Induced measures in the space of mixed quantum states

By Karol Zyczkowski, H. Sommers. 2000

470 citations recorded by the source corpus.

We analyse several product measures in the space of mixed quantum states. In particular, we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N×K composite system, induces a unique measure in the space of N×N mixed states (or in the space of K×K mixed states, if the reduction takes place with respect to the first subsystem). For K = N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of N×N pure states behaves as lnN-1/2.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e395f3b27eec9ffc6247a11f78a638e7cbab6964

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Citations in the corpus

Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices (2015, unknown): “:3) ψ | to N/ 2 qubits reads where the constant c depends on the choice of the ensemble used to generate random states [6,9].”
Generating random density matrices (2010, methodology): “This observation leads to a simple algorithm to generate a Hilbert-Schmidt random matrix [4]: i) Take a square complex random matrix A of size N pertaining to the Ginibre ensemble [18, 19] (with real and imaginary parts of each element being independent normal random variables); ii) Write down the r”
Genuinely multipartite entangled states and orthogonal arrays (2014, background): “Analyze generic random pure states of N qudits, distributed with respect to a Haar measure [57], to find, for which values of both parameters they are ( , k)–uniform.”
Aspects of Generic Entanglement (2004, background): “Instead, we follow an old proposal to induce probability measures on mixed states [4, 18, 59] by partial tracing.”