Is absolute separability determined by the partial transpose?

2015 ◽  
Vol 15 (7&8) ◽  
pp. 694-720 ◽  
Author(s):  
Srinivasan Arunachalam ◽  
Nathaniel Johnston ◽  
Vincent Russo
Keyword(s):  
Quantum States ◽  
Entanglement Witness ◽  
Unitary Matrices ◽  
Positive Partial Transpose ◽  
Entanglement Witnesses ◽  
The Family ◽  
Separability Criteria ◽  
Separability Problem ◽  
Partial Transpose

The absolute separability problem asks for a characterization of the quantum states $\rho \in M_m\otimes M_n$ with the property that $U\rho U^\dagger$ is separable for all unitary matrices $U$. We investigate whether or not it is the case that $\rho$ is absolutely separable if and only if $U\rho U^\dagger$ has positive partial transpose for all unitary matrices $U$. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer--Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.

scholarly journals Small sets of locally indistinguishable orthogonal maximally entangled states

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1098-1106
Author(s):  
Alessandro Cosentino ◽  
Vincent Russo
Keyword(s):  
Affirmative Answer ◽  
Semidefinite Program ◽  
Entangled States ◽  
Classical Communication ◽  
Fundamental Interest ◽  
Positive Partial Transpose ◽  
Partial Transpose ◽  
Maximally Entangled States ◽  
First Time

We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of $k \leq d$ orthogonal maximally entangled states in $\complex^{d}\otimes\complex^{d}$ that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012)] gives an affirmative answer for the case $k = d$. We give, for the first time, a proof that such sets of states indeed exist even in the case $k < d$. Our result is constructive and holds for an even wider class of operations known as positive-partial-transpose measurements (PPT). The proof uses the characterization of the PPT-distinguishability problem as a semidefinite program.

scholarly journals Entangling capacities and the geometry of quantum operations

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Jhih-Yuan Kao ◽  
Chung-Hsien Chou
Keyword(s):  
Quantum Information ◽  
Physical Significance ◽  
Quantum States ◽  
Important Class ◽  
Quantum Operation ◽  
Positive Partial Transpose ◽  
Quantum Operations ◽  
Partial Transpose ◽  
Ppt States

Abstract Quantum operations are the fundamental transformations on quantum states. In this work, we study the relation between entangling capacities of operations, geometry of operations, and positive partial transpose (PPT) states, which are an important class of states in quantum information. We show a method to calculate bounds for entangling capacity, the amount of entanglement that can be produced by a quantum operation, in terms of negativity, a measure of entanglement. The bounds of entangling capacity are found to be associated with how non-PPT (PPT preserving) an operation is. A length that quantifies both entangling capacity/entanglement and PPT-ness of an operation or state can be defined, establishing a geometry characterized by PPT-ness. The distance derived from the length bounds the relative entangling capability, endowing the geometry with more physical significance. We also demonstrate the equivalence of PPT-ness and separability for unitary operations.

scholarly journals Universal separability criterion for arbitrary density matrices from causal properties of separable and entangled quantum states

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Gleb A. Skorobagatko
Keyword(s):  
Quantum Systems ◽  
Many Body ◽  
Density Matrices ◽  
Positive Partial Transpose ◽  
Separability Criterion ◽  
Pair Density ◽  
Wide Range ◽  
The Family ◽  
Partial Transpose ◽  
General Physical

