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arXiv:2508.18644 · Open Systems & Information Dynamics

Equality Condition for a Matrix Inequality by Partial Transpose

Nalan Wang, Lin Chen

Published Jun 1, 2026• 0 citations
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Abstract

The partial transpose map is a linear map widely used in quantum information theory. We study the equality condition for a matrix inequality generated by partial transpose, namely [Formula: see text], where [Formula: see text]’s and [Formula: see text]’s are respectively the matrices of the same size, and [Formula: see text] is the Schmidt rank. We find explicit conditions that are equivalent to equality when the [Formula: see text]’s are column or row vectors, or [Formula: see text] matrices. For the case where the Schmidt rank equals the dimension of [Formula: see text], we extend the results from [Formula: see text] matrices to square matrices, and further to rectangular matrices. In detail, we show that [Formula: see text] is locally equivalent to an elegant block-diagonal form consisting solely of identity and zero matrices. We also study the general case for [Formula: see text], and it turns out that the key is to characterize the expression of matrices [Formula: see text]’s and [Formula: see text]’s.

DOI
10.1142/s1230161226500071

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The partial transpose map is a linear map widely used in quantum information theory. We study the equality condition for a matrix inequality generated by partial transpose, namely [Formula: see text], where [Formula: see text]’s and [Formula: see text]’s are respectively the matrices of the same size, and [Formula: see text] is the Schmidt rank. We find explicit conditions that are equivalent to equality when the [Formula: see text]’s are column or row vectors, or [Formula: see text] matrices. For the case where the Schmidt rank equals the dimension of [Formula: see text], we extend the results from [Formula: see text] matrices to square matrices, and further to rectangular matrices. In detail, we show that [Formula: see text] is locally equivalent to an elegant block-diagonal form consisting solely of identity and zero matrices. We also study the general case for [Formula: see text], and it turns out that the key is to characterize the expression of matrices [Formula: see text]’s and [Formula: see text]’s.
Quantum information theory has been greatly developed in the past decades, and many theoretical problems are related to matrix theory. We study the equality condition for a matrix inequality, [Formula: see text] rank([Formula: see text] ⊗[Formula: see text]) [Formula: see text] rank([Formula: see text] ⊗[Formula: see text] where [Formula: see text]’s are linearly independent matrices of the same size, and [Formula: see text]’s are linearly independent matrices of the same size. The inequality used to be a conjecture since 2013 and has recently been proven in the paper [Z. Song, L. Chen, Y. Sun and M. Hu, IEEE Trans. Inform. Theory 69 (2023) 2385]. We study several cases such as that [Formula: see text]’s are column vectors and [Formula: see text]’s are of various sizes. It turns out that some cases never satisfy the equality condition.
Hadamard matrices play a key role in the study of algebra and quantum information theory, and it is an open problem to characterize 6 × 6 Hadamard matrices. In this paper, we investigate the problem in terms of the Schmidt rank. The primary achievement of this paper lies in establishing a systematic approach to generate 6 × 6 Hadamard matrices and H-2 reducible matrices through partial transpose. First, if the Schmidt rank of a Hadamard matrix is at most three, then the partial transpose of the Hadamard matrix is also a Hadamard matrix. Conversely, if the Schmidt rank is four, then the partial transpose is no longer a Hadamard matrix. Second, we discuss the relationship between Schmidt rank and H-2 reducible matrices. We prove Hadamard matrices with Schmidt-rank-one are all H-2 reducible, and prove that some Schmidt-rank-two matrices are H-2 reducible. Finally, we confirm that the partial transpose of an H-2 reducible Schmidt-rank-one or two Hadamard matrix remains H-2 reducible.
A block matrix [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] is positive partial transpose (PPT) if both [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] and [AX∗XB]\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right][AX​X∗B​] are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.
Permutation matrices are an old and widely used family of matrices in mathematics and physics. In this paper, we classify bipartite permutation matrices under local permutation equivalence when the equivalence is in the form of the Kronecker product of two permutation matrices. We classify all [Formula: see text] bipartite permutation matrices of Schmidt rank up to three. We also study the equivalence for general bipartite permutation gates in terms of Schmidt rank two. We apply our result to study the entangling power of two-qubit permutation gates. Under the local unitary equivalence, we show that the five distinct two-qubit gates under local permutation equivalence can be reduced to exactly three gates. Our results present a physical understanding of the permutation group from the point of view of quantum information and experiments.
We express the positive-partial-transpose (PPT) separability criterion for symmetric states of multiqubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix from the tensor representation of the state and show that it is similar to the partial transpose of the density matrix written in the computational basis. Furthermore, the positivity of this matrix is equivalent to the positivity of a correlation matrix constructed from tensor products of Pauli operators. This allows for a more transparent experimental interpretation of the PPT criteria for an arbitrary spin-jjj state. The unitary matrices connecting our matrix to the partial transpose of the state generalize the so-called magic basis that plays a central role in Wootters' explicit formula for the concurrence of a two-qubit system and the Bell bases used for the teleportation of a one- or two-qubit state.
Abstract. We consider a family of vector and operator norms which we refer to as Schmidt norms. We show that these norms have several uses in quantum information theory – they can be used to help classify k-positive linear maps (and hence entanglement witnesses), they are useful for approaching the problem of finding non-positive partial transpose bound entangled Werner states, they are related to the quantum fidelity and trace distance measures, and they are connected to the recently-defined local numerical range. We show that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We show that one particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections, and we use our inequalities to study that family of projections. We also develop a family of semidefinite programs that can be used to further bound the operator norms. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states. 1.