Michał Studziński

Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics

The second law of thermodynamics places a limitation into which states a system can evolve into. For systems in contact with a heat bath, it can be combined with the law of energy conservation, and it says that a system can only evolve into another if the free energy goes down. Recently, its been shown that there are actually many second laws, and that it is only for large macroscopic systems that they all become equivalent to the ordinary one. These additional second laws also hold for quantum systems, and are, in fact, often more relevant in this regime. They place a restriction on how the probabilities of energy levels can evolve. Here, we consider additional restrictions on how the coherences between energy levels can evolve. Coherences can only go down, and we provide a set of restrictions which limit the extent to which they can be maintained. We find that coherences over energy levels must decay at rates that are suitably adapted to the transition rates between energy levels. We show that the limitations are matched in the case of a single qubit, in which case we obtain the full characterization of state-to-state transformations. For higher dimensions, we conjecture that more severe constraints exist. We also introduce a new class of thermodynamical operations which allow for greater manipulation of coherences and study its power with respect to a class of operations known as thermal operations.Piotr Ćwikliński1,2, Michał Studziński1,2, Michał Horodecki1,2 and Jonathan Oppenheim3 1 Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland 2 National Quantum Information Centre of Gdańsk, 81-824 Sopot, Poland 3 Department of Physics and Astronomy, University College of London, and London Interdisciplinary Network for Quantum Science, London WC1E 6BT, UK (Dated: February 2, 2015)

Structure and properties of the algebra of partially transposed permutation operators

We consider the structure of algebra of operators, acting in n-fold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semi-simple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two kinds of irreducible representations of the algebra. The first one is strictly connected with the representations of the group S(n − 1) induced by irreducible representations of the group S(n − 2). The second kind is structurally connected with irreducible representations of the group S(n − 1).

Darboux points and integrability analysis of Hamiltonian systems with homogeneous rational potentials

Abstract We study integrability in the Liouville sense for natural Hamiltonian systems with a homogeneous rational potential V ( q ) . Strong necessary conditions for the integrability of such systems were obtained by analysis of differential Galois groups of variational equations along certain particular solutions. These conditions have the form of arithmetic restrictions on eigenvalues of the Hessians V ″ ( d ) calculated at non-zero solutions d of the equation grad V ( d ) = d . Such solutions are called proper Darboux points. It was recently proved that for generic polynomial homogeneous potentials, there exist universal relations between eigenvalues of Hessians of the potential taken at all proper Darboux points. The existence of such relations for rational potentials seems to be a difficult question. One of the reasons is the presence of points of indeterminacy of the potential and its gradient. Nevertheless, for two degrees of freedom we prove that such a relation exists. This result is important because it allows us to show that the set of admissible values for Hessian eigenvalues at a proper Darboux point for potentials satisfying the necessary conditions for integrability is finite. In turn, this provides a tool for classification of integrable rational potentials. It was also recently shown that for polynomial homogeneous potentials, additional necessary conditions for integrability can be deduced from the existence of improper Darboux points, that is, points d that are non-zero solutions of grad V ( d ) = 0 . These new conditions also take the form of arithmetic restrictions imposed on eigenvalues of V ″ ( d ) . In this paper we prove that for rational potentials, improper Darboux points give the same necessary conditions for integrability.

Using non-positive maps to characterize entanglement witnesses

In this paper we present a new method for entanglement witnesses construction. We show that to construct such an object we can deal with maps which are not positive on the whole domain, but only on a certain sub-domain. In our approach crucial role play such maps which are surjective between sets

Local random quantum circuits are approximate polynomial-designs: numerical results

\mathcal{P}_{k}^d

Group-representation approach to1→Nuniversal quantum cloning machines

of

Region of fidelities for a 1→N universal qubit quantum cloner

k \leq d

Distillation of entanglement by projection on permutationally invariant subspaces

rank projectors and the set

Optimal port-based teleportation

\mathcal{P}_1^d

Structure of irreducibly covariant quantum channels for finite groups

of rank one projectors acting in the