Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces [Formula: see text] of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the metric property of the modified Bures angles. As a consequence, we can show that the modified Bures angles induced by the geometric mean, harmonic mean, minimum and maximum of two positive numbers are metrics on [Formula: see text]. In addition, we can also show that the metric property of the modified Bures angles is stronger than that of the modified Bures distances.Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces [Formula: see text] of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the metric property of the modified Bures angles. As a consequence, we can show that the modified Bures angles induced by the geometric mean, harmonic mean, minimum and maximum of two positive numbers are metrics on [Formula: see text]. In addition, we can also show that the metric property of the modified Bures angles is stronger than that of the modified Bures distances.

Infinite Dimensional Analysis Quantum Probability and Related TopicsApr 23, 20230 citations

The geometric phase

P. Hariharan

Progress in opticsJan 1, 200528 citations

Geometric interpretation for the A fidelity and its relation with the Bures fidelity

Zhihao Ma, Fulin Zhang, Jing‐Ling Chen

A geometric interpretation for the $A$ fidelity between two states of a qubit system is presented, which leads to an upper bound of the Bures fidelity. The metrics defined based on the $A$ fidelity are studied by the numerical method. An alternative generalization of the $A$ fidelity, which has the same geometric picture, to an $N$ -state quantum system is also discussed.A geometric interpretation for the $A$ fidelity between two states of a qubit system is presented, which leads to an upper bound of the Bures fidelity. The metrics defined based on the $A$ fidelity are studied by the numerical method. An alternative generalization of the $A$ fidelity, which has the same geometric picture, to an $N$-state quantum system is also discussed.

arXiv:0809.1786Physical Review ADec 29, 200819 citations

The geometric phase

J. Anandan

NatureNov 1, 1992360 citations

The geometry of entanglement: metrics, connections and the geometric phase

Péter Lévay

Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.

arXiv:quant-ph/0306115Journal of Physics A Mathematical and GeneralJan 19, 200483 citations

The Geometric Phase

Michael Berry

Scientific AmericanDec 1, 198889 citations

A geometric interpretation for the

A

fidelity between two states of a qubit system is presented, which leads to an upper bound of the Bures fidelity. The metrics defined based on the

A

fidelity are studied by the numerical method. An alternative generalization of the

A

fidelity, which has the same geometric picture, to an

N

-state quantum system is also discussed.

Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of

S^7

over the quaternionic projective space

{\bf HP}^1\simeq S^4

with an

SU(2)\simeq S^3

fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on

{\bf HP}^1

between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.