On some additivity problems in quantum information theory

Physical Review A, 2009

We study a natural generalization of the additivity problem in quantum information theory: given a pair of quantum channels, then what is the set of convex trace functions that attain their maximum on unentangled inputs, if they are applied to the corresponding output state? We prove several results on the structure of the set of those convex functions that are "additive" in this more general sense. In particular, we show that all operator convex functions are additive for the Werner-Holevo channel in 3 × 3 dimensions, which contains the well-known additivity results for this channel as special cases.

2004

We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. For this class of channels the norm of the output is maximized for the output being a normalized projection. We prove the additivity of the minimal output Renyi entropies with entropic parameters α ∈ [0, 2], generalizing an argument by Alicki and Fannes, and present a number of examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify several instances of weakly covariant channels for which we can infer the additivity of the classical information capacity. Both additivity results apply to the case of an arbitrary number of different channels. Finally, we relate the obtained results to instances of bi-partite quantum states for which the entanglement cost can be calculated.

2007

In this paper a notion of entropy transmission of quantum channels is introduced as a natural extension of Ohya's entropy. Here by quantum channel is meant unital completely positive mappings (ucp) of $B(H)$ into itself, where $H$ is an infinite dimensional Hilbert space. Using a representation theorem of ucp mapping we associate to every ucp map a uniquely determined state,

Journal of Mathematical Analysis and Applications, 2018

We investigate spectral properties of the tensor products of two completely positive and trace preserving linear maps (also known as quantum channels) acting on matrix algebras. This leads to an important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels the multiplicative domain of their tensor product splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of maximal unital proper *-subalgebras (MUPSA) of a matrix algebra we provide a non-trivial upper bound of the recentlyintroduced multiplicative index of a unital channel. This bound gives a criteria on when a channel cannot be factored into a product of two different channels. We construct examples of channels which cannot be realized as a tensor product of two channels in any way. With these techniques and results, we found some applications in quantum information theory.

2005

Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M d → M d with d < d , and different constructions may differ by conjugation with a partial isometry. We show that a channel and its conjugate have the same minimal output entropy and maximal output p-norm. It then follows that the additivity and multiplicativity conjectures for these measures of optimal output purity hold for a product of channels if and only if they also hold for the product of their conjugates. This allows us to reduce these conjectures to the special case of maps taking M d → M d 2 with a minimal representation of dimension at most d. We find explicit expressions for the conjugates for a number of well-known examples, including entanglement-breaking channels, unital qubit channels, the depolarizing channel, and a subclass of random unitary channels. For the entanglement-breaking channels, channels this yields a new class of channels for which additivity and multiplicativity of optimal output purity can be established. For random unitary channels using the generalized Pauli matrices, we obtain a new formulation of the multiplicativity conjecture. The conjugate of the completely noisy channel plays a special role and suggests a mechanism for using noise to transmit information.

A multiplicativity conjecture for quantum communication channels is formulated, validity of which for the values of parameter p close to 1 is related to the solution of the fundamental problem of additivity of the channel capacity in quantum information theory. The proof of the conjecture is given for the case of natural numbers p.

Journal of Physics A: Mathematical and Theoretical, 2011

We consider properties of quantum channels with use of unified entropies. Extremal unravelings of quantum channel with respect to these entropies are examined. The concept of map entropy is extended in terms of the unified entropies. The map (q, s)-entropy is naturally defined as the unified (q, s)-entropy of rescaled dynamical matrix of given quantum channel. Inequalities of Fannes type are obtained for introduced entropies in terms of both the trace and Frobenius norms of difference between corresponding dynamical matrices. Additivity properties of introduced map entropies are discussed. The known inequality of Lindblad with the entropy exchange is generalized to many of the unified entropies. For tensor product of a pair of quantum channels, we derive two-side estimating of the output entropy of a maximally entangled input state.

Journal of Mathematical Physics, 2018

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A 1 ,. .. A K). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A 1 ,. .. A K) one can predict the asymptotic dynamics for a class of operations.

Topics in Operator Theory, 2010

We introduce two additive invariants of output quantum channels. If the value of one these invariants is less than 1 then the logarithm of the inverse of its value is a positive lower bound for the regularized minimum entropy of an output quantum channel. We give a few examples in which one of these invariants is less than 1. We also study the special cases where the above both invariants are equal to 1.

IEEE Transactions on Information Theory, 2003

Coding problems of classical-quantum channels are considered in the most general setting, where no structural assumptions such as the stationary memoryless property are made on a channel. The channel capacity as well as the characterization of the strong converse property is given just in parallel with the corresponding classical results of Verdú-Han based on the so-called information-spectrum method. The general results are applied to the stationary memoryless case with or without cost constraint, whereby a deep relation between the channel coding theory and the hypothesis testing for two quantum states is elucidated.