic matrices

Chterental

Djoković

2008

Linear Algebra and its Applications

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We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by physicists. We also introduce quaternionic Hadamard matrices and quaternionic mutually unbiased bases (MUB). In particular we show that the number of MUB in an n-dimensional quaternionic Hilbert space is at most 2n + 1. The bound is attained for n = 2. We also determine all quaternionic Hadamard matrices of size n ≤ 4.

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Section: It Is Orthostochastic If and Only If The Vectorsmentioning

confidence: 78%

On orthostochastic, unistochastic and qustochastic matrices

Chterental

Djoković

2008

Linear Algebra and its Applications

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“…The fact that Hadamard matrices saturate the upper bound for the so-called entropy of a unitary matrix is well known [57]. Observe, however, that the analogous problem for real orthogonal matrices is highly non-trivial [23], [40], since real Hadamard matrices can exist only if d = 1, 2 or is a multiple of 4.…”

Section: Entropy-maximising Unitariesmentioning

confidence: 99%

Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Słomczyński

Szczepanek

2017

IEEE Trans. Inform. Theory

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We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM-and POVM-dynamical (quantum) entropies, which are analogues of the classical Kolmogorov-Sinai entropy rate. They quantify the maximal randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy, or equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity.

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“…The maximal unistochastic ball is centered at B ⋆ and touches the boundary at the hypocycloid, as one might guess from the picture; its radius was deduced from results presented in ref. [29]. To see that the boundary consists of orthostochastic matrices is the observation that when the chainlinks conditions are saturated the phases in U will equal ±1.…”

Section: Birkhoff's Polytopementioning

confidence: 99%

Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4

Bengtsson

Ericsson

Kuś

et al. 2005

Commun. Math. Phys.

119

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The set of bistochastic or doubly stochastic N × N matrices form a convex set called Birkhoff's polytope, that we describe in some detail. Our problem is to characterize the set of unistochastic matrices as a subset of Birkhoff's polytope. For N = 3 we present fairly complete results. For N = 4 partial results are obtained. An interesting difference between the two cases is that there is a ball of unistochastic matrices around the van der Waerden matrix for N = 3, while this is not the case for N = 4.

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Entropy and Hadamard matrices

Cited by 9 publications

References 2 publications

On orthostochastic, unistochastic and qustochastic matrices

On orthostochastic, unistochastic and qustochastic matrices

Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4

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