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Quantum Information and Computation

The necessary and sufficient condition for the uniqueness of Schmidt decomposition and the necessary condition for the maximal von Neumann entanglement entropy

Dafa Li, Maggie X. Cheng, Xiangrong Li, Shuwany Li

Published Sep 1, 2024• 1 citation
Publisher

Abstract

Schmidt decomposition (SD) has been proven to be an important tool in quantum information and computation. Ac{\' i}n et al. proposed the SD for three qubits, but in the meanwhile, they indicated that it is possible to have two different SDs for the same state. The more challenging quest of finding the sufficient and necessary condition for the uniqueness of SD has never been undertaken. In this paper, we propose a necessary and sufficient condition for the uniqueness of SD for three qubits. By examining the condition, one can tell what state has one SD and what state has two SDs without actually performing the Schmidt decomposition. We investigate the relation between the uniqueness of SD and the von Neumann entanglement entropy (vNEE). To this end, we prove that any state having the maximal vNEE $S(\rho _{x})=\ln 2$, $ x=A, B,$ or $C$ must have a unique SD. %Therefore, we should choose a state having a unique SD for its maximal vNEE for quantum information theory. This means if a state has two SDs, then the state does not have the maximal vNEE. Therefore, we should not choose a state having two SDs for its maximal vNEE for quantum information theory. In this paper, we also give all the SD states that have the maximal vNEE and a unique SD, as well as all the SD states that have a unique SD.

DOI
10.26421/qic24.9-10-5

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Schmidt decomposition (SD) has been proven to be an important tool in quantum information and computation. Ac{' i}n et al. proposed the SD for three qubits, but in the meanwhile, they indicated that it is possible to have two different SDs for the same state. The more challenging quest of finding the sufficient and necessary condition for the uniqueness of SD has never been undertaken. In this paper, we propose a necessary and sufficient condition for the uniqueness of SD for three qubits. By examining the condition, one can tell what state has one SD and what state has two SDs without actually performing the Schmidt decomposition. We investigate the relation between the uniqueness of SD and the von Neumann entanglement entropy (vNEE). To this end, we prove that any state having the maximal vNEE S(ρx)=ln⁡2S(\rho _{x})=\ln 2S(ρx​)=ln2, x=A,B, x=A, B,x=A,B, or CCC must have a unique SD. %Therefore, we should choose a state having a unique SD for its maximal vNEE for quantum information theory. This means if a state has two SDs, then the state does not have the maximal vNEE. Therefore, we should not choose a state having two SDs for its maximal vNEE for quantum information theory. In this paper, we also give all the SD states that have the maximal vNEE and a unique SD, as well as all the SD states that have a unique SD.
The bipartite entanglement of a pure quantum state is known to be characterized by its Schmidt decomposition. In particular the state is maximally entangled when all the Schmidt coefficients are equal. We point out a convenient method which always yields a single analytical condition for the state to be maximally entangled, in terms of its expansion coefficients in any basis. The method works even when the Schmidt coefficients cannot be calculated analytically, and does not require their calculation. As an example this technique is used to derive the Bell basis for a system of two qubits. In a second example the technique shows a particular state to \textit{never} be maximally entangled, a general conclusion that cannot be reached using the Schmidt decomposition.
We relate the notion of entanglement for quantum systems composed of two identical constituents to the impossibility of attributing a complete set of properties to both particles. This implies definite constraints on the mathematical form of the state vector associated with the whole system. We then analyze separately the cases of fermion and boson systems, and we show how the consideration of both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the global state vector and the von Neumann entropy of the one-particle reduced density operators can supply us with a consistent criterion for detecting entanglement. In particular, the consideration of the von Neumann entropy is particularly useful in deciding whether the correlations of the considered states are simply due to the indistinguishability of the particles involved or are a genuine manifestation of the entanglement. The treatment leads to a full clarification of the subtle aspects of entanglement of two identical constituents which have been a source of embarrassment and of serious misunderstandings in the recent literature.
We review two general criteria for deciding whether a pure bipartite quantum state describing a system of two identical particles is entangled or not. The first one considers the possibility of attributing a complete set of objective properties to each particle belonging to the composed system, while the second is based both on the consideration of the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators. 1
We review two general criteria for deciding whether a pure bipartite quantum state describing a system of two identical particles is entangled or not. The first one considers the possibility of attributing a complete set of objective properties to each particle belonging to the composed system, while the second is based both on the consideration of the Slater-Schmidt number of the fermionic and bosonic analogue of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators.
We explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyze properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entangle ment.