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arXiv:2203.13451 · Physical Review Letters

Quantum Dynamics is Not Strictly Bidivisible

David Davalos, Mário Ziman

Published Feb 21, 2023• 3 citations
arXivHTMLPDF

Abstract

We address the question of the existence of quantum channels that are divisible in two quantum channels but not in three or, more generally, channels divisible in n but not in n+1 parts. We show that for the qubit those channels do not exist, whereas for general finite-dimensional quantum channels the same holds at least for full Kraus rank channels. To prove these results, we introduce a novel decomposition of quantum channels which separates them into a boundary and Markovian part, and it holds for any finite dimension. Additionally, the introduced decomposition amounts to the well-known connection between divisibility classes and implementation types of quantum dynamical maps and can be used to implement quantum channels using smaller quantum registers.

DOI
10.1103/physrevlett.130.080801

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We address the question of the existence of quantum channels that are divisible in two quantum channels but not in three or, more generally, channels divisible in n but not in n+1 parts. We show that for the qubit those channels do not exist, whereas for general finite-dimensional quantum channels the same holds at least for full Kraus rank channels. To prove these results, we introduce a novel decomposition of quantum channels which separates them into a boundary and Markovian part, and it holds for any finite dimension. Additionally, the introduced decomposition amounts to the well-known connection between divisibility classes and implementation types of quantum dynamical maps and can be used to implement quantum channels using smaller quantum registers.
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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.
We present a simple, dimension-independent criterion which guarantees that some quantum channel Φ is divisible, i.e., that there exists a non-trivial factorization Φ = Φ1Φ2. The idea is to first define an “elementary” channel Φ2 and then to analyze when ΦΦ2−1 is completely positive. The sufficient criterion obtained this way—which even yields an explicit factorization of Φ—is that one has to find orthogonal unit vectors x, x⊥ such that 〈x⊥|KΦKΦ⊥|x〉=〈x|KΦKΦ⊥|x〉={0} where KΦ is the Kraus subspace of Φ and KΦ⊥ is its orthogonal complement. Of course, using linearity this criterion can be reduced to finitely many equalities. Generically, this division even lowers the Kraus rank which is why repeated application—if possible—results in a factorization of Φ into in some sense “simple” channels. Finally, be aware that our techniques are not limited to the particular elementary channel we chose.
We describe how to extend the notion of infinitesimal Markovian divisibility from quantum channels to general linear maps and compact and convex sets of generators. We give a general approach toward proving necessary criteria for (infinitesimal) Markovian divisibility. With it, we prove two necessary criteria for infinitesimal divisibility of quantum channels in any finite dimension d: an upper bound on the determinant in terms of a Θ(d)-power of the smallest singular value and in terms of a product of Θ(d) smallest singular values. These allow us to analytically construct, in any given dimension, a set of channels that contains provably non-infinitesimal Markovian divisible ones. Moreover, we show that, in general, no such non-trivial criteria can be derived for the classical counterpart of this scenario.
Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable, and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE, respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of ΦC. Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with “large” environments, in the sense that the minimal dimension dE>dB. Indeed, one can have dE>14dB2. These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.
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.