scholarly journals Partial traces and the geometry of entanglement: Sufficient conditions for the separability of Gaussian states

2021 ◽  
Author(s):  
Nuno Costa Dias ◽  
Maurice de Gosson ◽  
João Nuno Prata
Keyword(s):  
Density Operator ◽  
Sufficient Conditions ◽  
Quantum States ◽  
Partial Trace ◽  
Gaussian States ◽  
Geometrical Properties ◽  
Partial Transposition

The notion of partial trace of a density operator is essential for the understanding of the entanglement and separability properties of quantum states. In this paper, we investigate these notions putting an emphasis on the geometrical properties of the covariance ellipsoids of the reduced states. We thereafter focus on Gaussian states and we give new and easily numerically implementable sufficient conditions for the separability of all Gaussian states. Unlike the positive partial transposition criterion, none of these conditions is however necessary.

scholarly journals Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

2019 ◽  
Vol 35 ◽  
pp. 156-180 ◽  
Author(s):  
Nathaniel Johnston ◽  
Olivia MacLean
Keyword(s):  
Sufficient Conditions ◽  
Positive Semidefinite ◽  
Quantum States ◽  
Completely Positive ◽  
Positive Partial Transpose ◽  
Completely Positive Matrices ◽  
Partial Transposition ◽  
Partial Transpose ◽  
Positive Matrices ◽  
Necessary And Sufficient

A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable.

Compatibility between local and multipartite states

2004 ◽  
Vol 4 (1) ◽  
pp. 12-26
Author(s):  
S. Bravyi
Keyword(s):  
Hilbert Spaces ◽  
Pure State ◽  
Local Density ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Quantum States ◽  
Density Matrices ◽  
Partial Trace ◽  
Field State ◽  
Necessary And Sufficient

We consider a partial trace transformation which maps a multipartite quantum state to collection of local density matrices. We call this collection a mean field state. For the Hilbert spaces $(\CC^2)^{\otimes n}$ and $\CC^2\otimes\CC^2\otimes\CC^4$ the necessary and sufficient conditions under which a mean field state is compatible with at least one multipartite pure state are found. Compatibility of mean field states with more general classes of multipartite quantum states is discussed.

Minimum guesswork discrimination between quantum states

2015 ◽  
Vol 15 (9&10) ◽  
pp. 737-758
Author(s):  
Weien Chen ◽  
Yongzhi Cao ◽  
Hanpin Wang ◽  
Yuan Feng
Keyword(s):  
Error Probability ◽  
Sufficient Conditions ◽  
Optimization Criterion ◽  
Quantum States ◽  
Actual State ◽  
Information Theoretic ◽  
Minimum Error ◽  
State Discrimination ◽  
Semidefinite Programming Problem ◽  
Necessary And Sufficient

Error probability is a popular and well-studied optimization criterion in discriminating non-orthogonal quantum states. It captures the threat from an adversary who can only query the actual state once. However, when the adversary is able to use a brute-force strategy to query the state, discrimination measurement with minimum error probability does not necessarily minimize the number of queries to get the actual state. In light of this, we take Massey's guesswork as the underlying optimization criterion and study the problem of minimum guesswork discrimination. We show that this problem can be reduced to a semidefinite programming problem. Necessary and sufficient conditions when a measurement achieves minimum guesswork are presented. We also reveal the relation between minimum guesswork and minimum error probability. We show that the two criteria generally disagree with each other, except for the special case with two states. Both upper and lower information-theoretic bounds on minimum guesswork are given. For geometrically uniform quantum states, we provide sufficient conditions when a measurement achieves minimum guesswork. Moreover, we give the necessary and sufficient condition under which making no measurement at all would be the optimal strategy.

scholarly journals Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States

2020 ◽  
Vol 10 (16) ◽  
pp. 5475
Author(s):  
Yukito Mototake ◽  
Jun Suzuki
Keyword(s):  
Coherent State ◽  
Quantum System ◽  
Robust Statistics ◽  
Quantum States ◽  
Phase Estimation ◽  
Gaussian States ◽  
Statistical Framework

In this paper, we investigate the problem of estimating the phase of a coherent state in the presence of unavoidable noisy quantum states. These unwarranted quantum states are represented by outlier quantum states in this study. We first present a statistical framework of robust statistics in a quantum system to handle outlier quantum states. We then apply the method of M-estimators to suppress untrusted measurement outcomes due to outlier quantum states. Our proposal has the advantage over the classical methods in being systematic, easy to implement, and robust against occurrence of noisy states.

Entanglement conditions for four-mode states via the uncertainty relations in conjunction with partial transposition

Open Physics ◽  
2008 ◽  
Vol 6 (2) ◽  
Author(s):  
Lijun Song ◽  
Xiaoguang Wang ◽  
Dong Yan ◽  
Yongda Li
Keyword(s):  
Uncertainty Relation ◽  
Sufficient Conditions ◽  
Entangled States ◽  
Uncertainty Relations ◽  
Heisenberg Uncertainty Relation ◽  
Partial Transposition ◽  
Heisenberg Uncertainty ◽  
Entanglement Conditions

AbstractFrom the Heisenberg uncertainty relation in conjunction with partial transposition, we derive a class of inequalities for detecting entanglements in four-mode states. The sufficient conditions for bipartite entangled states are presented. We also discuss the generalization of the entanglement conditions via the Schrödinger-Robertson indeterminacy relation, which are in general stronger than those based on the Heisenberg uncertainty relation.

scholarly journals ENTANGLEMENT CRITERION FOR COHERENT SUBTRACTION AND COHERENT ADDITION OF BIPARTITE CONTINUOUS VARIABLE STATES

2013 ◽  
Vol 11 (06) ◽  
pp. 1350060
Author(s):  
LI-ZHEN JIANG ◽  
XIAO-YU CHEN ◽  
TIAN-YU YE ◽  
FANG-YU HONG ◽  
LIANG-NENG WU
Keyword(s):  
Thermal Noise ◽  
Photon Number ◽  
Sufficient Conditions ◽  
Continuous Variable ◽  
Entangled State ◽  
Gaussian States ◽  
Non Gaussian ◽  
Entanglement Criterion ◽  
Entanglement Conditions ◽  
Thermal States

We study the entanglement conditions of two quite general classes of two-mode non-Gaussian states. Using computable cross norm and realignment criterion, we obtain the sufficient conditions of entanglement for coherently added or subtracted two-mode squeezed thermal states, and the sufficient condition of entanglement for any photon number entangled state (PNES) evolving in a thermal noise and amplitude damping channel.

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