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arXiv:1009.0116

The RCCN criterion of separability for states in infinite-dimensional quantum systems

Yu Jing Guo, Jinchuan Hou

Published Sep 1, 2010• 4 citations
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Abstract

In this paper, the realignment criterion and the RCCN criterion of separability for states in infinite-dimensional bipartite quantum systems are established. Let $H_A$ and $H_B$ be complex Hilbert spaces with $\dim H_A\otimes H_B=+\infty$. Let $ρ$ be a state on $H_A\otimes H_B$ and $\{δ_k\}$ be the Schmidt coefficients of $ρ$ as a vector in the Hilbert space ${\mathcal C}_2(H_A)\otimes{\mathcal C}_2(H_B)$. We introduce the realignment operation $ρ^R$ and the computable cross norm $\|ρ\|_{\rm CCN}$ of $ρ$ and show that, if $ρ$ is separable, then $\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k\leq1.$ In particular, if $ρ$ is a pure state, then $ρ$ is separable if and only if $\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k=1$.

DOI
10.48550/arxiv.1009.0116

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In this paper, the realignment criterion and the RCCN criterion of separability for states in infinite-dimensional bipartite quantum systems are established. Let HAH_AHA​ and HBH_BHB​ be complex Hilbert spaces with dim⁡HA⊗HB=+∞\dim H_A\otimes H_B=+\inftydimHA​⊗HB​=+∞. Let ρρρ be a state on HA⊗HBH_A\otimes H_BHA​⊗HB​ and {δk}\{δ_k\}{δk​} be the Schmidt coefficients of ρρρ as a vector in the Hilbert space C2(HA)⊗C2(HB){\mathcal C}_2(H_A)\otimes{\mathcal C}_2(H_B)C2​(HA​)⊗C2​(HB​). We introduce the realignment operation ρRρ^RρR and the computable cross norm ∥ρ∥CCN\|ρ\|_{\rm CCN}∥ρ∥CCN​ of ρρρ and show that, if ρρρ is separable, then ∥ρR∥Tr=∥ρ∥CCN=∑kδk≤1.\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k\leq1.∥ρR∥Tr​=∥ρ∥CCN​=k∑​δk​≤1. In particular, if ρρρ is a pure state, then ρρρ is separable if and only if ∥ρR∥Tr=∥ρ∥CCN=∑kδk=1\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k=1∥ρR∥Tr​=∥ρ∥CCN​=k∑​δk​=1.
Inspired by the `computable cross norm' or `realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let MMM be the matrix of the amplitudes of ∣ψ⟩\ketψ∣ψ⟩, we prove MMM is a compact operator. We also prove ∣ψ⟩\ketψ∣ψ⟩ is separable if and only if MMM is a bounded linear operator with rank 1, that is the image of MMM is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting properties.
Let HHH and KKK be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from B(H){\mathcal B}(H)B(H) into B(K){\mathcal B}(K)B(K) is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state ρ\rhoρ on H⊗KH\otimes KH⊗K is separable if and only if (Φ⊗I)ρ≥0(\Phi\otimes I)\rho\geq 0(Φ⊗I)ρ≥0 for all positive finite rank elementary operators Φ\PhiΦ. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.