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arXiv:1006.1697 · Physical Review A

When different entanglement witnesses detect the same entangled states

Jinchuan Hou, Yu Jing Guo

Published Nov 2, 2010• 27 citations
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Abstract

The question of under what conditions different witnesses (e.g., ${W}_{1},{W}_{2}$) may detect some common entangled states [i.e., there exists some state $\ensuremath{\rho}$ so that $\mathrm{Tr}({W}_{1}\ensuremath{\rho})<0$ and $\mathrm{Tr}({W}_{2}\ensuremath{\rho})<0$] is answered for both finite-dimensional and infinite-dimensional bipartite systems. Finitely many different witnesses ${W}_{1},{W}_{2},\dots{},{W}_{n}$ can detect some common entangled states if and only if ${\ensuremath{\sum}}_{i=1}^{n}{d}_{i}{W}_{i}$ is still a witness for any nonnegative numbers ${d}_{1},{d}_{2},\dots{},{d}_{n}$ with ${\ensuremath{\sum}}_{i=1}^{n}{d}_{i}=1$; they cannot detect any common entangled state if and only if ${\ensuremath{\sum}}_{i=1}^{n}{c}_{i}{W}_{i}$ is a positive operator for some nonnegative numbers ${c}_{1},{c}_{2},\dots{},{c}_{n}$ with ${\ensuremath{\sum}}_{i=1}^{n}{c}_{i}=1$. For two witnesses ${W}_{1}$ and ${W}_{2}$ more can be said. First, ${W}_{1}$ and ${W}_{2}$ can detect the same set of entangled states if and only if ${W}_{1}=a{W}_{2}$ for some number $a>0$. Second, ${W}_{2}$ can detect more entangled states than ${W}_{1}$ can if and only if ${W}_{1}=a{W}_{2}+D$ for some number $a>0$ and a positive operator $D$. As an application, some characterizations of the optimal witnesses are given and some structural properties of the decomposable optimal witnesses are presented.

