Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by S lomczyński andŻyczkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions.2000 Mathematics Subject Classification. Primary: 46L55, 94A17 ; Secondary: 37M25, 60G99, 82C10.
We introduce the concept of group state transfer on graphs, summarize its relationship to other concepts in the theory of quantum walks, set up a basic theory, and discuss examples.Let X be a graph with adjacency matrix A and consider quantum walks on the vertex set V (X) governed by the continuous time-dependent unitary transition operator U (t) = exp(itA). For S, T ⊆ V (X), we says X admits "group state transfer" from S to T at time τ if the submatrix of U (τ ) obtained by restricting to columns in S and rows not in T is the all-zero matrix. As a generalization of perfect state transfer, fractional revival and periodicity, group state transfer satisfies natural monotonicity and transitivity properties. Yet non-trivial group state transfer is still rare; using a compactness argument, we prove that bijective group state transfer (the optimal case where |S| = |T |) is absent for almost all t. Focusing on this bijective case, we obtain a structure theorem, prove that bijective group state transfer is "monogamous", and study the relationship between the projections of S and T into each eigenspace of the graph.Group state transfer is obviously preserved by graph automorphisms and this gives us information about the relationship between the setwise stabilizer of S ⊆ V (X) and the stabilizers of naturally defined subsets obtained by spreading S out over time and crudely reversing this process. These operations are sufficiently well-behaved to give us a topology on V (X) which is likely to be simply the topology of subsets for which bijective group state transfer occurs at that time. We illustrate non-trivial group state transfer in bipartite graphs with integer eigenvalues, in joins of graphs, and in symmetric double stars. The Cartesian product allows us to build new examples from old ones.
In this article we study lossless compression of strings of pure quantum states of indeterminate-length quantum codes which were introduced by Schumacher and Westmoreland. Past work has assumed that the strings of quantum data are prepared to be encoded in an independent and identically distributed way. We introduce the notion of quantum stochastic ensembles, allowing us to consider strings of quantum states prepared in a general way. For any quantum stochastic ensemble we define an associated quantum dynamical system and prove that the optimal average codeword length via lossless coding is equal to the quantum dynamical entropy of the associated quantum dynamical system. Date: August 6, 2019. 2010 Mathematics Subject Classification. Primary: 81P45. Secondary: 81P70, 94A15, 37A35 . Key words and phrases. Kraft-McMillan inequality, quantum dynamical system, quantum Markov chain, quantum dynamical entropy.
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