Random pure quantum states via unitary Brownian motion

Ion Nechita (CNRS, Université de Toulouse)
Clément Pellegrini (Université de Toulouse)


We introduce a new family of probability distributions on the set of pure states of a finite dimensional quantum system. Without any a priori assumptions, the most natural measure on the set of pure state is the uniform (or Haar) measure. Our family of measures is indexed by a time parameter $t$ and interpolates between a deterministic measure ($t=0$) and the uniform measure ($t=\infty$). The measures are constructed using a Brownian motion on the unitary group $\mathcal U_N$. Remarkably, these measures have a $\mathcal U_{N-1}$ invariance, whereas the usual uniform measure has a $\mathcal U_N$ invariance. We compute several averages with respect to these measures using as a tool the Laplace transform of the coordinates.

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Pages: 1-13

Publication Date: April 15, 2013

DOI: 10.1214/ECP.v18-2426


  • Benaych-Georges, Florent. Central limit theorems for the Brownian motion on large unitary groups. Bull. Soc. Math. France 139 (2011), no. 4, 593--610. MR2869307
  • Bengtsson, Ingemar; Życzkowski, Karol. Geometry of quantum states. An introduction to quantum entanglement. Cambridge University Press, Cambridge, 2006. xii+466 pp. ISBN: 978-0-521-81451-5; 0-521-81451-0 MR2230995
  • Collins, Benoît; Nechita, Ion. Random quantum channels I: graphical calculus and the Bell state phenomenon. Comm. Math. Phys. 297 (2010), no. 2, 345--370. MR2651902
  • Collins, Benoît; Nechita, Ion. Random quantum channels II: entanglement of random subspaces, Rényi entropy estimates and additivity problems. Adv. Math. 226 (2011), no. 2, 1181--1201. MR2737781
  • Collins, Benoît; Nechita, Ion; Życzkowski, Karol. Random graph states, maximal flow and Fuss-Catalan distributions. J. Phys. A 43 (2010), no. 27, 275303, 39 pp. MR2658283
  • Hastings, M.B. Superadditivity of communication capacity using entangled inputs. Nature Physics 5, 255.
  • Hayden, Patrick; Winter, Andreas. Counterexamples to the maximal $p$-norm multiplicity conjecture for all $p>1$. Comm. Math. Phys. 284 (2008), no. 1, 263--280. MR2443305
  • Hiai, Fumio; Petz, Dénes. The semicircle law, free random variables and entropy. Mathematical Surveys and Monographs, 77. American Mathematical Society, Providence, RI, 2000. x+376 pp. ISBN: 0-8218-2081-8 MR1746976
  • Lévy, Thierry. Schur-Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218 (2008), no. 2, 537--575. MR2407946
  • Lévy, Thierry; Maïda, Mylène. Central limit theorem for the heat kernel measure on the unitary group. J. Funct. Anal. 259 (2010), no. 12, 3163--3204. MR2727643
  • Nechita, Ion. Asymptotics of random density matrices. Ann. Henri Poincaré 8 (2007), no. 8, 1521--1538. MR2374950
  • Życzkowski, Karol; Sommers, Hans-Jürgen. Induced measures in the space of mixed quantum states. Quantum information and computation. J. Phys. A 34 (2001), no. 35, 7111--7125. MR1863143

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