DOI
10.1103/physreva.82.052301

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The question of under what conditions different witnesses (e.g., W1,W2{W}_{1},{W}_{2}W1​,W2​) may detect some common entangled states [i.e., there exists some state \ensuremathρ\ensuremath{\rho}\ensuremathρ so that Tr(W1\ensuremathρ)<0\mathrm{Tr}({W}_{1}\ensuremath{\rho})<0Tr(W1​\ensuremathρ)<0 and Tr(W2\ensuremathρ)<0\mathrm{Tr}({W}_{2}\ensuremath{\rho})<0Tr(W2​\ensuremathρ)<0] is answered for both finite-dimensional and infinite-dimensional bipartite systems. Finitely many different witnesses W1,W2,…,Wn{W}_{1},{W}_{2},\dots{},{W}_{n}W1​,W2​,…,Wn​ can detect some common entangled states if and only if \ensuremath∑i=1ndiWi{\ensuremath{\sum}}_{i=1}^{n}{d}_{i}{W}_{i}\ensuremath∑i=1n​di​Wi​ is still a witness for any nonnegative numbers d1,d2,…,dn{d}_{1},{d}_{2},\dots{},{d}_{n}d1​,d2​,…,dn​ with \ensuremath∑i=1ndi=1{\ensuremath{\sum}}_{i=1}^{n}{d}_{i}=1\ensuremath∑i=1n​di​=1; they cannot detect any common entangled state if and only if \ensuremath∑i=1nciWi{\ensuremath{\sum}}_{i=1}^{n}{c}_{i}{W}_{i}\ensuremath∑i=1n​ci​Wi​ is a positive operator for some nonnegative numbers c1,c2,…,cn{c}_{1},{c}_{2},\dots{},{c}_{n}c1​,c2​,…,cn​ with \ensuremath∑i=1nci=1{\ensuremath{\sum}}_{i=1}^{n}{c}_{i}=1\ensuremath∑i=1n​ci​=1. For two witnesses W1{W}_{1}W1​ and W2{W}_{2}W2​ more can be said. First, W1{W}_{1}W1​ and W2{W}_{2}W2​ can detect the same set of entangled states if and only if W1=aW2{W}_{1}=a{W}_{2}W1​=aW2​ for some number a>0a>0a>0. Second, W2{W}_{2}W2​ can detect more entangled states than W1{W}_{1}W1​ can if and only if W1=aW2+D{W}_{1}=a{W}_{2}+DW1​=aW2​+D for some number a>0a>0a>0 and a positive operator DDD. As an application, some characterizations of the optimal witnesses are given and some structural properties of the decomposable optimal witnesses are presented.
We show the properties and characterization of coherence witnesses. We show methods for constructing coherence witnesses for an arbitrary coherent state. We investigate the problem of finding common coherence witnesses for certain class of states. We show that finitely many different witnesses W1,W2,⋯,Wn can detect some common coherent states if and only if ∑i=1ntiWi is still a witnesses for any nonnegative numbers ti(i=1,2,⋯,n). We show coherent states play the role of high-level witnesses. Thus, the common state problem is changed into the question of when different high-level witnesses (coherent states) can detect the same coherence witnesses. Moreover, we show a coherent state and its robust state have no common coherence witness and give a general way to construct optimal coherence witnesses for any comparable states.
Dvoretzky's theorem and the complexity of entanglement detection, Discrete Analysis 2017:1, 20 pp. Let HHH be a Hilbert space. A _state_ on HHH is a linear operator ρ:H→H\rho:H\to Hρ:H→H such that tr(ρ)=1(\rho)=1(ρ)=1 and tr(ρP)≥0(\rho P)\geq 0(ρP)≥0 for every orthogonal projection PPP. A linear operator satisfying just the second condition is called _positive_. If H=H1⊗H2H=H_1\otimes H_2H=H1​⊗H2​ then an important piece of information about ρ\rhoρ is the extent to which it can be split up into a part that acts on H1H_1H1​ and a part that acts on H2H_2H2​. An appropriate formal definition is the following: we say that ρ\rhoρ is _separable_ if it can be approximated in the trace-class norm by operators of the form ∑iciρi1⊗ρi2\sum_ic_i\rho^1_i\otimes\rho^2_i∑i​ci​ρi1​⊗ρi2​, where each ρi1\rho^1_iρi1​ is a state on H1H_1H1​, each ρi2\rho^2_iρi2​ is a state on H2H_2H2​, and the cic_ici​ are positive constants. An operator that is not separable is called _entangled_. Owing to the importance of entanglement, it is useful to have a criterion that detects it. A simple necessary condition is that if Φ\PhiΦ is a linear map from operators to operators that takes positive operators to positive operators, and if ρ\rhoρ is separable, then (Φ⊗I)ρ(\Phi\otimes I)\rho(Φ⊗I)ρ is a positive operator: that is simply because (Φ⊗I)(∑iciρi1⊗ρi2)=∑iciΦρi1⊗ρi2(\Phi\otimes I)(\sum_ic_i\rho^1_i\otimes\rho^2_i)=\sum_ic_i\Phi\rho^1_i\otimes\rho^2_i(Φ⊗I)(∑i​ci​ρi1​⊗ρi2​)=∑i​ci​Φρi1​⊗ρi2​ and each Φρi1\Phi\rho^1_iΦρi1​ is positive. Interestingly, however, the converse is true: if ρ\rhoρ is entangled, then there exists a positive Φ\PhiΦ such that (Φ⊗I)(ρ)(\Phi\otimes I)(\rho)(Φ⊗I)(ρ) is _not_ positive. This equivalence is called the _Horodecki criterion_ for entanglement. The Horodecki criterion gives us a witness Φ\PhiΦ for each entangled state ρ\rhoρ. The starting question for this paper is how many different witnesses one needs to detect all entangled states (as a function of the dimensions of H1H_1H1​ and H2H_2H2​). More precisely, since this number is known to be infinite, the paper considers the stronger notion of _robustly_ entangled states, which are states that remain entangled even when you average them with a suitable multiple of the identity. The main result of the paper is that the number needed is large: the authors obtain a bound of exp⁡(cd3/log⁡d)\exp(cd^3/\log d)exp(cd3/logd). A key observation that enables them to prove this is that there is a resemblance between the problem they are considering and a result from a famous paper of Figiel, Lindenstrauss and Milman concerning Dvoretzky's theorem [1]. It follows from the analysis in the FLM paper that the product of the logarithm of the number of faces of a symmetric convex body with the logarithm of the number of vertices has to be at least cncncn, so it is not possible for a symmetric convex body to have few faces and few vertices (in strong contrast with a general convex body, since an nnn-dimensional simplex has n+1n+1n+1 of each). Something like that occurs here. Roughly speaking, the set of separable states can be regarded as having "few" extreme points, and therefore "many" faces. A single witness Φ\PhiΦ cannot help with too many faces, so the number of witnesses needed must be large. These interesting ideas are new to the field of quantum information. [1] T. Figiel, J. Lindenstrauss and V. D. Milman, _The dimension of almost spherical sections of convex bodies,_ Acta Math. 139 (1-2) (1977), pp. 53-94.
This thesis deals with methods of the detection of entanglement.After recalling some facts and definitions concerning entanglement and separability, we investigate two methods of the detection of entanglement.In the first part of this thesis we consider so-called entanglement witnesses, mainly in view of the detection of multipartite entanglement.Entanglement witnesses are observables for which a negative expectation value indicates entanglement.We first present a simple method to construct these witnesses.Since witnesses are nonlocal observables, they are not easy to measure in a real experiment.However, as we will show, one can circumvent this problem by decomposing the witness into several local observables which can be measured separately.We calculate the local decompositions for several interesting witnesses for two, three and four qubits.Local decompositions can be optimized in the number of measurement settings which are needed for an experimental implementation.We present a method to prove that a given local decomposition is optimal and discuss with this the optimality of our decompositions.Then we present another method of designing witnesses which are by construction measurable with local measurements.Finally, we shortly report on experiments where some of the witnesses derived in this part have been used to detect three-and four-partite entanglement of polarized photons.The second part of this thesis deals with separability criteria which are written in terms of uncertainty relations.There are two different formulations of uncertainty relations since one can measure the uncertainty of an observable by its variance as well as by entropic quantities.We show that both formulations are useful tools for the derivation of separability criteria for finite-dimensional systems and investigate the resulting criteria.Our results in this part exhibit also some more fundamental properties of entanglement: We show how known separability criteria for infinitedimensional systems can be translated to finite-dimensional systems.Furthermore, we prove that any entropic uncertainty relation on one part of the system gives rise to a separability criterion on the composite system.
It is shown that every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form $\ensuremath{\alpha}I+T$ with $\ensuremath{\alpha}$ a non-negative number and TTT a finite rank self-adjoint operator. We also provide two methods of constructing an entanglement witness and apply them to obtain some entangled states that cannot be detected by the positive partial transpose criterion and the realignment criterion.
Entanglement witnesses (EWs) are nonpositive Hermitian operators which can detect the presence of entanglement. In this paper, we prove that for a given pair of entangled states ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$ an EW, WWW, common to both of them exists if and only if for all $0\ensuremath{\leqslant}\ensuremath{\lambda}\ensuremath{\leqslant}1$, $\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}$ is an entangled state. Based on this result, we find a sufficient condition for the presence of entanglement. We then show that an EW which can detect all bell states simultaneously does not exist.