Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0103086 · Journal of Mathematical Physics

Maximization of capacity and lp norms for some product channels

Christopher King

Published Mar 1, 2002• 52 citations
arXivHTMLPDF

Abstract

It is conjectured that the Holevo capacity of a product channel Ω⊗Φ is achieved when product states are used as input. Amosov, Holevo, and Werner have also conjectured that the maximal lp norm of a product channel is achieved with product input states. In this article we establish both of these conjectures in the case that Ω is arbitrary and Φ is a CQ or QC channel (as defined by Holevo). We also establish the Amosov, Holevo and Werner conjecture when Ω is arbitrary and either Φ is a qubit channel and p=2, or Φ is a unital qubit channel and p is integer. Our proofs involve a new conjecture for the norm of an output state of the half-noisy channel I⊗Φ, when Φ is a qubit channel. We show that this conjecture in some cases also implies additivity of the Holevo capacity.

DOI
10.1063/1.1433943

Related papers

Additivity for unital qubit channels

Christopher King

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.

arXiv:quant-ph/0103156•Journal of Mathematical Physics•Sep 26, 2002•202 citations

Qubit channels which require four inputs to achieve capacity: Implications for additivity conjectures

Masahito Hayashi, H. Imai, Keiji Matsumoto +2 more

An example is given of a qubit quantum channel which requires four inputs to maximize the Holevo capacity. The example is one of a family of channels which are related to 3-state channels. The capacity of the product channel is studied and numerical evidence presented which strongly suggests additivity. The numerical evidence also supports a conjecture about the concavity of output entropy as a function of entanglement parameters. However, an example is presented which shows that for some channels this conjecture does not hold for all input states. A numerical algorithm for finding the capacity and optimal inputs is presented and its relation to a relative entropy optimization discussed.

arXiv:quant-ph/0403176•Quantum Information and Computation•Jan 1, 2005•48 citations

The capacity of the quantum depolarizing channel

Christopher King

The information-carrying capacity of the d-dimensional depolarizing channel is computed. It is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space, and using measurements which are products of projections onto this same orthonormal basis. In other words, neither entangled signal states nor entangled measurements give any advantage for information capacity. The result follows from an additivity theorem for the product channel /spl Delta//spl ominus//spl Psi/, where /spl Delta/ is the depolarizing channel and /spl Psi/ is a completely arbitrary channel. We establish the Amosov-Holevo-Werner(see Probl. Inform. Transm., vol.36, p.305-313, 2000) p-norm conjecture for this product channel for all p/spl ges/1, and deduce from this the additivity of the minimal entropy and of the Holevo quantity /spl chi//sup */.

arXiv:quant-ph/0204172•IEEE Transactions on Information Theory•Jan 1, 2003•290 citations

Conditions for multiplicativity of maximal ℓp-norms of channels for fixed integer p

Vittorio Giovannetti, Seth Lloyd, Mary Beth Ruskai

We introduce a condition for memoryless quantum channels which, when satisfied guarantees the multiplicativity of the maximal ℓp-norm with p a fixed integer. By applying the condition to qubit channels, it can be shown that it is not a necessary condition, although some known results for qubits can be recovered. When applied to the Werner-Holevo channel, which is known to violate multiplicativity when p is large relative to the dimension d, the condition suggests that multiplicativity holds when d⩾2p−1. This conjecture is proved explicitly for p=2,3,4. Finally, a new class of channels is considered which generalizes the depolarizing channel to maps which are combinations of the identity channel and a noisy one whose image is an arbitrary density matrix. It is shown that these channels are multiplicative for p=2.

arXiv:quant-ph/0408103•Journal of Mathematical Physics•Mar 22, 2005•28 citations

Maximal p-norms of entanglement breaking channels

Christopher King

It shown that when one of the components of a product channel is entanglement breaking, the output state with maximal p-norm is always a product state. This result complements Shor's theorem that both minimal entropy and Holevo capacity are additive for entanglement breaking channels. It is also shown how Shor's results can be recovered from the p-norm results by considering their behavior for p close to one.

arXiv:quant-ph/0212057•Quantum Information and Computation•Mar 1, 2003•68 citations

Minimal entropy of states emerging from noisy quantum channels

Christopher King, Mary Beth Ruskai

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.

arXiv:quant-ph/9911079•IEEE Transactions on Information Theory•Jan 1, 2001•345 citations
It is conjectured that the Holevo capacity of a product channel Ω⊗Φ is achieved when product states are used as input. Amosov, Holevo, and Werner have also conjectured that the maximal lp norm of a product channel is achieved with product input states. In this article we establish both of these conjectures in the case that Ω is arbitrary and Φ is a CQ or QC channel (as defined by Holevo). We also establish the Amosov, Holevo and Werner conjecture when Ω is arbitrary and either Φ is a qubit channel and p=2, or Φ is a unital qubit channel and p is integer. Our proofs involve a new conjecture for the norm of an output state of the half-noisy channel I⊗Φ, when Φ is a qubit channel. We show that this conjecture in some cases also implies additivity of the Holevo capacity.
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.
An example is given of a qubit quantum channel which requires four inputs to maximize the Holevo capacity. The example is one of a family of channels which are related to 3-state channels. The capacity of the product channel is studied and numerical evidence presented which strongly suggests additivity. The numerical evidence also supports a conjecture about the concavity of output entropy as a function of entanglement parameters. However, an example is presented which shows that for some channels this conjecture does not hold for all input states. A numerical algorithm for finding the capacity and optimal inputs is presented and its relation to a relative entropy optimization discussed.
The information-carrying capacity of the d-dimensional depolarizing channel is computed. It is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space, and using measurements which are products of projections onto this same orthonormal basis. In other words, neither entangled signal states nor entangled measurements give any advantage for information capacity. The result follows from an additivity theorem for the product channel /spl Delta//spl ominus//spl Psi/, where /spl Delta/ is the depolarizing channel and /spl Psi/ is a completely arbitrary channel. We establish the Amosov-Holevo-Werner(see Probl. Inform. Transm., vol.36, p.305-313, 2000) p-norm conjecture for this product channel for all p/spl ges/1, and deduce from this the additivity of the minimal entropy and of the Holevo quantity /spl chi//sup */.
We introduce a condition for memoryless quantum channels which, when satisfied guarantees the multiplicativity of the maximal ℓp-norm with p a fixed integer. By applying the condition to qubit channels, it can be shown that it is not a necessary condition, although some known results for qubits can be recovered. When applied to the Werner-Holevo channel, which is known to violate multiplicativity when p is large relative to the dimension d, the condition suggests that multiplicativity holds when d⩾2p−1. This conjecture is proved explicitly for p=2,3,4. Finally, a new class of channels is considered which generalizes the depolarizing channel to maps which are combinations of the identity channel and a noisy one whose image is an arbitrary density matrix. It is shown that these channels are multiplicative for p=2.
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.