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Register a new userDissipative generators, divisible dynamical maps and Kadison-Schwarz inequality
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Added by
Dariusz Chruscinski
Publication date
2019
fields
Physics
and research's language is
English
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We introduce a concept of Kadison-Schwarz divisible dynamical maps. It turns out that it is a natural generalization of the well known CP-divisibility which characterizes quantum Markovian evolution. It is proved that Kadison-Schwarz divisible maps are fully characterized in terms of time-local dissipative generators. Simple qubit evolution illustrates the concept.
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We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M_N of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N-cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in M_N induced by partial trace of mixed quantum states distributed uniformly with respect to HS-measure in $M_{N^2}$.
In this paper, Wielandts inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zero--error capacity of quantum channels and for the Matrix Product State (MPS) dimension of ground states of frustration-free Hamiltonians are derived. The obtained inequalities also imply new bounds on the required interaction-range of Hamiltonians with unique MPS ground state.
We elaborate on the notion of generalized tomograms, both in the classical and quantum domains. We construct a scheme of star-products of thick tomographic symbols and obtain in explicit form the kernels of classical and quantum generalized tomograms. Some of the new tomograms may have interesting applications in quantum optical tomography.
The Cauchy-Schwarz (CS) inequality -- one of the most widely used and important inequalities in mathematics -- can be formulated as an upper bound to the strength of correlations between classically fluctuating quantities. Quantum mechanical correlations can, however, exceed classical bounds.Here we realize four-wave mixing of atomic matter waves using colliding Bose-Einstein condensates, and demonstrate the violation of a multimode CS inequality for atom number correlations in opposite zones of the collision halo. The correlated atoms have large spatial separations and therefore open new opportunities for extending fundamental quantum-nonlocality tests to ensembles of massive particles.
We associate with k hermitian Ntimes N matrices a probability measure on R^k. It is supported on the joint numerical range of the k-tuple of matrices. We call this measure the joint numerical shadow of these matrices. Let k=2. A pair of hermitian Ntimes N matrices defines a complex Ntimes N matrix. The joint numerical range and the joint numerical shadow of the pair of hermitian matrices coincide with the numerical range and the numerical shadow, respectively, of this complex matrix. We study relationships between the dynamics of quantum maps on the set of quantum states, on one hand, and the numerical ranges, on the other hand. In particular, we show that under the identity resolution assumption on Kraus operators defining the quantum map, the dynamics shrinks numerical ranges.
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