scholarly journals The rigorous stochastic matrix multiplication scheme for the calculations of reduced equilibrium density matrices of open multilevel quantum systems

2014 ◽  
Vol 140 (15) ◽  
pp. 154101 ◽  
Author(s):  
Xin Chen
Keyword(s):  
Stochastic Matrix ◽  
Matrix Multiplication ◽  
Quantum Systems ◽  
Equilibrium Density ◽  
Density Matrices

scholarly journals RIGOROUS RESULTS IN NON-EXTENSIVE THERMODYNAMICS

2000 ◽  
Vol 12 (10) ◽  
pp. 1305-1324 ◽  
Author(s):  
JAN NAUDTS
Keyword(s):  
Density Matrix ◽  
Degrees Of Freedom ◽  
Energy Levels ◽  
High Energy ◽  
Quantum Systems ◽  
Equilibrium Density ◽  
Unique Equilibrium ◽  
Density Matrices ◽  
Rigorous Results ◽  
Entropic Index

This paper studies quantum systems with a finite number of degrees of freedom in the context of non-extensive thermodynamics. A trial density matrix, obtained by heuristic methods, is proved to be the equilibrium density matrix. If the entropic index q is larger than 1 then existence of the trial equilibrium density matrix requires that q is less than some critical value qc which depends on the rate by which the eigenvalues of the Hamiltonian diverge. Existence of a unique equilibrium density matrix is proved if in addition q < 2 holds. For q between 0 and 1, such that 2 < q + qc, the free energy has at least one minimum in the set of trial density matrices. If a unique equilibrium density matrix exists then it is necessarily one of the trial density matrices. Note that this is a finite rank operator, which means that in equilibrium high energy levels have zero probability of occupancy.

CANONICAL FORM AND SEPARABILITY OF PPT STATES IN ${\mathcal C}^2\otimes {\mathcal C}^M\otimes {\mathcal C}^N$ COMPOSITE QUANTUM SYSTEMS

2003 ◽  
Vol 01 (03) ◽  
pp. 337-347
Author(s):  
XIAO-HONG WANG ◽  
SHAO-MING FEI ◽  
ZHI-XI WANG ◽  
KE WU
Keyword(s):  
Canonical Form ◽  
General Expression ◽  
Quantum Systems ◽  
Canonical Forms ◽  
Density Matrices ◽  
Partial Transposition ◽  
Separability Condition ◽  
Ppt States

We investigate the canonical forms of positive partial transposition (PPT) density matrices in [Formula: see text] composite quantum systems with rank N. A general expression for these PPT states are explicitly obtained. From this canonical form a sufficient separability condition is presented.

scholarly journals Separable approximations of density matrices of composite quantum systems

2001 ◽  
Vol 34 (35) ◽  
pp. 6919-6937 ◽  
Author(s):  
Sinisa Karnas ◽  
Maciej Lewenstein
Keyword(s):  
Quantum Systems ◽  
Density Matrices

scholarly journals Multiplication of medium-density matrices using TensorFlow on multicore CPUs

2019 ◽  
Vol 13 (4) ◽  
pp. 286-290
Author(s):  
Siraphob Theeracheep ◽  
Jaruloj Chongstitvatana
Keyword(s):  
Machine Learning ◽  
Image Processing ◽  
Linear Algebra ◽  
Matrix Multiplication ◽  
Medium Density ◽  
Density Matrices ◽  
Input Matrix ◽  
Programming Paradigm ◽  
Dataflow Programming ◽  
Multicore Cpus

Matrix multiplication is an essential part of many applications, such as linear algebra, image processing and machine learning. One platform used in such applications is TensorFlow, which is a machine learning library whose structure is based on dataflow programming paradigm. In this work, a method for multiplication of medium-density matrices on multicore CPUs using TensorFlow platform is proposed. This method, called tbt_matmul, utilizes TensorFlow built-in methods tf.matmul and tf.sparse_matmul. By partitioning each input matrix into four smaller sub-matrices, called tiles, and applying an appropriate multiplication method to each pair depending on their density, the proposed method outperforms the built-in methods for matrices of medium density and matrices of significantly uneven distribution of non-zeros.

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 Gaussian resolutions for equilibrium density matrices

2003 ◽  
Vol 381 (1-2) ◽  
pp. 117-122 ◽  
Author(s):  
Pavel Frantsuzov ◽  
Arnold Neumaier ◽  
Vladimir A. Mandelshtam
Keyword(s):  
Equilibrium Density ◽  
Density Matrices

Parametrizing Density Matrices for Composite Quantum Systems

2008 ◽  
Vol 15 (04) ◽  
pp. 397-408 ◽  
Author(s):  
Erwin Brüning ◽  
Dariusz Chruściński ◽  
Francesco Petruccione
Keyword(s):  
Unitary Group ◽  
Quantum Systems ◽  
Density Matrices ◽  
Density Operators ◽  
Quantum Properties ◽  
Bipartite Quantum Systems

A parametrization of density operators for bipartite quantum systems is proposed. It is based on the particular parametrization of the unitary group found recently by Jarlskog. It is expected that this parametrization will find interesting applications in the study of quantum properties of multipartite systems.

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