arXiv:2607.16878

Bargmann invariants and local unitary equivalence

Minghui Ma, Rui Shi

quant-phmath.OASubmitted Jul 18, 2026• Announced Jul 21, 2026• 0 citations

Abstract

In this paper, we study the local unitary equivalence of quantum states on $\mathbb{C}^n\otimes\mathbb{C}^n$, which is an important notion in quantum information theory. For the case $n=2$, the local unitary orbits of two-qubit states are completely determined by their local unitary Bargmann invariants. We show that the local unitary Bargmann invariants do not form a complete set of invariants of local unitary orbits for $n\geqslant 3$, which negatively answers a problem proposed by L. Zhang and B. Xie.

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arXiv:2607.16878

Bargmann invariants and local unitary equivalence

Minghui Ma, Rui Shi

quant-phmath.OASubmitted Jul 18, 2026• Announced Jul 21, 2026• 0 citations

arXiv HTML PDF

Abstract

In this paper, we study the local unitary equivalence of quantum states on

\mathbb{C}^n\otimes\mathbb{C}^n

, which is an important notion in quantum information theory. For the case

n=2

, the local unitary orbits of two-qubit states are completely determined by their local unitary Bargmann invariants. We show that the local unitary Bargmann invariants do not form a complete set of invariants of local unitary orbits for

n\geqslant 3

, which negatively answers a problem proposed by L. Zhang and B. Xie.

Related papers

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Research on quantum states often focuses on the correlation between nonlocal effects and local unitary invariants, among which local unitary equivalence plays a significant role in quantum state classification and resource theories. This paper focuses on the local unitary equivalence of multipartite quantum states in quantum information theory, aiming to determine a complete set of invariants that identify their local unitary orbits; these invariants are crucial for deriving polynomial invariants and describing the physical properties preserved under local unitary transformations. The study deeply explores the characterization of local unitary equivalence and the method of detecting entanglement using local unitary Bargmann invariants. Taking two-qubit systems as an example, it verifies the measurability of invariants that determine equivalence and establishes a connection between Makhlin fundamental invariants (a complete set of 18 local unitary invariants for two-qubit states) and local unitary Bargmann invariants. These Bargmann invariants, related to the traces of products of density operators and marginal states, can be measured through cycle tests (an extended form of swap tests).Research on quantum states often focuses on the correlation between nonlocal effects and local unitary invariants, among which local unitary equivalence plays a significant role in quantum state classification and resource theories. This paper focuses on the local unitary equivalence of multipartite quantum states in quantum information theory, aiming to determine a complete set of invariants that identify their local unitary orbits; these invariants are crucial for deriving polynomial invariants and describing the physical properties preserved under local unitary transformations. The study deeply explores the characterization of local unitary equivalence and the method of detecting entanglement using local unitary Bargmann invariants. Taking two-qubit systems as an example, it verifies the measurability of invariants that determine equivalence and establishes a connection between Makhlin fundamental invariants (a complete set of 18 local unitary invariants for two-qubit states) and local unitary Bargmann invariants. These Bargmann invariants, related to the traces of products of density operators and marginal states, can be measured through cycle tests (an extended form of swap tests).

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Since two quantum states that are local unitary (LU) equivalent have the same amount of entanglement, it is meaningful to find a practical method to determine the LU equivalence of given quantum states. In this paper, we present a valid process to find the unitary tensor product decomposition for an arbitrary unitary matrix. Then, by using this process, the conditions for determining the local unitary equivalence of quantum states are obtained. A numerical verification is carried out, which shows the practicability of our protocol. We also present a property of LU invariants by using the universality of quantum gates which can be used to construct the complete set of LU invariants.Since two quantum states that are local unitary (LU) equivalent have the same amount of entanglement, it is meaningful to find a practical method to determine the LU equivalence of given quantum states. In this paper, we present a valid process to find the unitary tensor product decomposition for an arbitrary unitary matrix. Then, by using this process, the conditions for determining the local unitary equivalence of quantum states are obtained. A numerical verification is carried out, which shows the practicability of our protocol. We also present a property of LU invariants by using the universality of quantum gates which can be used to construct the complete set of LU invariants.

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Caihong Wang, Jiangtao Yuan, Mao-Sheng Li +2 more

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Bargmann Invariants Are Insufficient for Complete Local-Unitary Orbit Discrimination

Lin Zhang, Bing Xie

Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In $2\otimes2$ and $2\otimes3$ systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decompositionBargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In $2\otimes2$ and $2\otimes3$ systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition

quant-pharXiv:2609.11090Sep 11, 20260 citations

In this paper, we study the local unitary equivalence of quantum states on

\mathbb{C}^n\otimes\mathbb{C}^n

, which is an important notion in quantum information theory. For the case

n=2

n\geqslant 3

, which negatively answers a problem proposed by L. Zhang and B. Xie.

Local unitary classification of sets of generalized Bell states in

\mathbb{C}^{d}\otimes \mathbb{C}^{d}

l\ (\geq 2)

generalized Bell states (GBSs), high-dimensional generalizations of Bell states, in a bipartite system

\mathbb{C}^{d}\otimes \mathbb{C}^{d}

with

d\geq 3

. Based on this classification, we determine the minimal cardinality of indistinguishable GBS sets in

\mathbb{C}^{6}\otimes \mathbb{C}^{6}

under one-way local operations and classical communication (one-way LOCC). We first propose two classification methods based on LU-equivalence for all sets of l GBSs (l-GBS sets). We then establish LU-classification for all 2-GBS, 3-GBS, 4-GBS and 5-GBS sets in

\mathbb{C}^{6}\otimes \mathbb{C}^{6}

. Since LU-equivalent sets share identical local distinguishability, it suffices to examine representative GBS sets from equivalent classes. Notably, we identify a one-way LOCC indistinguishable 4-GBS set among these representatives, thereby resolving the case of

d = 6

for the problem of determining the minimum cardinality of one-way LOCC indistinguishable GBS sets in (Yuan et al. in Quantum Inf Process. 18:145, 2019) or (Zhang et al. in Phys Rev A 91:012329, 2015).

2\otimes2

and

2\otimes3

systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition