[Paper Review] Super-Activation of Zero-Error Capacity of Noisy Quantum Channels

Runyao Duan|ArXiv.org|Jun 15, 2009

Quantum Computing Algorithms and Architecture53 citations

TL;DR

This paper demonstrates that certain noisy quantum channels can have zero classical zero-error capacity individually yet achieve perfect classical communication when used jointly with entangled inputs across multiple uses. The key contribution is the first explicit construction of a single-sender, single-receiver quantum channel that exhibits super-activation: one use is ineffective, but two uses with entanglement enable error-free transmission of at least one classical bit—impossible in classical channels and highlighting the non-classical power of quantum entanglement in zero-error communication.

ABSTRACT

We study various super-activation effects in the following zero-error communication scenario: One sender wants to send classical or quantum information through a noisy quantum channel to one receiver with zero probability of error. First we show that there are quantum channels of which a single use is not able to transmit classical information perfectly yet two uses can. This is achieved by employing entangled input states between different uses of the given channel and thus cannot happen for classical channels. Second we exhibit a class of quantum channel with vanishing zero-error classical capacity such that when a noiseless qubit channel or one ebit shared entanglement are available, it can be used to transmit $\log_2 d$ noiseless qubits, where 2d is the dimension of input state space. Third we further construct quantum channels with vanishing zero-error classical capacity when assisted with classical feedback can be used to transmit both classical and quantum information perfectly. These striking findings not only indicate both the zero-error quantum and classical capacities of quantum channels satisfy a strong super-additivity beyond any classical channels, but also highlight the activation power of auxiliary physical resources in zero-error communication.

Motivation & Objective

• To resolve the open question of whether a single-sender, single-receiver quantum channel can exhibit super-activation of zero-error classical capacity.
• To investigate how entanglement between channel uses enables perfect classical communication despite individual uses being ineffective.
• To explore the role of auxiliary resources—such as shared entanglement, classical feedback, and noiseless qubit channels—in enhancing zero-error capacity.
• To establish that zero-error classical and quantum capacities in quantum channels are strongly super-additive beyond classical limits.

Proposed method

• Construct two quantum channels 𝒪 and 𝒫 such that each individually has zero zero-error classical capacity, but their joint use with entangled input states enables error-free classical communication.
• Use the Choi-Jamiołkowski isomorphism to map quantum channels to bipartite quantum states and analyze unextendibility properties of subspaces associated with the channels.
• Employ matrix subspaces S₀ and S₁ that are unextendible and satisfy specific symmetry and self-adjointness conditions to ensure no single-use transmission is possible.
• Demonstrate that the tensor product S₀ ⊗ S₁ is unextendible, implying that the joint channel can transmit one bit perfectly via entangled input states.
• Introduce a class of quantum channels with vanishing zero-error classical capacity that become capable of transmitting log₂d noiseless qubits when assisted by a noiseless qubit channel or one ebit of shared entanglement.
• Construct a channel that, when combined with classical feedback, enables perfect transmission of both classical and quantum information despite having zero zero-error classical capacity on its own.

Experimental results

Research questions

• RQ1 Can a single-sender, single-receiver quantum channel exhibit super-activation of zero-error classical capacity, where one use is ineffective but two uses with entangled input enable perfect classical communication?
• RQ2 Is it possible for two quantum channels, each with zero zero-error classical capacity, to jointly achieve positive zero-error capacity when used together with entangled inputs across uses?
• RQ3 How do auxiliary resources such as shared entanglement, classical feedback, or noiseless qubit channels affect the zero-error capacity of quantum channels with initially vanishing capacity?
• RQ4 Can the zero-error quantum capacity be super-activated in a way analogous to the classical capacity, and is there a super-multiplicative behavior in the unextendibility of quantum state subspaces?
• RQ5 What is the strongest form of super-additivity possible for zero-error classical capacity, i.e., can two channels with zero capacity jointly achieve positive capacity?

Key findings

• An explicit quantum channel 𝒢 with one sender and one receiver is constructed such that a single use cannot transmit classical information perfectly, but two uses with entangled input states can transmit at least one classical bit with zero error.
• The super-activation effect is achieved through entanglement between different uses of the channel, a phenomenon impossible in classical channels and not observable under product-state input constraints.
• A class of quantum channels with vanishing zero-error classical capacity can transmit log₂d noiseless qubits when assisted by a noiseless qubit channel or one ebit of shared entanglement.
• Classical feedback enables perfect transmission of both classical and quantum information through a quantum channel that has zero zero-error classical capacity when used alone.
• The zero-error classical and quantum capacities of quantum channels exhibit strong super-additivity, indicating a fundamental quantum advantage beyond classical communication limits.
• The paper leaves open the question of whether the strongest super-additivity—two channels with zero capacity jointly achieving positive capacity—can be realized, though candidate channels are identified.