Quantum Reverse Shannon Theorem (0912.5537v5)

Abstract: Dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one, reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones, and more generally the use of one noisy channel to simulate another. For channels of nonzero capacity, this simulation is always possible, but for it to be efficient, auxiliary resources of the proper kind and amount are generally required. In the classical case, shared randomness between sender and receiver is a sufficient auxiliary resource, regardless of the nature of the source, but in the quantum case the requisite auxiliary resources for efficient simulation depend on both the channel being simulated, and the source from which the channel inputs are coming. For tensor power sources (the quantum generalization of classical IID sources), entanglement in the form of standard ebits (maximally entangled pairs of qubits) is sufficient, but for general sources, which may be arbitrarily correlated or entangled across channel inputs, additional resources, such as entanglement-embezzling states or backward communication, are generally needed. Combining existing and new results, we establish the amounts of communication and auxiliary resources needed in both the classical and quantum cases, the tradeoffs among them, and the loss of simulation efficiency when auxiliary resources are absent or insufficient. In particular we find a new single-letter expression for the excess forward communication cost of coherent feedback simulations of quantum channels (i.e. simulations in which the sender retains what would escape into the environment in an ordinary simulation), on non-tensor-power sources in the presence of unlimited ebits but no other auxiliary resource. Our results on tensor power sources establish a strong converse to the entanglement-assisted capacity theorem.

• The paper presents a framework for simulating quantum channels using noiseless quantum communication assisted by entanglement, detailing resource tradeoffs.
• A key finding is the Quantum Reverse Shannon Theorem, which explores the resources needed for quantum channel simulation, contrasting with classical requirements.
• The research highlights the significance of entanglement spread and the need for complex resource management, distinguishing quantum simulation from its classical counterpart.

Quantum Reverse Shannon Theorem and Resource Tradeoffs for Simulating Quantum Channels

The paper "The Quantum Reverse Shannon Theorem and Resource Tradeoffs for Simulating Quantum Channels" by Bennett, Devetak, Harrow, Shor, and Winter provides a comprehensive framework for understanding the simulation of quantum channels using various resources. This work parallels classical information theory while highlighting the complexities introduced by quantum mechanics.

Overview and Motivation

In classical information theory, the noisy channel coding theorem establishes how a noisy channel can simulate an ideal noiseless one. Conversely, the classical reverse Shannon theorem discusses using a noiseless channel to simulate a noisy one, assuming shared randomness between sender and receiver. This duality simplifies the characterization of classical channels. However, quantum channels present more complexity due to the possible non-commuting nature of quantum operations and entangled inputs.

Key Concepts and Results

1. Quantum Reverse Shannon Theorem (QRST):

The QRST explores how quantum channels can be simulated using noiseless quantum communication, assisted by entanglement. Unlike classical channels, quantum channels may require additional resources due to entanglement spread and the preservation of quantum coherence.

2. Resource Tradeoffs:

The paper investigates various tradeoffs involving quantum and classical communication resources, including shared entanglement (ebits), forward and backward quantum communication (qubits), and shared randomness (rbits). For instance, while classical shared randomness suffices for simulating classical channels, quantum simulations often require more complex entangled states or entanglement-embezzling states.

3. Entanglement Spread:

A significant finding is the concept of entanglement spread, which quantifies the variability in entanglement across different inputs and the necessity for coherence in superpositions of protocols. Efficient simulation while preserving quantum coherence demands addressing entanglement spread, often requiring embezzling states or backward communication.

4. Tensor Power vs. General Sources:

For tensor power sources, entanglement and communication costs can be precisely characterized. However, for general, potentially correlated inputs, the resource requirements are more complex, necessitating new strategies to handle entanglement spread without excessive resource consumption.

Implications and Future Directions

This research advances the understanding of quantum channel capacities by characterizing the resources needed for their simulation. The findings suggest that simply scaling classical methods does not directly extend to the quantum field due to issues like entanglement spread.

Future research might focus on:

• Developing efficient algorithms to manage entanglement spread without excessive resource use.
• Exploring simulation methods for specific quantum channels, like those used in quantum error correction or quantum networks.
• Investigating the implications of these results on emerging quantum technologies, such as quantum internet and distributed quantum computing.

In conclusion, this paper lays a critical foundation for understanding the simulation of quantum channels, highlighting the nuanced tradeoffs and resource requirements distinct from classical information theory.