Phase transitions for random states and a semicircle law for the partial transpose

Guillaume Aubrun; Stanisław J. Szarek; Deping Ye

doi:10.1103/physreva.85.030302

THE ABSOLUTE POSITIVE PARTIAL TRANSPOSE PROPERTY FOR RANDOM INDUCED STATES

Random Matrices Theory and Application ◽

10.1142/s2010326312500025 ◽

2012 ◽

Vol 01 (03) ◽

pp. 1250002 ◽

Cited By ~ 10

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.

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PARTIAL TRANSPOSITION OF RANDOM STATES AND NON-CENTERED SEMICIRCULAR DISTRIBUTIONS

Random Matrices Theory and Application ◽

10.1142/s2010326312500013 ◽

2012 ◽

Vol 01 (02) ◽

pp. 1250001 ◽

Cited By ~ 20

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.

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Electron Microscopy of Graphite Intercalation Compounds

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100055564 ◽

1982 ◽

Vol 40 ◽

pp. 544-545

Author(s):

G. Timp ◽

L. Salamanca-Riba ◽

L.W. Hobbs ◽

G. Dresselhaus ◽

M.S. Dresselhaus

Keyword(s):

Electron Microscopy ◽

Phase Transitions ◽

Domain Wall ◽

Intercalation Compounds ◽

Lattice Plane ◽

Wall Model ◽

Graphite Intercalation Compounds ◽

Fundamental Symmetry ◽

Graphite Intercalation

Electron microscopy can be used to study structures and phase transitions occurring in graphite intercalations compounds. The fundamental symmetry in graphite intercalation compounds is the staging periodicity whereby each intercalate layer is separated by n graphite layers, n denoting the stage index. The currently accepted model for intercalation proposed by Herold and Daumas assumes that the sample contains equal amounts of intercalant between any two graphite layers and staged regions are confined to domains. Specifically, in a stage 2 compound, the Herold-Daumas domain wall model predicts a pleated lattice plane structure.

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Multiple phase transitions investigated by high-speed TEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100175880 ◽

1990 ◽

Vol 48 (4) ◽

pp. 550-551

Author(s):

Oleg Bostanjoglo ◽

Peter Thomsen-Schmidt

Keyword(s):

Phase Transitions ◽

High Speed ◽

Short Exposure ◽

Semiconductor Film ◽

Time Resolved ◽

Short Exposure Time ◽

Multiple Phase ◽

Model Substance ◽

History Of ◽

Electron Image

Thin GexTe1-x (x = 0.15-0.8) were studied as a model substance of a composite semiconductor film, in addition being of interest for optical storage material. Two complementary modes of time-resolved TEM were used to trace the phase transitions, induced by an attached Q-switched (50 ns FWHM) and frequency doubled (532 nm) Nd:YAG laser. The laser radiation was focused onto the specimen within the TEM to a 20 μm spot (FWHM). Discrete intermediate states were visualized by short-exposure time doubleframe imaging /1,2/. The full history of a transformation was gained by tracking the electron image intensity with photomultiplier and storage oscilloscopes (space/time resolution 100 nm/3 ns) /3/. In order to avoid radiation damage by the probing electron beam to detector and specimen, the beam is pulsed in this continuous mode of time-resolved TEM,too.Short events ( <2 μs) are followed by illuminating with an extended single electron pulse (fig. 1c)

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