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Maximal output purity and capacity for asymmetric unital qudit channels

Nilanjana Datta, Mary Beth Ruskai

Published May 8, 2005• 23 citations
OpenAlex

Abstract

We consider generalizations of depolarizing channels to maps which are convex combinations of unitary conjugations (rather than the identity) and the completely noisy map. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this condition holds, but the channel has no evident covariance properties.

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Maximal output purity and capacity for asymmetric unital qudit channels

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We consider generalizations of depolarizing channels to maps in which the identity channel is replaced by a convex combinations of unitary conjugations. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this condition holds, but the channel has no evident covariance properties.

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Some new additivity results on quantum channels

Motohisa Fukuda

In this paper we introduce a lemma. This lemma states that for a unitary covariant channel $\\Psi$ and a channel $M$ such that $M(\\rho_0)$ is of rank one for some state $\\rho_0$ the additivity of the minimal output entropy of $\\Psi$ implies that of $\\Psi \\circ M$, and the multiplicativity of the maximal output $p$-norm of $\\Psi$, that of $\\Psi \\circ M$. Applying the lemma to the depolarising channel we produce a new class of memoryless quantum channels for which the additivity of the minimal output entropy and the multiplicativity of the maximal output $p$-norm hold. This class includes the shifted depolarising channel.

•May 4, 2005•1 citation
We consider generalizations of depolarizing channels to maps which are convex combinations of unitary conjugations (rather than the identity) and the completely noisy map. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this condition holds, but the channel has no evident covariance properties.
We consider generalizations of depolarizing channels to maps in which the identity channel is replaced by a convex combinations of unitary conjugations. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this condition holds, but the channel has no evident covariance properties.
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 to 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 to M_d' 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.
We consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, namely, that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. We give a separate argument, based on the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps. We also show that the maximal norm of the output states is multiplicative for most product maps on n-qubit states, including all those for which at least one map is unital. For the class of unital channels on C/sup 2/, we show that additivity of minimal entropy implies that the Holevo (see IEEE Trans. Inform. Theory, vol.44, p.269-73, 1998 and Russian Math. Surv., p.1295-1331, 1999) capacity of the channel is additive over two inputs, achievable with orthogonal states, and equal to the Shannon capacity. This implies that superadditivity of the capacity is possible only for nonunital channels.
Additivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is a unital qubit channel, with the other completely arbitrary. As a byproduct this proves that the Holevo bound is the classical information capacity of such qubit channels, and provides an explicit formula for this classical capacity. Additivity of minimal entropy and multiplicativity of p-norms are also proved under the same assumptions. The proof relies on a new bound for the p-norm of an output state from the half-noisy phase-damping channel.
In this paper we introduce a lemma. This lemma states that for a unitary covariant channel Psi\\PsiPsi and a channel MMM such that M(rho0)M(\\rho_0)M(rho0​) is of rank one for some state rho0\\rho_0rho0​ the additivity of the minimal output entropy of Psi\\PsiPsi implies that of PsicircM\\Psi \\circ MPsicircM, and the multiplicativity of the maximal output ppp-norm of Psi\\PsiPsi, that of PsicircM\\Psi \\circ MPsicircM. Applying the lemma to the depolarising channel we produce a new class of memoryless quantum channels for which the additivity of the minimal output entropy and the multiplicativity of the maximal output ppp-norm hold. This class includes the shifted depolarising channel.