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International Journal of Quantum Information

Classification of bipartite permutation matrices under local per mutation equivalence

Jian Yan, Lin Chen

Published Apr 26, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0219749925500121

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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.
Local unitary equivalence plays a crucial role in quantifying and categorizing the entanglement of quantum states. When it comes to two isospectral diagonal bipartite quantum states, determining whether they are locally unitary equivalent turns out to be a rather challenging problem. Intuitively speaking, we conjecture that for two diagonal quantum states, local unitary equivalence and local permutation equivalence between them are the same. By taking into account the cosets of the fixed point subgroup and the tensor decomposability of permutation matrices, we present the classifications of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text] under local permutation equivalence, respectively. Furthermore, relying on the classification results of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text], we demonstrate that if two diagonal quantum states are not locally permutation equivalent, then they cannot be locally unitary equivalent. This offers significant backing for the conjecture that two general bipartite diagonal quantum states are locally unitary equivalent if and only if they are locally permutation equivalent.
Abstract The multipartite unitary gates are called genuine if they are not product unitary operators across any bipartition. We mainly investigate the classification of genuine multipartite unitary gates of Schmidt rank two (SR-2), by focusing on the multiqubit scenario. For genuine multipartite (excluding bipartite) unitary gates of SR-2, there is an essential fact that their Schmidt decompositions are unique. Based on this fact, we propose a key notion named as singular number (SN) to classify the unitary gates concerned. The SN is defined as the number of local singular operators in the Schmidt decomposition. We then determine the accurate range of SN. For each SN, we formulate the parametric Schmidt decompositions of genuine multiqubit unitary gates under local equivalence. Finally, we extend the study to three-qubit diagonal unitary gates due to the close relation between diagonal unitary gates and SR-2 unitaries. We start with discussing two typical examples of SR-2, one of which is a fundamental three-qubit unitary gate, i.e. the CCZ gate. Then we characterize the diagonal unitary gates of SR greater than two. We show that a three-qubit diagonal unitary gate has SR at most three, and present a necessary and sufficient condition for such a unitary gate of SR-3. This completes the characterization of all genuine three-qubit diagonal unitary gates.
Abstract We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. Our scrutiny reveals that these operators can be used to simulate any bipartite unitary gate via stochastic local operations and classical communication. Furthermore, we establish a one-to-one connection between the set of local diagonal unitary invariant dual unitary operators with maximum entangling power and the set of complex Hadamard matrices. Finally, we discuss distinguishability of unitary operators in the setting of the stated diagonal symmetry.
The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂd1 ⊗ ℂd2 are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.
Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.
In quantum information, pure states correspond to tensors. Two states equivalent under local unitary group or local general linear group can be used for the same quantum-information tasks, and it is thus an essential problem to determine whether two tensors are equivalent. In this paper, we determine the two sorts of equivalence for 2-tensors in terms of the Schmidt decomposition and singular value decomposition of matrices. We also extend the equivalence to some 3-tensors, proving that Schmidt decomposition is unachievable for 3-tensors. However, we present a method to express 3-tensors using the logic behind Schmidt decomposition. Furthermore, we generalized the scenario to higher dimensions and discussed the characteristic behaviors of different groups in tensor transformations.