# Geometry of quantum state space and quantum correlations

Geometry of quantum state space and quantum correlations Quantum state space is endowed with a metric structure, and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical considerations on the quantum state space. In this article, considering the quantum state space being spanned by $$2\times 2$$ 2 × 2 density matrices, we determine a particular Riemannian metric for a state $$\rho$$ ρ and show that if $$\rho$$ ρ gets entangled with another quantum state, the negativity of the generated entangled state is, upto a constant factor, equal to square root of that particular Riemannian metric . Our result clearly relates a geometric quantity to a measure of entanglement. Moreover, the result establishes the possibility of understanding quantum correlations through geometric approach. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Geometry of quantum state space and quantum correlations

, Volume 15 (4) – Jan 12, 2016
10 pages

/lp/springer_journal/geometry-of-quantum-state-space-and-quantum-correlations-2xApty0ZeK
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1227-2
Publisher site
See Article on Publisher Site

### Abstract

Quantum state space is endowed with a metric structure, and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical considerations on the quantum state space. In this article, considering the quantum state space being spanned by $$2\times 2$$ 2 × 2 density matrices, we determine a particular Riemannian metric for a state $$\rho$$ ρ and show that if $$\rho$$ ρ gets entangled with another quantum state, the negativity of the generated entangled state is, upto a constant factor, equal to square root of that particular Riemannian metric . Our result clearly relates a geometric quantity to a measure of entanglement. Moreover, the result establishes the possibility of understanding quantum correlations through geometric approach.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Jan 12, 2016

### References

• Monotone metrics on matrix spaces
Petz, D
• Wigner–Yanase information on quantum state space: the geometric approach
Gibilisco, P; Isola, T
• Quantum entanglement
Horodecki, R; Horodecki, P; Horodecki, M; Horodecki, K

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