Volume of the quantum mechanical state space

Abstract

The volume of the quantum mechanical state space over $n$-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endowed with a monotone metric or a pull-back metric is considered too, we give formulas to compute the volume of the state space with respect to the given Riemannian metric. We present the volume of the space of qubits with respect to various monotone metrics. It turns out that the volume of the space of qubits can be infinite too. We characterize those monotone metrics which generates infinite volume.

17 pages

References in corpus (9)

• Bures volume of the set of mixed quantum states
• Random quantum correlations and density operator distributions
• Hilbert--Schmidt volume of the set of mixed quantum states
• A priori Probabilities of Separable Quantum States
• Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems
• Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems
• Comparative Noninformativities of Quantum Priors Based on Monotone Metrics
• On the monotonicity of scalar curvature in classical and quantum information geometry
• Monotone Riemannian metrics on density matrices with non-monotone scalar curvature

Cited by in corpus (21)

• The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states
• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities
• Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities
• Invariance of separability probability over reduced states in 4x4 bipartite systems
• Efficient Bayesian credible-region certification for quantum-state tomography
• Subnormalized states and trace-nonincreasing maps
• Volume of the space of qubit-qubit channels and state transformations under random quantum channels
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• Bayesian error regions in quantum estimation I: analytical reasonings
• Numerical Computations of Separability Probabilities
• Incompatibility probability of random quantum measurements
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures
• Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems
• Volume of the set of locally diagonalizable bipartite states
• Calculation of the unitary part of the Bures measure for N-level quantum systems
• Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities
• One application of Duistermaat-Heckman measure in quantum information theory