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Physical Review A

Mutually unbiased unitary bases

Jesni Shamsul Shaari, Rinie N. M. Nasir, Stefano Mancini

Published Nov 21, 2016• 15 citations
Publisher

Abstract

We consider the notion of unitary transformations forming bases for subspaces of $M(d,\mathbb{C})$ such that the square of the Hilbert-Schmidt inner product of matrices from the differing bases is a constant. Moving from the qubit case, we construct the maximal number of such bases for the four- and two-dimensional subspaces while proving the nonexistence of such a construction for the three-dimensional case. Extending this to higher dimensions, we commit to such a construct for the case of qutrits and provide evidence for the existence of such unitaries for prime dimensional quantum systems. Focusing on the qubit case, we show that the average fidelity for estimating any such transformation is equal to the case for estimating a completely unknown unitary from $\text{SU}(2)$. This is then followed by a quick application for such unitaries in a quantum cryptographic setup.

DOI
10.1103/physreva.94.052328

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We consider the notion of unitary transformations forming bases for subspaces of M(d,C)M(d,\mathbb{C})M(d,C) such that the square of the Hilbert-Schmidt inner product of matrices from the differing bases is a constant. Moving from the qubit case, we construct the maximal number of such bases for the four- and two-dimensional subspaces while proving the nonexistence of such a construction for the three-dimensional case. Extending this to higher dimensions, we commit to such a construct for the case of qutrits and provide evidence for the existence of such unitaries for prime dimensional quantum systems. Focusing on the qubit case, we show that the average fidelity for estimating any such transformation is equal to the case for estimating a completely unknown unitary from SU(2)\text{SU}(2)SU(2). This is then followed by a quick application for such unitaries in a quantum cryptographic setup.
We consider the notion of unitary transformations forming bases for subspaces of M(d,C)M(d,\mathbb{C})M(d,C) such that the square of Hilbert-Schmidt inner product of matrices from the differing bases is a constant. Moving from the qubit case, weconstruct the maximal number of such bases for the 4 and 2 dimensional subspaces while proving the nonexistence of such a construction for the 3 dimensional case. Extending this to higher dimensions, we commit to such a construct for the case of qutrits and provide evidence for the existence of such unitaries for prime dimensional quantum systems. Focusing on the qubit case, we show that the average fidelity for estimating any of such a transformation is equal to the case for estimating a completely unknown unitary from SU(2)\text{SU}(2)SU(2). This is then followed by a quick application for such unitaries in a quantum cryptographic setup.
We study sets of unitary transformations on qubits forming bases for subspaces of M(2,C)M(2,\mathbb{C})M(2,C) such that the square of Hilbert-Schmidt inner product of matrices from the differing bases is a constant. We construct the maximal number of such bases for the 4 and 2 dimensional subspaces while proving the nonexistence of such a construction for the 3 dimensional case. We also provide the average fidelity for estimating any of such a transformation and show that it is equal to the case for estimating a completely unknown unitary from SU(2)\text{SU}(2)SU(2).
Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, Mðd; CÞ, acting on such Hilbert spaces. The notion of MUUB re°ects the equiprobable guesses of unitary operators in one basis of Mðd; CÞ when estimating a unitary operator in another. Though, for prime dimension d, the maximal number of MUUBs is known to be d2 1, there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of d. MUUBs can also exist for some d-dimensional subspace of Mðd; CÞ with the maximal number being d.
Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].
Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.
A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are complementary, i.e. a measurement of one observable implies maximal uncertainty about the possible outcome of a subsequent measurement of a second observable. MU bases have attracted interest in recent years because their properties seem to depend dramatically on the dimension d of the quantum system in hand. If the dimension is given by a prime or prime-power, the state space is known to accommodate a complete set of d+1 MU bases. However, for composite dimensions, such as d = 6, 10, 12, ..., complete sets seem to be absent and it is not understood why. In this thesis we carry out a comprehensive study of MU product bases in dimension six. In particular, we construct all MU bases in dimension six consisting of product states only. The exhaustive classification leads to several non-existence results. We also present a new construction of complex Hadamard matrices of composite order, which is a consequence of our work on MU product bases. Based on this construction we obtain several new isolated Hadamard matrices of Butson-type. \n