Witness of mixed separable states useful for entanglement creation

Ganguly, Nirman; Chatterjee, Jyotishman; Majumdar, A. S.

doi:10.1103/physreva.89.052304

Cited by 27 publications

(34 citation statements)

References 40 publications

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“…A possible implementation of the proposed witnesses for detecting mixedness here could be through techniques involving measurement of two-photon polarizationentangled modes for qutrits 71 . Since the class of all pure states is not convex, the witnesses proposed for detecting mixedness do not arise from the separability criterion that holds for the widely studied entanglement witnesses 37 , as well as the recently proposed teleportation witnesses 38 , and witnesses for absolutely separable states 39 . However, a similar prescription of distinction of categories of quantum states based on the measurement of expectation values of Hermitian operators is followed.…”

Section: Discussionmentioning

confidence: 99%

“…An observational scheme which can detect mixedness of qutrit systems unambiguously, requires less resources compared to tomography, and is implementable through the measurement of Hermitian witness-like operators 28 . It may be relevant to note here though that the set of pure states is not convex, and hence, such witness-like operators do not arise from any geometrical separability criterion inherent to the theory of entanglement witnesses 37 , that has been applied more recently to the cases of teleportation witnesses 38 , as well as for witnesses of absolutely separable states 39 . The operational determination of purity using the RS relation requires a few additional steps.…”

Section: Determining Purity Of States Using the Robertson-schrodingermentioning

confidence: 99%

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Some applications of uncertainty relations in quantum information

Majumdar

Pramanik

2016

Int. J. Quantum Inform.

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We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the EPR paradox. Entropic uncertainty relations are used to reveal quantum steering for non-Gaussian continuous variable states. Entropic uncertainty relations for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncertainty and the nonlocality of retrieval games for bipartite and tipartite systems. The RobertsonSchrodinger uncertainty relation is applied for distinguishing pure and mixed states of discrete variables.

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Section: Discussionmentioning

confidence: 99%

Section: Determining Purity Of States Using the Robertson-schrodingermentioning

confidence: 99%

Some applications of uncertainty relations in quantum information

Majumdar

Pramanik

2016

Int. J. Quantum Inform.

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“…It is worth noting that these results show that the sets Conv σ(BP m,n ) and Conv σ(DBP m,n ) are the sets of spectra of what might be called "absolute separability witnesses" and "absolute PPT witnesses", respectively. These types of witnesses were considered in [26].…”

Section: Absolute Separability and Absolute Pptmentioning

confidence: 99%

The inverse eigenvalue problem for entanglement witnesses

Johnston

Patterson

2018

Linear Algebra and its Applications

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We consider the inverse eigenvalue problem for entanglement witnesses, which asks for a characterization of their possible spectra (or equivalently, of the possible spectra resulting from positive linear maps of matrices). We completely solve this problem in the two-qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions. We also establish a natural duality relationship with the set of absolutely separable states, and we completely characterize witnesses (i.e., separating hyperplanes) of that set when one of the local dimensions is 2.From a mathematical perspective, quantum information theory (and more specifically, quantum entanglement theory) is largely concerned with properties of (Hermitian) positive semidefinite matrices and the tensor product. A pure quantum state |v ∈ C n is a unit (column) vector and a mixed quantum state ρ ∈ M n (C) is a (Hermitian) positive semidefinite matrix with Tr(ρ) = 1 (we use Tr(·) to denote the trace and M n (R) or M n (C) to denote the set of n × n real or complex matrices, respectively). Whenever we use "ket" notation like |v or |w , or lowercase Greek letters like ρ or σ, we are implicitly assuming that they represent pure or mixed quantum states, respectively.where "⊗" is the tensor (Kronecker) product, v| is the dual (row) vector of |v , so |v v| is the rank-1 projection onto |v , and p 1 , p 2 , . . . form a probability distribution (i.e., they are non-negative and add up to 1). Equivalently, ρ is separable if and only if it can be written in the formwhere each X j ∈ M m (C) and Y j ∈ M n (C) is a (Hermitian) positive semidefinite matrix. If ρ is not separable then it is called entangled.Determining whether or not a mixed state is separable is a hard problem [10,11], so in practice numerous one-sided tests are used to demonstrate separability or entanglement (see [12,13] and the references therein for a more thorough introduction to this problem). The most well-known such test says that if we define the partial transpose of a matrix A = ∑ j B j ⊗ C j ∈ M m (C) ⊗ M n (C) viathen ρ being separable implies that ρ Γ is positive semidefinite (so we write ρ Γ O) [14]. This test follows simply from the fact that if ρ is separable then ρ = ∑ j X j ⊗ Y j with each X j , Y j O, sowhich is still positive semidefinite since each Y T j is positive semidefinite, and tensoring and adding positive semidefinite matrices preserves positive semidefiniteness. If a mixed state ρ is such that ρ Γ O then we say that it has positive partial transpose (PPT), and the previous discussion shows that the set of separable states is a subset of the set of PPT states. Entanglement Witnesses and Positive MapsA straightforward generalization of the partial transpose test for entanglement is based on positive linear matrix-valued maps. A linear map Φ : M m (C) → M n (C) is called positive if X O implies Φ(X) O, and the transpose is an example of one such map. Based on positive maps, we can define block-positive matrices, which are matrices of the form W := ...

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“…Such states are called absolutely separable states [31][32][33]. Geometrically one may think of the absolutely separable states as a convex and compact [34] subset A of the separable states S, much like S forms a convex subset of all states. In particular, when dim( H A ) = dim( H B ) = d, one may inscribe a ball of maximal radius r max = 1/ d √ d 2 − 1 into the set S, where the distance of a state ρ from the central maximally mixed state ρ mix = 1 d 2 /d 2 is measured by the trace distance ||ρ − ρ mix || = Tr(ρ − ρ mix ) 2 .…”

Section: A Entanglement and Separabilitymentioning

confidence: 99%

Geometry of two-qubit states with negative conditional entropy

Friis

Bulusu

Bertlmann

2017

J. Phys. A: Math. Theor.

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We review the geometric features of negative conditional entropy and the properties of the conditional amplitude operator proposed by Cerf and Adami for two qubit states in comparison with entanglement and nonlocality of the states. We identify the region of negative conditional entropy in the tetrahedron of locally maximally mixed two-qubit states. Within this set of states, negative conditional entropy implies nonlocality and entanglement, but not vice versa, and we show that the Cerf-Adami conditional amplitude operator provides an entanglement witness equivalent to the Peres-Horodecki criterion. Outside of the tetrahedron this equivalence is generally not true.

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Witness of mixed separable states useful for entanglement creation

Cited by 27 publications

References 40 publications

Some applications of uncertainty relations in quantum information

Some applications of uncertainty relations in quantum information

The inverse eigenvalue problem for entanglement witnesses

Geometry of two-qubit states with negative conditional entropy

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