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  • AIP Publishing  (17)
  • Englisch  (17)
  • 1
    Online-Ressource
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    Generating random density matrices
    AIP Publishing ; 2011
    In:  Journal of Mathematical Physics Vol. 52, No. 6 ( 2011-06-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 52, No. 6 ( 2011-06-01)
    Kurzfassung: We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/1.3595693
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2011
    ZDB Id: 1472481-9
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  • 2
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    Application of Shemesh theorem to quantum channels
    AIP Publishing ; 2018
    In:  Journal of Mathematical Physics Vol. 59, No. 10 ( 2018-10-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 59, No. 10 ( 2018-10-01)
    Kurzfassung: Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/1.5027616
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2018
    ZDB Id: 1472481-9
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  • 3
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    Cones of positive maps and their duality relations
    AIP Publishing ; 2009
    In:  Journal of Mathematical Physics Vol. 50, No. 6 ( 2009-06-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 50, No. 6 ( 2009-06-01)
    Kurzfassung: The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable, and k-entangled operators due to the Jamiołkowski–Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/1.3155378
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2009
    ZDB Id: 1472481-9
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  • 4
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    Generating random quantum channels
    AIP Publishing ; 2021
    In:  Journal of Mathematical Physics Vol. 62, No. 6 ( 2021-06-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 62, No. 6 ( 2021-06-01)
    Kurzfassung: Several techniques of generating random quantum channels, which act on the set of d-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show that they are mathematically equivalent. We discuss under which conditions they give the uniform Lebesgue measure on the convex set of quantum operations and compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute the mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity, and the 2-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed, and their spectral properties are studied using the Bloch representation of a classical probability vector.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/5.0038838
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2021
    ZDB Id: 1472481-9
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  • 5
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    Log-convex set of Lindblad semigroups acting on N -level system
    AIP Publishing ; 2021
    In:  Journal of Mathematical Physics Vol. 62, No. 7 ( 2021-07-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 62, No. 7 ( 2021-07-01)
    Kurzfassung: We analyze the set ANQ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an N-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set ANQ is shown to be log-convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that for mixed Weyl channels, the super-decoherence commutes with the dynamics so that decohering a quantum accessible channel, we obtain a bistochastic matrix from the set ANC of classical maps accessible by a semigroup. Focusing on three-level systems, we investigate the geometry of the sets of quantum accessible maps, its classical counterpart, and the support of their spectra. We demonstrate that the set A3Q is not included in the set U3Q of quantum unistochastic channels, although an analogous relation holds for N = 2. The set of transition matrices obtained by super-decoherence of unistochastic channels of order N ≥ 3 is shown to be larger than the set of unistochastic matrices of this order and yields a motivation to introduce the larger sets of k-unistochastic matrices.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/5.0009745
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2021
    ZDB Id: 1472481-9
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  • 6
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    Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant
    AIP Publishing ; 2009
    In:  Journal of Mathematical Physics Vol. 50, No. 12 ( 2009-12-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 50, No. 12 ( 2009-12-01)
    Kurzfassung: A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that Bij=|Uij|2 for i,j=1,…,N. The set U3 of all unistochastic matrices of order N=3 forms a proper subset of the Birkhoff polytope, which contains all bistochastic (doubly stochastic) matrices. We compute the volume of the set U3 with respect to the flat (Lebesgue) measure and analytically evaluate the mean entropy of an unistochastic matrix of this order. We also analyze the Jarlskog invariant J, defined for any unitary matrix of order three, and derive its probability distribution for the ensemble of matrices distributed with respect to the Haar measure on U(3) and for the ensemble which generates the flat measure on the set of unistochastic matrices. For both measures the probability of finding |J| smaller than the value observed for the Cabbibo–Kobayashi–Maskawa matrix, which describes the violation of the CP parity, is shown to be small. Similar statistical reasoning may also be applied to the Maki–Nakagawa–Sakata matrix, which plays role in describing the neutrino oscillations. Some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/1.3272543
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2009
    ZDB Id: 1472481-9
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  • 7
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    Quantum chaos: An entropy approach
    AIP Publishing ; 1994
    In:  Journal of Mathematical Physics Vol. 35, No. 11 ( 1994-11-01), p. 5674-5700
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 35, No. 11 ( 1994-11-01), p. 5674-5700
    Kurzfassung: A new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov–Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent states entropy, is defined. It may be used to measure chaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent states entropy of a quantum map as ℏ→0 is less than or equal to the KS-entropy of the corresponding classical map. ‘‘Chaos umpire sits, And by decision more imbroils the fray By which he reigns: next him high arbiter Chance governs all.’’ John Milton, Paradise Lost, Book II
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/1.530704
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 1994
    ZDB Id: 1472481-9
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  • 8
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    Algebraic and geometric structures inside the Birkhoff polytope
    AIP Publishing ; 2022
    In:  Journal of Mathematical Physics Vol. 63, No. 1 ( 2022-01-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 63, No. 1 ( 2022-01-01)
    Kurzfassung: The Birkhoff polytope Bd consisting of all bistochastic matrices of order d assists researchers from many areas, including combinatorics, statistical physics, and quantum information. Its subset Ud of unistochastic matrices, determined by squared moduli of unitary matrices, is of particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantized. In order to investigate the problem of unistochasticity, we introduce the set Ld of bracelet matrices that forms a subset of Bd, but a superset of Ud. We prove that for every dimension d, this set contains the set of factorizable bistochastic matrices Fd and is closed under matrix multiplication by elements of Fd. Moreover, we prove that both Ld and Fd are star-shaped with respect to the flat matrix. We also analyze the set of d × d unistochastic matrices arising from circulant unitary matrices and show that their spectra lie inside d-hypocycloids on the complex plane. Finally, applying our results to small dimensions, we fully characterize the set of circulant unistochastic matrices of order d ≤ 4 and prove that such matrices form a monoid for d = 3.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/5.0046581
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2022
    ZDB Id: 1472481-9
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  • 9
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    Coarse-grained entanglement classification through orthogonal arrays
    AIP Publishing ; 2018
    In:  Journal of Mathematical Physics Vol. 59, No. 7 ( 2018-07-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 59, No. 7 ( 2018-07-01)
    Kurzfassung: Classification of entanglement in multipartite quantum systems is an open problem solved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of N subsystems with an arbitrary number of internal levels each, based on the properties of orthogonal arrays with N columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four and five qubits, as well as heterogeneous tripartite systems consisting of two qubits and one qutrit or one qubit and two qutrits.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/1.5006890
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2018
    ZDB Id: 1472481-9
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  • 10
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    Asymptotic entropic uncertainty relations
    AIP Publishing ; 2016
    In:  Journal of Mathematical Physics Vol. 57, No. 3 ( 2016-03-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 57, No. 3 ( 2016-03-01)
    Kurzfassung: We analyze entropic uncertainty relations for two orthogonal measurements on a N-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than those obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations, which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of independent interest.
    Materialart: Online-Ressource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/1.4944425
    RVK:
    UA 4660
    Sprache: Englisch
    Verlag: AIP Publishing
    Publikationsdatum: 2016
    ZDB Id: 1472481-9
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