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Mutually Unbiased Unitaries for Qubits

Jesni Shamsul Shaari, Stefano Mancini

Published Mar 20, 2016• 0 citations
OpenAlex

Abstract

We study sets of unitary transformations on qubits forming bases for subspaces of $M(2,\mathbb{C})$ 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 $\text{SU}(2)$.

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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).
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.
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.
Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if ||^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.