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arXiv:1108.5065

Entropy of quantum channel in the theory of quantum information

Wojciech Roga

Published Aug 25, 2011• 2 citations
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Abstract

Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of properties of quantum channels and different entropies used to quantify the decoherence introduced into the system by a given operation. Part I of the thesis provides a general introduction to the subject. In Part II, the action of a quantum channel is treated as a process of preparation of a quantum ensemble. The Holevo information associated with this ensemble is shown to be bounded by the entropy exchanged during the preparation process between the initial state and the environment. A relation between the Holevo information and the entropy of an auxiliary matrix consisting of square root fidelities between the elements of the ensemble is proved in some special cases. Weaker bounds on the Holevo information are also established. The entropy of a channel, also called the map entropy, is defined as the entropy of the state corresponding to the channel by the Jamiolkowski isomorphism. In Part III of the thesis, the additivity of the entropy of a channel is proved. The minimal output entropy, which is difficult to compute, is estimated by an entropy of a channel which is much easier to obtain. A class of quantum channels is specified, for which additivity of channel capacity is conjectured. The last part of the thesis contains characterization of Davies channels, which correspond to an interaction of a state with a thermal reservoir in the week coupling limit, under the condition of quantum detailed balance and independence of rotational and dissipative evolutions. The Davies channels are characterized for one-qubit and one-qutrit systems.

DOI
10.48550/arxiv.1108.5065

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Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of properties of quantum channels and different entropies used to quantify the decoherence introduced into the system by a given operation. Part I of the thesis provides a general introduction to the subject. In Part II, the action of a quantum channel is treated as a process of preparation of a quantum ensemble. The Holevo information associated with this ensemble is shown to be bounded by the entropy exchanged during the preparation process between the initial state and the environment. A relation between the Holevo information and the entropy of an auxiliary matrix consisting of square root fidelities between the elements of the ensemble is proved in some special cases. Weaker bounds on the Holevo information are also established. The entropy of a channel, also called the map entropy, is defined as the entropy of the state corresponding to the channel by the Jamiolkowski isomorphism. In Part III of the thesis, the additivity of the entropy of a channel is proved. The minimal output entropy, which is difficult to compute, is estimated by an entropy of a channel which is much easier to obtain. A class of quantum channels is specified, for which additivity of channel capacity is conjectured. The last part of the thesis contains characterization of Davies channels, which correspond to an interaction of a state with a thermal reservoir in the week coupling limit, under the condition of quantum detailed balance and independence of rotational and dissipative evolutions. The Davies channels are characterized for one-qubit and one-qutrit systems.
Continuity properties of the output entropy of positive linear maps between Banach spaces of trace class operators are investigated with the special attention to the classes of quantum channels and operations. It is shown that finiteness of the output entropy of a positive map on the whole input state space implies its continuity. Sufficient conditions for this property expressed in terms of the Kraus representation of quantum channels and operations are presented. The characterization of a positive map "preserving continuity of the entropy" (in the sense that continuity of the entropy on a set of input states implies continuity of its output entropy on this set) is obtained and its applications to the class of quantum operations are considered. The special relation between continuity properties of the output entropies of complementary quantum operations is established and its corollaries are discussed. Continuity conditions for the output entropy of positive maps considered as a function of a pair (map, input state) are also obtained. Several applications to the quantum information theory are presented in the last part of the paper.
The entropy of a quantum operation, defined as the von Neumann entropy of the corresponding Choi–Jamiołkowski state, characterizes the coupling of the principal system with the environment. For any quantum channel acting on a state of a given size, one defines the complementary channel, which sends the input state into the state of the environment after the operation. Making use of subadditivity of entropy, we show that for any dimension the sum of both entropies is bounded from below. This result characterizes the trade-off between the information on the initial quantum state accessible to the principal system and the information leaking to the environment. For one qubit maps we describe the interpolating family of depolarizing maps, for which the sum of both entropies gives the lower boundary of the region allowed in the space spanned by both entropies.
Quantum channels are functions that map density matrices to density matrices. They can be used to describe a vast array of system environment interactions for quantum mechanical systems. The focus in quantum information theory lies on the study of the information content -- the quantum entropy -- of input and output system. The property of how much information a channel can convey is called capacity. When two quantum channels are combined they act on a bigger, possibly entangled state in parallel. The effects of entanglement on the capacity of a pair of channels have been widely studied. Still, recent work showed unexpected behavior. Hastings found that the capacities of two channels to transmit classical information does not simply add, but can exceed the sum of the individual quantum capacities, a property called superadditivity. We study Hastings' work and find estimates for how large the dimensions of such channels would have to be. The question of capacity relates to the behavior of quantum channels in extremal cases, when the output entropy is minimized. It is natural to wonder about the "standard" behavior of quantum channels and consider their average output entropy. We have studied this behavior in particular in cases where many relatively simple quantum channels act in parallel. When combining quantum channels in parallel, the parts of the system evolve independently of each other -- up to entanglement -- and only interact with the environment. Expanding on this idea one can think of a system where the parts not only interact with the environment but also directly with each other, a quantum channel network. Interesting cases of such systems arise naturally in the study of large biological molecules. In this case, information is often transmitted via energy pulses. Thus, the study of entropy and the study of energy overlap. Of particular interest to us are antennas in photo-active systems, they absorb light energy and transports it to a chemical reaction site. Recently, it has been discovered in the FMO molecule that this process evolves coherently, i.e. shows quantum oscillatory behavior. We study such quantum channel networks and how they can be approximated by kinetic networks. When considering the transport efficiency, this helps separating possible coherent effects from incoherent hopping.
We investigate decoherence induced by a quantum channel in terms of minimal output entropy and of map entropy. The latter is the von Neumann entropy of the Jamiolkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1 acting on a finite dimensional system there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 x Ψ2 holds.
We investigate decoherence induced by a quantum channel in terms of minimal output entropy and map entropy. The latter is the von Neumann entropy of the Jamiołkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1acting on a finite dimensional system, there exists a class of channels Φ2sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1⊗ Ψ2holds.
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity: