Electronic Communications in Probability

Random pure quantum states via unitary Brownian motion

Ion Nechita and Clément Pellegrini

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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.

Article information

Electron. Commun. Probab. Volume 18 (2013), paper no. 27, 13 pp.

Accepted: 15 April 2013
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 39A50: Stochastic difference equations
Secondary: 81P45: Quantum information, communication, networks [See also 94A15, 94A17]

quantum states unitary Brownian motion

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Nechita, Ion; Pellegrini, Clément. Random pure quantum states via unitary Brownian motion. Electron. Commun. Probab. 18 (2013), paper no. 27, 13 pp. doi:10.1214/ECP.v18-2426. https://projecteuclid.org/euclid.ecp/1465315566

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  • Benaych-Georges, Florent. Central limit theorems for the Brownian motion on large unitary groups. Bull. Soc. Math. France 139 (2011), no. 4, 593–610.
  • Bengtsson, Ingemar; Å»yczkowski, 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
  • Collins, Benoît; Nechita, Ion. Random quantum channels I: graphical calculus and the Bell state phenomenon. Comm. Math. Phys. 297 (2010), no. 2, 345–370.
  • 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.
  • Collins, Benoît; Nechita, Ion; Å»yczkowski, Karol. Random graph states, maximal flow and Fuss-Catalan distributions. J. Phys. A 43 (2010), no. 27, 275303, 39 pp.
  • 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.
  • 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
  • Lévy, Thierry. Schur-Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218 (2008), no. 2, 537–575.
  • 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.
  • Nechita, Ion. Asymptotics of random density matrices. Ann. Henri Poincaré 8 (2007), no. 8, 1521–1538.
  • Å»yczkowski, 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.