Quantum capacity

Introduction

Quantum capacity is a fundamental concept in the field of quantum information theory, which quantifies the maximum rate at which quantum information can be reliably transmitted over a quantum channel. This capacity is crucial for understanding the limits of quantum communication and plays a significant role in the development of quantum computing and quantum cryptography.

Quantum Channels

A quantum channel is a physical medium through which quantum information is transmitted. It can be described mathematically as a completely positive, trace-preserving (CPTP) map acting on a density matrix. Quantum channels can be affected by various types of noise, such as decoherence and depolarization, which degrade the transmitted information.

Quantum Capacity Definition

The quantum capacity of a quantum channel is defined as the maximum number of qubits that can be transmitted per channel use with an arbitrarily low error rate, in the limit of many channel uses. Mathematically, it is given by the regularized coherent information:

\[ Q(\mathcal{N}) = \lim_{n \to \infty} \frac{1}{n} I_c(\mathcal{N}^{\otimes n}), \]

where \( \mathcal{N} \) is the quantum channel, \( \mathcal{N}^{\otimes n} \) denotes \( n \) uses of the channel, and \( I_c \) is the coherent information.

Coherent Information

Coherent information is a measure of the amount of quantum information that can be preserved through a quantum channel. For a given quantum state \( \rho \) and a quantum channel \( \mathcal{N} \), the coherent information is defined as:

\[ I_c(\rho, \mathcal{N}) = S(\mathcal{N}(\rho)) - S(\mathcal{N}, \rho), \]

where \( S \) denotes the von Neumann entropy.

Quantum Error Correction

To achieve the quantum capacity, it is essential to employ quantum error correction techniques. These techniques protect quantum information from errors induced by noise in the quantum channel. Quantum error correction codes, such as Shor code and Steane code, are designed to detect and correct errors without measuring the quantum information directly.

Entanglement-Assisted Quantum Capacity

The entanglement-assisted quantum capacity is a variant of quantum capacity that considers the use of pre-shared entanglement between the sender and receiver. This capacity is given by the quantum mutual information:

\[ Q_E(\mathcal{N}) = \max_{\rho} I(\rho, \mathcal{N}), \]

where \( I(\rho, \mathcal{N}) \) is the quantum mutual information of the state \( \rho \) through the channel \( \mathcal{N} \).

Superadditivity and Regularization

One of the unique features of quantum capacity is the phenomenon of superadditivity, where the capacity of multiple uses of a channel can be greater than the sum of the capacities of individual uses. This necessitates the use of regularization in the definition of quantum capacity, making its computation challenging.

Practical Implications

Understanding quantum capacity has significant implications for the development of quantum networks, quantum internet, and secure communication protocols. It provides insights into the fundamental limits of quantum communication and guides the design of efficient quantum communication systems.

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