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arXiv:math-ph/0611057 · Communications in Mathematical Physics

Dividing Quantum Channels

Michael M. Wolf, J. I. Cirac

Published Jan 31, 2008• 289 citations
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Abstract

We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘ indivisible ’ channels which can not be written as non-trivial products of other channels and study the set of ‘ infinitesimal divisible ’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.

DOI
10.1007/s00220-008-0411-y

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We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘ indivisible ’ channels which can not be written as non-trivial products of other channels and study the set of ‘ infinitesimal divisible ’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.
We present two projects concerning the main part of my PhD work. In the first one we study quantum channels, which are the most general operations mapping quantum states into quantum states, from the point of view of their divisibility properties. We introduced tools to test if a given quantum channel can be implemented by a process described by a Lindblad master equation. This in turn defines channels that can be divided in such a way that they form a one-parameter semigroup, thus introducing the most restricted studied divisibility type of this work. Using our results, together with the study of other types of divisibility that can be found in the literature, we characterized the space of qubit quantum channels. We found interesting results connecting the concept of entanglement-breaking channel and infinitesimal divisibility. Additionally we proved that infinitely divisible channels are equivalent to the ones that are implementable by one-parameter semigroups, opening this question for more general channel spaces. In the second project we study the functional forms of one-mode Gaussian quantum channels in the position state representation, beyond Gaussian functional forms. We perform a black-box characterization using complete positivity and trace preserving conditions, and report the existence of two subsets that do not have a functional Gaussian form. The study covers as particular limit the case of singular channels, thus connecting our results with the known classification scheme based on canonical forms. Our full characterization of Gaussian channels without Gaussian functional form is completed by showing how Gaussian states are transformed under these operations, and by deriving the conditions for the existence of master equations for the non-singular cases.
In this paper we study quantum dynamical mappings called also quantum processes.The set of values of such mapping is a one-parameter family of completely positive trace-preserving linear operators defined on a finite-dimensional Hilbert space.In quantum information theory such operators are referred to as quantum channels.An important concept for quantum dynamical mappings is their divisibility.There are different types of this concept.The present paper deals with so-called completely positive divisible quantum processes.For two such processes, which are bijective and satisfy a commutativity condition, we construct a compound quantum process.It is shown that this compound quantum process is also completely positive divisible.Endowing a set of quantum channels with the norm topology, we consider continuous quantum processes and continuous completely positive evolutions.The latter are defined as two-parameter families of quantum channels satisfying additional properties.We prove that a continuous bijective completely positive divisible quantum process generates a continuous completely positive evolution.In order to illustrate the considered concepts and the results on them, we provide examples of quantum dynamical mappings with values in the set of qubit channels.In particular, a completely positive divisible compound quantum process is constructed for two bijective commuting quantum processes.Geometric and physical interpretations of this compound quantum process are given.
We investigate the semigroup structure of bosonic Gaussian quantum channels. Particular focus lies on the sets of channels which are divisible, idempotent or Markovian (in the sense of either belonging to one-parameter semigroups or being infinitesimal divisible). We show that the non-compactness of the set of Gaussian channels allows for remarkable differences when comparing the semigroup structure with that of finite dimensional quantum channels. For instance, every irreversible Gaussian channel is shown to be divisible in spite of the existence of Gaussian channels which are not infinitesimal divisible. A simpler and known consequence of non-compactness is the lack of generators for certain reversible channels. Along the way we provide new representations for classes of Gaussian channels: as matrix semigroup, complex valued positive matrices or in terms of a simple form describing almost all one-parameter semigroups.
The concept of divisibility of dynamical maps is used to introduce an analogous concept for quantum channels by analyzing the simulability of channels by means of dynamical maps. In particular, this is addressed for Lindblad divisible, completely positive divisible and positive divisible dynamical maps. The corresponding L-divisible, CP-divisible and P-divisible subsets of channels are characterized (exploiting the results by Wolf et al. \cite{cirac}) and visualized for the case of qubit channels. We discuss the general inclusions among divisibility sets and show several equivalences for qubit channels. To this end we study the conditions of L-divisibility for finite dimensional channels, especially the cases with negative eigenvalues, extending and completing the results of Ref.~\cite{Wolf2008}. Furthermore we show that transitions between every two of the defined divisibility sets are allowed. We explore particular examples of dynamical maps to compare these concepts. Finally, we show that every divisible but not infinitesimal divisible qubit channel (in positive maps) is entanglement breaking, and open the question if something similar occurs for higher dimensions.
We investigate the semigroup structure of bosonic Gaussian quantum channels. Particular focus lies on the sets of channels which are divisible, idempotent or Markovian (in the sense of either belonging to one-parameter semigroups or being infinitesimal divisible). We show that the non-compactness of the set of Gaussian channels allows for remarkable differences when comparing the semigroup structure with that of finite dimensional quantum channels. For instance, every irreversible Gaussian channel is shown to be divisible in spite of the existence of Gaussian channels which are not infinitesimal divisible. A simpler and known consequence of non-compactness is the lack of generators for certain reversible channels. Along the way we provide new representations for classes of Gaussian channels: as matrix semigroup, complex valued positive matrices or in terms of a simple form describing almost all one-parameter semigroups.
Phase covariant qubit dynamics describes an evolution of a two-level system under simultaneous action of pure dephasing, energy dissipation, and energy gain with time-dependent rates γz(t)\gamma_{z}(t)γz​(t) , γ−(t)\gamma_{-}(t)γ−​(t) , and γ+(t)\gamma_{+}(t)γ+​(t) , respectively. Non-negative rates correspond to completely positive divisible dynamics, which can still exhibit such peculiarities as non-monotonicity of populations for any initial state. We find a set of quantum channels attainable in the completely positive divisible phase covariant dynamics and show that this set coincides with the set of channels attainable in semigroup phase covariant dynamics. We also construct new examples of eternally indivisible dynamics with γz(t)<0\gamma_{z}(t)<0γz​(t)<0 for all t>0t>0t>0 that is neither unital nor commutative. Using the quantum Sinkhorn theorem, we for the first time derive a restriction on the decoherence rates under which the dynamics is positive divisible, namely, γ±(t)≥0\gamma_{\pm}(t)\geq 0γ±​(t)≥0 , γ+(t)γ−(t)+2γz(t)>0\sqrt{\gamma_{+}(t)\gamma_{-}(t)}+2\gamma_{z}(t)>0γ+​(t)γ−​(t)​+2γz​(t)>0 . Finally, we consider phase covariant convolution master equations and find a class of admissible memory kernels that guarantee complete positivity of the dynamical map.