Bures volume of the set of mixed quantum states

By H. Sommers, Karol Zyczkowski. 2003

175 citations recorded by the source corpus.

We compute the volume of the (N2 − 1)-dimensional set N of density matrices of size N with respect to the Bures measure and show that it is equal to that of an (N2 − 1)-dimensional hyper-hemisphere of radius 1/2. For N = 2 we obtain the volume of the Uhlmann hemisphere, ½S3 ⊂ 4. We find also the area of the boundary of the set N and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures–Hall normalization constants is derived for an arbitrary N.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2ec0d66196c2ac069906ad2ce6b7a3ce028d47f2

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In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

In Mathematical Physics

Citations in the corpus

Quantum-enhanced measurements without entanglement (2017, unknown): “(15) An expansion of ds2Bures(ρθ, ρθ+dθ) leads to the quantum Fisher information matrix (Paris, 2009; Sommers and Zyczkowski, 2003), ds2Bures(ρθ, ρθ+dθ) = Iθk,θk′ 4 dθkdθk′ , (16) where Iθk,θk′ = trρθ(LθkLθk′ +Lθ′kLθk)/2, and Lθk is the symmetric logarithmic derivative with respect to parameter θk.”
Generating random density matrices (2010, background): “was obtained in [13, 14] for small N and in [29] in the general case.”
Geometry of quantum phase transitions (2019, background): “ restriction p j > 0;p k > 0 on the summation derives from the auxiliary condition (131). Casting this expression into Eq. (133) leads to the following explicit form for the Bures metric tensor [286, 287], g = 1 2 X p j>0;p k>0 h j@ ˆkih i p j + p k : (135) Now, let’s cast the express”