will trigger further interest. We are currently investi- gating strategies, developed well along the lines of this paper, to study the interplay between quantum correla- tions and thermodynamical properties of a multipartite system. where E$_{i}$ is an n × n matrix. Matrices E$_{i}$ ’s are readily constructed using products of proper rotation matrices R ( i,j $^{)}$( φ$_{i j}$, ψ$_{i j}$, χ$_{i j}$ ), each depending on the respective set of Euler’s angles { φ$_{i j}$, ψ$_{i j}$, χ$_{i j}$ } as follows 9 MP thanks Dr. M. Guta for interesting discussions on the topic of entanglement in random-matrix set- tings. This work has been supported by PRIN 2006 “Quantum noise in mesoscopic systems”, EUROTECH, DEL, the British Council/MIUR British-Italian Part- nership Programme 2007-2008 and the UK EPSRC (EP/G004579/1). SC gratefully acknowledges the hospi- tality of the DSFSA, Universita’ degli Studi di Palermo, where part of this work has been performed. Here we briefly outline the recipe to generate a random unitary matrix. The algorithm has been extensively used in recent works [11, 12, 31–33]. The parameterisation is based on the original work presented by Hurwitz in 1897 [10]. Any unitary matrix, U$_{r}$ , of dimension n , can be decomposed as and zero otherwise. The angles are drawn from the ranges 0 ≤ φ$_{i j}$ ≤ π/ 2 , 0 ≤ ψ$_{i j}$ ≤ 2 π, 0 ≤ χ$_{i j}$ ≤ 2 π and 0 ≤ α ≤ 2 π, uniformly with respect to the corre- sponding (and properly normalized) Haar measure [11]. The matrix elements are taken as Acknowledgments APPENDIX Construction of Random Unitary Matrices E$_{1}$ = R (1 , $^{2)}$( φ$_{1 2}$, ψ$_{1 2}$, χ$_{1 2}$ ) , E$_{2}$ = R (2 , $^{3)}$( φ$_{2 3}$, ψ$_{2 3}$, 0) R (1 , $^{3)}$( φ$_{1 3}$, ψ$_{1 3}$, χ$_{1 3}$ ) , E$_{3}$ = R (3 , $^{4)}$( φ$_{3 4}$, ψ$_{3 4}$, 0) R (2 , $^{4)}$( φ$_{2 4}$, ψ$_{2 4}$, 0) , × R (1 , $^{4)}$( φ$_{1 4}$, ψ$_{1 4}$, χ$_{1 4}$ ) , . E$_{n}$$_{−}$$_{1}$ = R ( n − 1 ,n $^{)}$( φ$_{n}$$_{−}$$_{1}$$_{n}$, ψ$_{n}$$_{−}$$_{1}$$_{n}$, 0) × . . . × R (1 ,n $^{)}$( φ$_{1}$$_{n}$, ψ$_{1}$$_{n}$, χ$_{1}$$_{n}$ ) . (A-2) R ( i,j ) k,k = 1 (for k = i, j ) , R ( i,j ) i,i = e iψ cos φ, R ( i,j ) i,j = e $^{iχ}$sin φ, R ( i,j ) j,i = − e − $^{iχ}$sin φ R ( i,j ) i,j = e − iψ cos φ, (A-3) U$_{r}$ = e $^{iα}$E$_{1}$ E$_{2}$ . . . E$_{n}$$_{−}$$_{1}$ (A-1) * These authors contributed equally to this paper. [1] I. Kim and G. Mahler, Phys. Lett. A 263 , 268 (1999); Phys. Rev. A 60 , 692 (1999); J. Mod. Opt. 47 , 177 (2000). [2] M. J. Henrich, G. Mahler, and M. Michel, Phys. Rev. E 75 , 051118 (2007). [3] G. Gennaro, G. Benenti, and G. M. Palma, Europhys. Lett. 82 , 20006 (2008). [4] G. Benenti and G. M. Palma, Phys. Rev. A 75 , 052110 (2007). [5] D. P. DiVincenzo, D. W. Leung, and B. M. Terhal, IEEE Trans. Inform. Theory 48 , 580 (2002); P. Sen, arXiv:quant-ph/0512085v1; A. Harrow, P. Hayden, and D. Leung, Phys. Rev. Lett. 92 , 187901 (2004); J. Emer- son, E. Livine, and S. Lloyd, Phys. Rev. A 72 , 060302 (2005). [6] T. Gorin, C. Pineda, and T. H. Seligman, Phys. Rev. Lett. 99 , 240405 (2007); C. Pineda, T. Gorin, and T. H. Seligman, New J. Phys. 9 , 106 (2007); C. Pineda, and T. H. Seligman, Phys. Rev. A 75 , 012106 (2007). [7] G. Gennaro, G. Benenti, and G. M. Palma Phys. Rev. A 79 , 022105 (2009). [8] V. Scarani, M. Ziman, P. Stelmachoviˇc, N. Gisin, and V. Buˇzek, Phys. Rev. Lett. 88 , 097905 (2002); M. Ziman, P. ˇ Stelmachoviˇc, V. Buˇzek, M. Hillery, V. Scarani, and N. Gisin, Phys. Rev. A 65 , 042105 (2002); M. Ziman, P. ˇ Stelmachoviˇc, and V. Buˇzek, J. Opt. B: Quantum Semi- classical Opt. 5 , S439 (2003); Open Syst. Inf. Dyn. 12 , 81 (2005); M. Ziman and V. Buˇzek, Phys. Rev. A 72 , 022110 (2005). [9] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information , (Cambridge University Press, Cambridge, 2000). [10] A. Hurwitz, Nachr. Ges Wiss. G¨ ott. Math-Phys. Kl. 71 1059 (1897). [11] K. ˙ Zyczkowski and M. Ku´s, J. Phys. A: Math. Gen. 27 , 4235 (1994). [12] M. Ku´s and K. ˙ Zyczkowski, Phys. Rev. A 44 , 956 (1991). [13] V. Kendon, K. Nemoto and W. Munro, J. Mod. Opt. 49 , 1709 (2002). [14] It is sufficient to consider the following simple example to see that the dimension of a multipartite environment prepared in an entangled state can affect the bipartite quantum correlations between an ancilla A and the qubit e of the environment it interacts with. For the sake of argument, let us assume the e − A interaction is embod- ied, in this example, by a controlled-NOT gate [9]. While A is prepared in a superposition state | + 〉 A such that