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Cited by 74Quantum Information and CryptographyQuantum Mechanics and ApplicationsQuantum many-body systems

Tensor products of convex sets and the volume of separable states onNqudits

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Aubrun, Guillaume, Szarek, Stanisław J.·2006·Physical Review A
DOI 10.1103/physreva.73.022109
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Abstract

This paper deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is superexponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound ${6}^{\ensuremath{-}N∕2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on $N$ qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces. One relatively nonstandard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept.

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  • In the case where N/k = s ∈ N, a speed-up can be obtained for pure states using the time-independent Hamiltonian H = √s ∑_{μ=1}^s h_μ, with h_μ = |1⟩^{⊗k}⟨d|^{⊗k} + h.c., assuming each h_μ acts on a completely different set of k batteries.

    “In this particular case, such a speed-up can be obtained for pure states using the time-independent Hamiltonian H = √ s ∑ s µ =1 h µ , with h µ = | 1 〉 ⊗ k 〈 d | ⊗ k + h.c. , assuming that each h µ acts on a completely different set of k batteries, i.e. , [ h µ , h µ ′ ] = 0 ∀ µ, µ ′ .”

    From Enhancing the Charging Power of Quantum Batteries (2017) · cites this paper as [36]

  • Γ_C1 = √k and Γ_C2 = k are achievable if the total number of batteries N can be divided by k, i.e., if N/k = s ∈ N.

    “For comparison, it has been shown elsewhere that Γ C1 = √ k and Γ C2 = k are achievable if the total number of batteries N can be divided by k , i.e. , if N/k = s ∈ N [15, 36].”

    From Enhancing the Charging Power of Quantum Batteries (2017) · cites this paper as [36]

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Research works printed but not linked: 4

  1. [2][2] M. Horodecki, P. Horodecki, and R. Horodecki, in Quantum Information , edited by G. Alber et al. , Springer Tracts in Modern Physics Vol. 173, (Springer, Berlin, 2001), p. 151. -
  2. [4][4] L. Gurvits and H. Barnum, arXiv.org e-print quant-ph/0409095 . - · arXiv
  3. [16][16] K. Ball, In Flavors of geometry , vol. 31 of Math. Sci. Res. Inst. Publ. (Cambridge Univ. Press, Cambridge, 1997). -
  4. [22][22] G. Aubrun, S. J. Szarek, and E. Werner, preprint. -

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Authors
Aubrun, G., & Szarek, S. J.
Affiliations
Université Pierre-et-Marie-Curie (FR)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2006-02-14
Volume
73
Issue
2
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.73.022109
OpenAlex
W1983042148
Cited by
74
References
26
Citations by year
2026: 2 · 2024: 1 · 2023: 1 · 2022: 2 · 2021: 4 · 2020: 5 · 2019: 3 · 2018: 4 · 2017: 6 · 2016: 4 · 2015: 3 · 2014: 3 · 2013: 3 · 2012: 6
FWCI
6.46
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
Keywords
separable states, qudits, projective tensor product, convex bodies, high-dimensional probability, Banach space geometry, mixed quantum states

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