AbstractGeneral physical background of famous Peres–Horodecki positive partial transpose (PH- or PPT-) separability criterion is revealed. Especially, the physical sense of partial transpose operation is shown to be equivalent to what one could call as the “local causality reversal” (LCR-) procedure for all separable quantum systems or to the uncertainty in a global time arrow direction in all entangled cases. Using these universal causal considerations brand new general relations for the heuristic causal separability criterion have been proposed for arbitrary $$ D^{N} \times D^{N}$$ D N × D N density matrices acting in $$ {\mathcal {H}}_{D}^{\otimes N} $$ H D ⊗ N Hilbert spaces which describe the ensembles of N quantum systems of D eigenstates each. Resulting general formulas have been then analyzed for the widest special type of one-parametric density matrices of arbitrary dimensionality, which model a number of equivalent quantum subsystems being equally connected (EC-) with each other to arbitrary degree by means of a single entanglement parameter p. In particular, for the family of such EC-density matrices it has been found that there exists a number of N- and D-dependent separability (or entanglement) thresholds$$ p_{th}(N,D) $$ p th ( N , D ) for the values of the corresponded entanglement parameter p, which in the simplest case of a qubit-pair density matrix in $$ {\mathcal {H}}_{2} \otimes {\mathcal {H}}_{2} $$ H 2 ⊗ H 2 Hilbert space are shown to reduce to well-known results obtained earlier independently by Peres (Phys Rev Lett 77:1413–1415, 1996) and Horodecki (Phys Lett A 223(1–2):1–8, 1996). As the result, a number of remarkable features of the entanglement thresholds for EC-density matrices has been described for the first time. All novel results being obtained for the family of arbitrary EC-density matrices are shown to be applicable to a wide range of both interacting and non-interacting (at the moment of measurement) multi-partite quantum systems, such as arrays of qubits, spin chains, ensembles of quantum oscillators, strongly correlated quantum many-body systems with the possibility of many-body localization, etc.

scholarly journals Quantum states with strong positive partial transpose

2008 ◽  
Vol 77 (2) ◽  
Author(s):  
Dariusz Chruściński ◽  
Jacek Jurkowski ◽  
Andrzej Kossakowski
Keyword(s):  
Quantum States ◽  
Positive Partial Transpose ◽  
Partial Transpose

scholarly journals Characterization of Combinatorially Independent Permutation Separability Criteria

2005 ◽  
Vol 12 (04) ◽  
pp. 331-345 ◽  
Author(s):  
Paweł Wocjan ◽  
Michał Horodecki
Keyword(s):  
Density Matrix ◽  
The State ◽  
Necessary Condition ◽  
Trace Norm ◽  
Mixed States ◽  
Graphical Notation ◽  
Operational Conditions ◽  
Separability Criteria ◽  
Partial Transpose

The so-called permutation separability criteria are simple operational conditions that are necessary for separability of mixed states of multipartite systems: (1) permute the indices of the density matrix and (2) check if the trace norm of at least one of the resulting operators is greater than one. If it is greater than one then the state is necessarily entangled. A shortcoming of the permutation separability criteria is that many permutations give rise to equivalent separability criteria. Therefore, we introduce a necessary condition for two permutations to yield independent criteria called combinatorial independence. This condition basically means that the map corresponding to one permutation cannot be obtained by concatenating the map corresponding to the second permutation with a norm-preserving map. We characterize completely combinatorially independent criteria, and determine simple permutations that represent all independent criteria. The representatives can be visualized by means of a simple graphical notation. They are composed of three basic operations: partial transpose, and two types of so-called reshufflings. In particular, for a four-partite system all criteria except one are composed of partial transpose and only one type of reshuffling; the exceptional one requires the second type of reshuffling. Furthermore, we show how to obtain efficiently a simple representative for every permutation. This method allows to check easily if two permutations are combinatorially equivalent or not.

scholarly journals THE ABSOLUTE POSITIVE PARTIAL TRANSPOSE PROPERTY FOR RANDOM INDUCED STATES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250002 ◽  
Author(s):  
BENOIT COLLINS ◽  
ION NECHITA ◽  
DEPING YE
Keyword(s):  
Quantum States ◽  
Wishart Matrix ◽  
Sharp Transition ◽  
Positive Partial Transpose ◽  
Large Probability ◽  
Convergence Results ◽  
Partial Transpose ◽  
Algebraic Formula ◽  
Random Pure State ◽  
Random States

