Singular Value Decomposition and Matrix Reorderings in Quantum Information Theory

Miszczak, Jarosław Adam

doi:10.1142/s0129183111016683

Cited by 47 publications

(53 citation statements)

References 24 publications

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“…and it can be calculated efficiently with Schmidt decomposition [70]. In the ergodic phase the entanglement entropy of a typical Hamiltonian eigenstate grows with the size of the subsystem A -a volume law scaling.…”

Section: A Bipartite Entanglement and Number Uncertaintymentioning

confidence: 99%

Probing the many-body localization phase transition with superconducting circuits

et al. 2019

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Chains of superconducting circuit devices provide a natural platform for studies of synthetic bosonic quantum matter. Motivated by the recent experimental progress in realizing disordered and interacting chains of superconducting transmon devices, we study the bosonic many-body localization phase transition using the methods of exact diagonalization as well as matrix product state dynamics. We estimate the location of transition separating the ergodic and the many-body localized phases as a function of the disorder strength and the many-body on-site interaction strength. The main difference between the bosonic model realized by superconducting circuits and similar fermionic model is that the effect of the on-site interaction is stronger due to the possibility of multiple excitations occupying the same site. The phase transition is found to be robust upon including longer-range hopping and interaction terms present in the experiments. Furthermore, we calculate experimentally relevant local observables and show that their temporal fluctuations can be used to distinguish between the dynamics of Anderson insulator, many-body localization, and delocalized phases. While we consider unitary dynamics, neglecting the effects of dissipation, decoherence and measurement back action, the timescales on which the dynamics is unitary are sufficient for observation of characteristic dynamics in the many-body localized phase. Moreover, the experimentally available disorder strength and interactions allow for tuning the many-body localization phase transition, thus making the arrays of superconducting circuit devices a promising platform for exploring localization physics and phase transition.

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Section: A Bipartite Entanglement and Number Uncertaintymentioning

confidence: 99%

Probing the many-body localization phase transition with superconducting circuits

et al. 2019

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“…Then, the Choi-Jamiołkowski state  is obtained from equation (25). Also, we can use reshaping operation [52,53], which is defined as, given an m × m matrix = A a [ ] ij with elements a ij ,…”

Section: Appendix B Simulation Example For Qutrit Channelsmentioning

confidence: 99%

Quantum circuit design for accurate simulation of qudit channels

Wang

Sanders

2015

New J. Phys.

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We construct a classical algorithm that designs quantum circuits for algorithmic quantum simulation of arbitrary qudit channels on fault-tolerant quantum computers within a pre-specified error tolerance with respect to diamond-norm distance. The classical algorithm is constructed by decomposing a quantum channel into a convex combination of generalized extreme channels by convex optimization of a set of nonlinear coupled algebraïc equations. The resultant circuit is a randomly chosen generalized extreme channel circuit whose run-time is logarithmic with respect to the error tolerance and quadratic with respect to Hilbert space dimension, which requires only a single ancillary qudit plus classical dits.

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“…2 the second-order expansion found using Eqs. (45) and (46) about an assortment of time points to show that our equations capture the negativity dynamics even in regions where negativity is constant, in intervals when ρ T B is positive semi-definite, and in the presence of bound entanglement. One may also use our method to analyze how negativity changes with respect to the system parameters ∆ and g in order to explore how entanglement in the JCM is sensitive to the entire parameter landscape.…”

Section: Negativity Growth In Various Systemsmentioning

confidence: 99%

“…A we show how (17) can be obtained through the action of the partial transposition map on the standard basis, and we also present a convenient form for K B by identifying its eigenvectors. For other representations of the transposition and partial transposition maps, see, for example, [46]. With this brief introduction to aspects of algebra on vectorized matrices, we now turn to calculus in order to identify ∂ ρ T B 1 /∂ vec T ρ T B , the remaining factor in (7).…”

Section: Perturbative Expansion Of Negativitymentioning

confidence: 99%

Perturbative expansion of entanglement negativity using patterned matrix calculus

2019

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Negativity is an entanglement monotone frequently used to quantify entanglement in bipartite states. Because negativity is a non-analytic function of a density matrix, existing methods used in the physics literature are insufficient to compute its derivatives. To this end we develop techniques in the calculus of complex, patterned matrices and use them to conduct a perturbative analysis of negativity in terms of arbitrary variations of the density operator. The result is an easy-to-implement expansion that can be carried out to all orders. On the way we provide convenient representations of the partial transposition map appearing in the definition of negativity. Our methods are well-suited to study the growth and decay of entanglement in a wide range of physical systems, including the generic linear growth of entanglement in many-body systems, and have broad relevance to many functions of quantum states and observables.1 By most definitions, entanglement measures ought to reduce to the entanglement entropy [17,24] in the pure state case; negativity does not. Nevertheless, we alternate between calling it a monotone and measure depending on the subsystem dimensions.arXiv:1809.07772v2 [quant-ph]

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Singular Value Decomposition and Matrix Reorderings in Quantum Information Theory

Cited by 47 publications

References 24 publications

Probing the many-body localization phase transition with superconducting circuits

Probing the many-body localization phase transition with superconducting circuits

Quantum circuit design for accurate simulation of qudit channels

Perturbative expansion of entanglement negativity using patterned matrix calculus

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