Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Abstract

We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM-and POVM-dynamical (quantum) entropies, which are analogues of the classical Kolmogorov-Sinai entropy rate. They quantify the maximal randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy, or equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity.

Orthogonal projections on hyperplanes intertwined with unitaries

“…to(8) results in(5) and concludes the proof of Theorem 2.A Financial support by the Polish National Science Centre under Project No. DEC-2015/18/A/ST2/00274 is gratefully acknowledged.…”

“…with p jl := P l U z j 2 = | z l , U z j | 2 for j, l ∈ {0, 1} being the probability that we obtain l as the measurement outcome, provided that the preceding measurement yielded the result j. The combined evolution of states (and of measurement outcomes) is then Markovian with two states: z 0 , z 1 , the initial distribution concentrated at z 0 , and the transition bistochastic matrix P := (p jl ) j,l=0,1 [7,8,9]. In particular,…”

On the nonlinearity of quantum dynamical entropy

“…In [20] the authors establish a class of instruments which have positive dynamical SZ entropy and we give further such examples in Section 6. Therefore Proposition 4.8 does establish the nonlinearity of dynamical SZ entropy in time.…”

“…Next we split the SZ entropy of (Θ, T , u) with respect to C into two different causes for randomness. The first cause of randomness is that caused by the choice of instrument, is referred to as the measurement SZ entropy and is given by (20) h SZ meas (T , u, C) := h SZ (½, T , u, C). The second cause of randomness is given by the dynamics; i.e.…”

Thermodynamic Formalism for Quantum Channels: Entropy, Pressure, Gibbs channels and generic properties

“…A different approach appears in [26] and [27] where the authors present their own definition of quantum dynamical entropy (see also [4]).…”