In this paper, we first obtain an algebraic formula for the moments of a centered Wishart matrix, and apply it to obtain new convergence results in the large dimension limit when both parameters of the distribution tend to infinity at different speeds. We use this result to investigate APPT (absolute positive partial transpose) quantum states. We show that the threshold for a bipartite random induced state on Cd = Cd1 ⊗ Cd2, obtained by partial tracing a random pure state on Cd ⊗ Cs, being APPT occurs if the environmental dimension s is of order s0 = min (d1, d2)3 max (d1, d2). That is, when s ≥ Cs0, such a random induced state is APPT with large probability, while such a random states is not APPT with large probability when s ≤ cs0. Besides, we compute effectively C and c and show that it is possible to replace them by the same sharp transition constant when min (d1, d2)2 ≪ d.

scholarly journals Boundary of the set of separable states

2015 ◽  
Vol 471 (2181) ◽  
pp. 20150102 ◽  
Author(s):  
Lin Chen ◽  
Dragomir Ž Ðoković
Keyword(s):  
Dimensional Simplex ◽  
Separable States ◽  
Finite Dimensional ◽  
Entanglement Witnesses ◽  
The Family ◽  
Full State ◽  
The Face ◽  
Separability Problem ◽  
Arbitrary Quantum ◽  
Finite Dimensional Hilbert Space

Motivated by the separability problem in quantum systems 2⊗4, 3⊗3 and 2⊗2⊗2, we study the maximal (proper) faces of the convex body, S 1 , of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space H = H 1 ⊗ H 2 ⊗ ⋯ ⊗ H n . To any subspace V ⊆ H , we associate a face F V of S 1 consisting of all states ρ ∈ S 1 whose range is contained in V . We prove that F V is a maximal face if and only if V is a hyperplane. If V =| ψ 〉 ⊥ , where | ψ 〉 is a product vector, we prove that Dim   F V = d 2 − 1 − ∏ ( 2 d i − 1 ) , where d i = Dim   H i and d = ∏ d i . We classify the maximal faces of S 1 in the cases 2⊗2 and 2⊗3. In particular, we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for 2⊗2, and 20 and 24 for 2⊗3. The boundary, ∂ S 1 , of S 1 is the union of all maximal faces. When d >6, it is easy to show that there exist full states on ∂ S 1 , i.e. states ρ ∈ ∂ S 1 such that all partial transposes of ρ (including ρ itself) have rank d . Ha and Kye have recently constructed explicit such states in 2⊗4 and 3⊗3. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter b >0, b ≠1. Each face in the family is a nine-dimensional simplex, and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses for these faces and analyse the three limiting cases b = 0 , 1 , ∞ .

scholarly journals Strategies for Positive Partial Transpose (PPT) States in Quantum Metrologies with Noise

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 685
Author(s):  
Arunava Majumder ◽  
Harshank Shrotriya ◽  
Leong-Chuan Kwek
Keyword(s):  
Quantum States ◽  
Entangled States ◽  
Heisenberg Uncertainty Principle ◽  
Positive Partial Transpose ◽  
Partial Transpose ◽  
Heisenberg Uncertainty ◽  
Noise Limit ◽  
Entangled Quantum States ◽  
Standard Precision ◽  
Shot Noise Limit

Quantum metrology overcomes standard precision limits and has the potential to play a key role in quantum sensing. Quantum mechanics, through the Heisenberg uncertainty principle, imposes limits on the precision of measurements. Conventional bounds to the measurement precision such as the shot noise limit are not as fundamental as the Heisenberg limits, and can be beaten with quantum strategies that employ `quantum tricks’ such as squeezing and entanglement. Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered to be too weakly entangled for applications. Since no pure entanglement can be distilled from them, they are also called bound entangled states. We provide strategies, using which multipartite quantum states that have a positive partial transpose with respect to all bi-partitions of the particles can still outperform separable states in linear interferometers.

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.

Sign in / Sign up

Export Citation Format

Share Document