Fortran Code for Generating Random Probability Vectors, Unitaries, and Quantum States

2017

Quantum Inf Process

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Using Bloch's parametrization for qudits (d-level quantum systems), we write the Hilbert-Schmidt distance (HSD) between two generic n-qudit states as an Euclidean distance between two vectors of observables mean values in R Π n s=1 d 2 s −1 , where ds is the dimension for qudit s. Then, applying the generalized Gell Mann's matrices to generate SU (ds), we use that result to obtain the HilbertSchmidt quantum coherence (HSC) of n-qudit systems. As examples, we consider in details onequbit, one-qutrit, two-qubit, and two copies of one-qubit states. In this last case, the possibility for controlling local and non-local coherences by tuning local populations is studied and the contrasting behaviors of HSC, l1-norm coherence, and relative entropy of coherence in this regard are noticed. We also investigate the decoherent dynamics of these coherence functions under the action of qutrit dephasing and dissipation channels. At last, we analyze the non-monotonicity of HSD under tensor products and report the first instance of a consequence (for coherence quantification) of this kind of property of a quantum distance measure.

Section: Discussionmentioning

confidence: 99%

“…Figure3: (color online) Percentage of the random quartets of states leading to the non-monotonic behavior under tensor products of the Hilbert-Schmidt distance. Using the standard method from the library described in Ref [62]. (the associated Fortran code can be accessed freely in[63]), we generated one million quartets of states for each dimension d.…”

mentioning

confidence: 99%

Hilbert–Schmidt quantum coherence in multi-qudit systems

2017

Quantum Inf Process

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“…By generating some random density matrices [45], we checked that the expressions and the corresponding code for the unoptimized and optimized versions of a, b, and C agree. Additional tests shall be presented in the next section, where we calculate some quantum discord quantifiers.…”

Section: Computing Coherence Vectors and Correlation Matricesmentioning

confidence: 99%

Computing Coherence Vectors and Correlation Matrices with Application to Quantum Discord Quantification

Advances in Mathematical Physics

2016

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Coherence vectors and correlation matrices are important functions frequently used in physics. The numerical calculation of these functions directly from their definitions, which involves Kronecker products and matrix multiplications, may seem to be a reasonable option. Notwithstanding, as we demonstrate in this article, some algebraic manipulations before programming can reduce considerably their computational complexity. Besides, we provide Fortran code to generate generalized Gell Mann matrices and to compute the optimized and unoptimized versions of the associated Bloch's vectors and correlation matrix, in the case of bipartite quantum systems. As a code test and application example, we consider the calculation of Hilbert-Schmidt quantum discords.

“…In spite of the physical interpretation of the partial trace not being a trivial matter [36,37], it's mathematical and operational meaning is well established [32][33][34][35]. Besides, the PTr appears very frequently, for instance, in the context of correlation quantifiers (mutual information [38,39], quantum entanglement [40][41][42][43], quantum discord [44][45][46], etc), in the generation of random density matrices [47,48], and in investigations regarding phase transitions [49,50]. It is also a fundamental ingredient in the quantum marginal and extension problems [51][52][53][54][55][56], for the strong subadditivity property of von Neumann entropy and related results [57][58][59], and in the theories of quantum measurement and decoherence [60][61][62][63].…”

Section: Introductionmentioning

confidence: 99%

Computing partial traces and reduced density matrices

2017

Int. J. Mod. Phys. C

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Taking partial traces for computing reduced density matrices, or related functions, is a ubiquitous procedure in the quantum mechanics of composite systems. In this article, we present a thorough description of this function and analyze the number of elementary operations (ops) needed, under some possible alternative implementations, to compute it on a classical computer. As we notice, it is worthwhile doing some analytical developments in order to avoid making null multiplications and sums, what can considerably reduce the ops. For instance, for a bipartite system Ha ⊗ H b with dimensions da = dim Ha and d b = dim H b and for da, d b ≫ 1, while a direct use of partial trace definition applied toIn the sequence, we regard the computation of partial traces for general multipartite systems and describe Fortran code provided to implement it numerically. We also consider the calculation of reduced density matrices via Bloch's parametrization with generalized Gell Mann's matrices.