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 PARTIAL TRANSPOSITION OF RANDOM STATES AND NON-CENTERED SEMICIRCULAR DISTRIBUTIONS

2012 ◽  
Vol 01 (02) ◽  
pp. 1250001 ◽  
Author(s):  
GUILLAUME AUBRUN
Keyword(s):  
Wishart Distribution ◽  
Quantum Information Theory ◽  
Positive Partial Transpose ◽  
Partial Transposition ◽  
Moments Method ◽  
The Matrix ◽  
Partial Transpose ◽  
Random Pure State ◽  
Random States ◽  
Random State

Let W be a Wishart random matrix of size d2 × d2, considered as a block matrix with d × d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d → ∞. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "When does a random state have a positive partial transpose (PPT)?". We answer this question and exhibit a strong threshold when the parameter from the Wishart distribution equals 4. When d gets large, a random state on Cd ⊗ Cd obtained after partial tracing a random pure state over some ancilla of dimension αd2 is typically PPT when α > 4 and typically non-PPT when α < 4.

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 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 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.

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.

ENTANGLEMENT OF CONVEX LINEAR COMBINATION AND CONSTRUCTION OF PPT ENTANGLED STATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350002 ◽  
Author(s):  
WEI CHENG ◽  
FANG XU ◽  
HUA LI ◽  
GANG WANG
Keyword(s):  
Linear Combination ◽  
Quantum States ◽  
Entangled States ◽  
Positive Partial Transpose ◽  
Partial Transpose ◽  
Convex Linear Combination ◽  
Ppt Criterion

Given two bipartite quantum states and the convex linear combination of them, we discuss the relation between the entanglement of the convex linear combination state and the entanglement of states being combined. This is achieved by characterizing quantum states quantitatively via the positive partial transpose (PPT) criterion and the computable cross-norm or realignment (CCNR) criterion. Inspired by the Horodecki's 3 ⊗ 3 quantum states, we also give explicit examples to illustrate all possible cases of convex linear combination. Finally, as an application of this method, we show how to construct new bipartite PPT entangled states from known PPT entangled states by convex linear combination.

Sign in / Sign up

Export Citation Format

Share Document