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Statistical distance and Hilbert space

Wootters, W. K.·1981·Physical Review D
DOI 10.1103/physrevd.23.357
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Abstract

A concept of "statistical distance" is defined between different preparations of the same quantum system, or in other words, between different rays in the same Hilbert space. Statistical distance is determined entirely by the size of statistical fluctuations occurring in measurements designed to distinguish one state from another. It is not related, a priori, to the usual distance (or angle) between rays. One finds, however, that these two kinds of distance are in fact the same, a result which depends on certain peculiarities of quantum mechanics.

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Crossref · 10.1103/physrevd.23.357

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  • For pure states, the angle between initial and final states is the arccosine of the absolute overlap.

    “for pure states the angle is simply given by [61] L (|ψ 0 , |ψ τ ) = arccos (| ψ 0 |ψ τ |), (23”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [61]

  • Wootters concluded that the shortest path connecting vectors in Hilbert space is identical to the geodesic under the Fisher-Rao metric.

    “As a main conclusion Wootters showed that the shortest path connecting vectors in Hilbert space, i.e. the angle between these vectors, is identical to the geodesic under the Fisher-Rao metric.”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [61]

  • All three approaches to survival probability have the common lower bound (2).

    “All three approaches, therefore, have the common lower bound (2).”

    From Kind of entanglement that speeds up quantum evolution (2012) · cites this paper as [19]

  • The quantum Fisher information satisfies F(ρ,H) ≤ 4(∆H)^2_ρ/ħ^2, with equality if ρ is pure.

    “Note that F ( ρ, H ) can be further estimated as F ( ρ, H ) ≤ 4 (∆ H ) 2 ρ / ℏ 2 [12, 19]. The equality holds if ρ is pure.”

    From Kind of entanglement that speeds up quantum evolution (2012) · cites this paper as [19]

  • Wootters' statistical distance is the only Riemannian metric, up to a constant factor, that is invariant under all unitary transformations.

    “Wootters' statistical distance is the only Riemannian metric, up to a constant factor, which is invariant under all unitary transformations.”

    From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [31]

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Authors
Wootters, W. K.
Affiliations
The University of Texas at Austin (US)
Corresponding author
Wootters, W. K.
ORCID
Wootters, W. K.
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
1981-01-15
Volume
23
Issue
2
Pages
357-362
ISSN
0556-2821 (ISSN-L)
DOI
10.1103/physrevd.23.357
OpenAlex
W1964348979
Cited by
793
References
5
Citations by year
2026: 24 · 2025: 28 · 2024: 35 · 2023: 37 · 2022: 34 · 2021: 45 · 2020: 49 · 2019: 29 · 2018: 45 · 2017: 40 · 2016: 34 · 2015: 30 · 2014: 40 · 2013: 30 · 2012: 29
FWCI
1.15
Type
journal article
Language
en
Open access
closed
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
Keywords
statistical distance, Hilbert spaces, quantum state discrimination, statistical fluctuations, quantum measurement, quantum state distinguishability

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