Statistical distance and the geometry of quantum states
Abstract
By finding measurements that optimally resolve neighboring quantum states, we use statistical distinguishability to define a natural Riemannian metric on the space of quantum-mechanical density operators and to formulate uncertainty principles that are more general and more stringent than standard uncertainty principles.
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This result is now firmly established and comes out naturally within this new framework.
“This is a result that is now firmly established [33, 36] and that comes out in a rather natural fashion within this new framework.”
From Quantum Speed Limits in Open System Dynamics (2013) · cites this paper as [33]
Braunstein and Caves expressed the Bures angle for infinitesimally close density operators as dL^2 = tr[dρ R^{-1}_ρ(dρ)].
“Braunstein and Caves showed that the Bures angle for two infinitesimally close density operators, ρ = ρ + dρ can be written as [68], dL 2 = tr dρ R -1 ρ (dρ) ,(26)”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [68]
For an isolated mixed state with a given spectral decomposition, the quantum Fisher information is time-independent and bounded above by four times the squared energy variance.
“the QFI is time-independent and can be shown to be bounded [68]”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [68]
For any Hamiltonian that is a sum of one-particle terms, k-producible states satisfy F(ρ^(k), H/(2ε)) ≤ s k^2 + (N - s k)^2 with s=⌊N/k⌋.
“The authors of Ref. [18] found that -for any H as a sum of one-particle termsF ( ρ ( k ) , H/ (2 ϵ )) ≤ sk 2 +( N -sk ) 2 with s = ⌊ N k ⌋ .”
From Kind of entanglement that speeds up quantum evolution (2012) · cites this paper as [18]
The connection of the Fisher information to entanglement is more intimate than the relation between the Hamiltonian standard deviation and the entanglement of the state.
“since the connection of the Fisher information to entanglement is much more intimate than the relation between (∆ H ) ρ and the entanglement of ρ [7, 17, 18].”
From Kind of entanglement that speeds up quantum evolution (2012) · cites this paper as [18]
With the upper bound on F for k-producible quantum states, the survival probability can be directly connected with k.
“With the upper bound [18] on F for k -producible quantum states, we can directly connect the survival probability with k .”
From Kind of entanglement that speeds up quantum evolution (2012) · cites this paper as [18]
Among MC functions, there is a minimal f_min(t)=2t/(1+t) and maximal f_max(t)=(1+t)/2, and any MC function satisfies f_min ≤ f ≤ f_max.
“As pointed out by Kubo and Ando [55], among the MC functions, there exists a minimal one, f min ðtÞ ¼ 2t=ð1 þ tÞ, and a maximal one, f max ðtÞ ¼ ð1 þ tÞ=2, such that a generic MC function fðtÞ must satisfy f min ðtÞ ≤ fðtÞ ≤ f max ðtÞ.”
From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [55]
The Bures metric for two infinitesimally close density operators is given by the Braunstein-Caves expression.
“According to Braunstein and Caves, the Bures metric for two infinitesimally close density operators, ρ ′ = ρ + dρ , is [37] (see also Ref. [38]),”
From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [37]
The Bures length is a natural generalization of Wootters' statistical distance for arbitrary mixed quantum states.
“The Bures length (3) has been shown to be a natural generalization of Wootters' statistical distance for arbitrary mixed quantum states [37].”
From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [37]
In the case that the rank of ρ(x) is unchanged with varying x, the QFIM is related to the infinitesimal Bures distance as D_B^2(ρ(x), ρ(x+dx)) = 1/4 Σ F_μν dx^μ dx^ν.
“In the case that the rank of ρ( x) is unchanged with the varying of x , the QFIM is related to the infinitestmal Bures distance in the same way as the QFIM related to the Fubinistudy metricfoot_5 D 2 B (ρ( x), ρ( x + d x)) = 1 4 µν F µν dx µ dx ν .(77)”
From Quantum Fisher information matrix and multiparameter estimation (2020) · cites this paper as [30]
The most general analytical expression for the QFIM requires diagonalizing the density matrix.
“Still, known expressions for the QFIM appear quite complicated, and the most general analytical expression requires diagonalizing the density matrix.”
