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Entropy

Regression of Concurrence via Local Unitary Invariants

Ming Li, Wenjun Wang, Xiaoyu Zhang, Jing Wang, Lei Li, Shu‐Qian Shen

Published Oct 29, 2024• 2 citations
Publisher

Abstract

Concurrence is a crucial entanglement measure in quantum theory used to describe the degree of entanglement between two or more qubits. Local unitary (LU) invariants can be employed to describe the relevant properties of quantum states. Compared to quantum state tomography, observing LU invariants can save substantial physical resources and reduce errors associated with tomography. In this paper, we use LU invariants as explanatory variables and employ methods such as multiple regression, tree models, and BP neural network models to fit the concurrence of 2-qubit quantum states. For pure states and Werner states, by analyzing the correlation between data, a functional formula for concurrence in terms of LU invariants is obtained. Additionally, for any two-qubit quantum states, the prediction accuracy for concurrence reaches 98.5%.

DOI
10.3390/e26110917

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Concurrence is a crucial entanglement measure in quantum theory used to describe the degree of entanglement between two or more qubits. Local unitary (LU) invariants can be employed to describe the relevant properties of quantum states. Compared to quantum state tomography, observing LU invariants can save substantial physical resources and reduce errors associated with tomography. In this paper, we use LU invariants as explanatory variables and employ methods such as multiple regression, tree models, and BP neural network models to fit the concurrence of 2-qubit quantum states. For pure states and Werner states, by analyzing the correlation between data, a functional formula for concurrence in terms of LU invariants is obtained. Additionally, for any two-qubit quantum states, the prediction accuracy for concurrence reaches 98.5%.
We propose a measure of two-qubit entanglement that is invariant under local unitary transformations, and which is based on local measurements covariances. It measures the Hilbert-Schmidt distance between the state and the product state obtained by multiplying the local density matrices. The measure has the benefit that the experimentalist need not have any a priori knowledge of the state to make the measurements. For pure states, the measure provides the state's concurrence directly, without resorting to state tomography. For statistically mixed states, the measure provides bounds for the concurrence. The two-qutrit case is also studied.
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).
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.
It is widely recognized that concurrence can be regarded as a measure of two-qubit entanglement. This paper studies the geometry of concurrence for a two-qubit system to show that the concurrence serves as a coordinate of the factor space , where is the space of normalized two-qubit states, and where G = U (1) × SU (2) × SU (2). Any monotonically increasing function of the concurrence can serve as a measure for entanglement. From the viewpoint of Riemannian geometry, a state with concurrence r is shown to be distant from the separable states by , where r ranges over 0 ⩽ r ⩽ 1. In addition, measures of entanglement for n -qubits are discussed on the basis of a bipartite decomposition with ℓ + m = n . They are invariant under the local unitary transformation group U (2 ℓ ) × U (2 m ).
Quantum entanglement is a crucial resource in quantum information processing, advancing quantum technologies. The greater the uncertainty in subsystems' pure states, the stronger the quantum entanglement between them. From the dual form of qqq-concurrence (q≥2q\geq 2q≥2) we introduce the total concurrence. A bona fide measure of quantum entanglement is introduced, the Cqt\mathcal{C}^{t}_qCqt​-concurrence (q≥2q \geq 2q≥2), which is based on the total concurrence. Analytical lower bounds for the Cqt\mathcal{C}^{t}_qCqt​-concurrence are derived. In addition, an analytical expression is derived for the Cqt\mathcal{C}^{t}_qCqt​-concurrence in the cases of isotropic and Werner states. Furthermore, the monogamy relations that the Cqt\mathcal{C}^{t}_qCqt​-concurrence satisfies for qubit systems are examined. Additionally, based on the parameterized ααα-concurrence and its complementary dual, the Cαt\mathcal{C}^{t}_αCαt​-concurrence (0≤α≤12)(0\leqα\leq\frac{1}{2})(0≤α≤21​) is also proposed.
We define the concurrence hierarchy as d − 1 independent invariants under local unitary transformations in d -level quantum system. The first one is the original concurrence defined by Wootters (1998 Phys. Rev. Lett. 80 2245) and Hill and Wootters (1997 Phys. Rev. Lett. 78 5022) in a two-level quantum system and generalized to the d -level pure quantum state case. We propose to use this concurrence hierarchy as a measurement of entanglement. This measurement does not increase under local quantum operations and classical communication.