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 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 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.

scholarly journals Nongeneric positive partial transpose states of rank five in 3×3 dimensions

2017 ◽  
Vol 15 (07) ◽  
pp. 1750054
Author(s):  
Leif Ove Hansen ◽  
Jan Myrheim
Keyword(s):  
Special Method ◽  
Positive Partial Transpose ◽  
Analytical Construction ◽  
Partial Transposition ◽  
Boundary States ◽  
The Face ◽  
State Formula ◽  
Partial Transpose ◽  
Ppt States ◽  
Symmetric States

In [Formula: see text] dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. These rank four states are completely understood. We say that they have rank [Formula: see text] since both a state [Formula: see text] and its partial transpose [Formula: see text] have rank four. The next problem is to understand the extremal PPT states of rank [Formula: see text]. We call two states [Formula: see text]-equivalent if they are related by a product transformation. A generic rank [Formula: see text] PPT state [Formula: see text] is extremal, and both [Formula: see text] and [Formula: see text] have six product vectors in their ranges, and no product vectors in their kernels. The three numbers [Formula: see text] are [Formula: see text]-invariants that help us classify the state. There is no analytical understanding of such states. We have studied numerically a few types of nongeneric rank five PPT states, in particular, states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to [Formula: see text]-equivalence, where they appear as boundary states on one single five-dimensional face on the set of normalized PPT states. The interior of the face consists of rank [Formula: see text] states with four common product vectors in their kernels, it is a simplex of separable states surrounded by entangled PPT states. We say that a state [Formula: see text] is [Formula: see text]-symmetric if [Formula: see text] and [Formula: see text] are [Formula: see text]-equivalent, and is genuinely [Formula: see text]-symmetric if it is [Formula: see text]-equivalent to a state [Formula: see text] with [Formula: see text]. Genuine [Formula: see text]-symmetry implies a special form of [Formula: see text]-symmetry. We have produced numerically, by a special method, a random sample of rank [Formula: see text] [Formula: see text]-symmetric states. About 50 of these are of type [Formula: see text], among those all are extremal and about half are genuinely [Formula: see text]-symmetric. All these genuinely [Formula: see text]-symmetric states can be transformed to have a circulant form. We find however that this is not a generic property of genuinely [Formula: see text]-symmetric states. The remaining [Formula: see text]-symmetric states found in the search have product vectors in their kernels, and they inspired us to study such states without regard to [Formula: see text]-symmetry.

scholarly journals Postponing Distillability Sudden Death in a Correlated Dephasing Channel

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 827
Author(s):  
Guanghao Xue ◽  
Liang Qiu
Keyword(s):  
Sudden Death ◽  
Quantum Channel ◽  
Positive Partial Transpose ◽  
Partial Correlations ◽  
Partial Transpose ◽  
Correlated Channel

We investigated the dynamics of a two-qutrit system in a correlated quantum channel. The partial correlations between consecutive actions of the channel can effectively postpone the phenomenon of distillability sudden death (DSD) and broaden the range of the time cutoff that indicates entanglement of the positive partial transpose states. Particularly, the negativity of the system will revive and DSD will disappear in the fully correlated channel.

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