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arXiv:2504.15576 · Journal of Physics A Mathematical and Theoretical

Reply to Comment on ‘Product states and Schmidt rank of mutually unbiased bases in dimension six’

Mengfan Liang, Lin Chen

Published Apr 17, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract McNulty and Weigert (2024 J. Phys. A: Math. Theor. submitted) voiced suspicions to the lemma 11(v) Part 6 in Chen and Yu (2017 J. Phys. A: Math. Theor. 50 475304) and three theorems derived in later publications (Liang et al 2019 Quantum Inf. Process. 18 352; Liang et al 2019 Linear Multilinear Algebra 69 2908–25; Chen et al 2021 Quantum Inf. Process. 20 353). For these suspicions, we reprove that any 6 × 6 complex Hadamard matrix whose number of real elements more than 22 does not belong to a set of four mutually unbiased bases. We show that the number of 2 × 2 complex Hadamard submatrices of any H 2-reducible matrix is not 10 , … , 16 , 18 . We also put forward some possible directions for further development.

DOI
10.1088/1751-8121/adc885

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Comment on ‘Product states and Schmidt rank of mutually unbiased bases in dimension six’

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Abstract A lemma by Chen et al (2017 J. Phys. A: Math. Theor. 50 475304 ) provides a necessary condition on the structure of any complex Hadamard matrix in a set of four mutually unbiased bases in C 6 . The proof of the lemma is shown to contain a mistake, ultimately invalidating three theorems derived in later publications.

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Product states and Schmidt rank of mutually unbiased bases in dimension six

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Real entries of complex Hadamard matrices and mutually unbiased bases in dimension six

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We investigate the number of real entries of an n×n complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22.

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Abstract McNulty and Weigert (2024 J. Phys. A: Math. Theor. submitted) voiced suspicions to the lemma 11(v) Part 6 in Chen and Yu (2017 J. Phys. A: Math. Theor. 50 475304) and three theorems derived in later publications (Liang et al 2019 Quantum Inf. Process. 18 352; Liang et al 2019 Linear Multilinear Algebra 69 2908–25; Chen et al 2021 Quantum Inf. Process. 20 353). For these suspicions, we reprove that any 6 × 6 complex Hadamard matrix whose number of real elements more than 22 does not belong to a set of four mutually unbiased bases. We show that the number of 2 × 2 complex Hadamard submatrices of any H 2-reducible matrix is not 10 , … , 16 , 18 . We also put forward some possible directions for further development.
Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I 2 ⊗ V and I 2 ⊗ W . We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.
Abstract We show that if a set of four mutually unbiased bases (MUBs) in C 6 exists and contains the identity, then any other basis in the set contains at most two product states and at the same time has Schmidt rank at least three. Here both the product states and the Schmidt rank are defined over the bipartite space C 2 ⊗ C 3 . We also investigate the connection of the Sinkhorn normal form of unitary matrices to the fact that there is at least one vector unbiased to any two orthonormal bases in any dimension.
It is conjectured that four mutually unbiased bases in dimension 6 do not exist in quantum information. The conjecture is equivalent to the nonexistence of some three 6×6 complex Hadamard matrices (CHMs) with Schmidt rank at least 3. We investigate the 6×6 CHM U of Schmidt rank 3 containing two nonintersecting identical 3×3 submatrices V, i.e. U=12[WVVX]. We show that such U exists, V, W, X have rank 2 or 3, and they have rank 2 at the same time. We construct the analytical expressions of U when V is, respectively, of rank 2, unitary and normal. We apply our results to the conjecture by showing that U with some normal V is not one of the three 6×6 CHMs.
Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct q+1q+1q+1 bases in dimension d=q(q+1)d = q(q+1)d=q(q+1) for every prime power q≡3(mod4)q\equiv3\pmod4q≡3(mod4). This count grows as d\sqrt dd​ and cannot be exceeded by tensor products of smaller sets.
We investigate the number of real entries of an n×n complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22.