From Simple expression for the quantum Fisher information matrix (2018) · cites this paper as [4]
The quantum Fisher information matrix gives the ultimate precision bound on the estimation of parameters encoded in a quantum state.
“Quantum Fisher information matrix (QFIM) gives the ultimate precision bound on the estimation of parameters encoded in a quantum state.”
From Simple expression for the quantum Fisher information matrix (2018) · cites this paper as [4]
The quantum Fisher information and the Bures metric are the same when two infinitesimally close states being compared are pure, or when both are described by a full rank density matrix.
“It has been shown that these quantities are the same when two infinitesimally close states that are being compared are pure [17], or when they are both described by a full rank density matrix [1, 18].”
From Discontinuities of the quantum Fisher information and the Bures metric (2017) · cites this paper as [1]
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Multiple phase estimation for arbitrary pure states under white noise
Yao, Y., Ge, L., Xiao, X., Wang, X., & Sun, C. P. (2014) · Physical Review A · doi:10.1103/physreva.90.062113 · Cited by 42 · see citing works
Quantum plasmonic N00N state in a silver nanowire and its use for quantum sensing
Chen, Y., Lee, C., Lu, L., Liu, D., Wu, Y.-K., Feng, L.-T., Li, M., Rockstuhl, C., Guo, G.-P., Guo, G.-C., et al. (2018) · Optica · doi:10.1364/optica.5.001229 · Cited by 43 · see citing works
On the natural gradient for variational quantum eigensolver
Yamamoto, N. (2019) · arXiv · doi:10.48550/arxiv.1909.05074 · Cited by 39 · see citing works
Quantum information and physics: Some future directions
Preskill, J. (2000) · Journal of Modern Optics · doi:10.1080/095003400148097 · Cited by 154 · see citing works
Quantum metrology of spatial deformation using arrays of classical and quantum light emitters
Sidhu, J. S., & Kok, P. (2017) · Physical Review A · doi:10.1103/physreva.95.063829 · Cited by 31 · see citing works
Multiple phase estimation in quantum cloning machines
Yao, Y., Ge, L., Xiao, X., Wang, X.-G., & Sun, C.-P. (2014) · Physical Review A · doi:10.1103/physreva.90.022327 · Cited by 31 · see citing works
Quantum Weiss–Weinstein bounds for quantum metrology
Lu, X.-M., & Tsang, M. (2016) · Quantum Science and Technology · doi:10.1088/2058-9565/1/1/015002 · Cited by 31 · see citing works
Does large quantum Fisher information imply Bell correlations?
Fröwis, F., Fadel, M., Treutlein, P., Gisin, N., & Brunner, N. (2019) · Physical Review A · doi:10.1103/physreva.99.040101 · Cited by 25 · see citing works
Information geometry of density matrices and state estimation
Brody, D. C. (2011) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8113/44/25/252002 · Cited by 32 · see citing works
Estimation of gradients in quantum metrology
Altenburg, S., Oszmaniec, M., Wölk, S., & Gühne, O. (2017) · Physical Review A · doi:10.1103/physreva.96.042319 · Cited by 23 · see citing works
Berry curvature as a lower bound for multiparameter estimation
Guo, W., Zhong, W., Jing, X.-X., Fu, L.-B., & Wang, X. (2016) · Physical Review A · doi:10.1103/physreva.93.042115 · Cited by 19 · see citing works
- Authors
- Braunstein, S. L., & Caves, C. M.
- Affiliations
- University of New Mexico (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1994-05-30
- Volume
- 72
- Issue
- 22
- Pages
- 3439-3443
- ISSN
- 0031-9007 (ISSN-L)
- OpenAlex
- W2083423624
- Cited by
- 3,176
- References
- 13
- Citations by year
- 2026: 250 · 2025: 285 · 2024: 254 · 2023: 240 · 2022: 242 · 2021: 233 · 2020: 238 · 2019: 192 · 2018: 174 · 2017: 174 · 2016: 171 · 2015: 110 · 2014: 119 · 2013: 90 · 2012: 71
- FWCI
- 8.57
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (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, quantum state distinguishability, Riemannian metrics, density matrix, quantum geometry, uncertainty principle, quantum information geometry
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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