Quantum discord

Introduction

Quantum discord is a measure of quantum correlations in a bipartite quantum system. It quantifies the amount of non-classical correlations between two subsystems, which includes but is not limited to entanglement. The concept was introduced by Harold Ollivier and Wojciech Zurek in 2001 and independently by Henderson and Vedral in the same year. Quantum discord has since become a significant topic in the study of quantum information theory.

Definition and Mathematical Formulation

Quantum discord is defined as the difference between two expressions of mutual information in a quantum system. In classical information theory, mutual information is a measure of the total correlations between two random variables. In the quantum realm, mutual information can be generalized in two different ways, leading to the concept of quantum discord.

Let \(\rho_{AB}\) be the density matrix of a bipartite quantum system composed of subsystems \(A\) and \(B\). The total mutual information \(I(A:B)\) is given by: \[ I(A:B) = S(\rho_A) + S(\rho_B) - S(\rho_{AB}), \] where \(S(\rho)\) is the von Neumann entropy of the density matrix \(\rho\).

The quantum conditional entropy \(S(A|B)\) is defined as: \[ S(A|B) = S(\rho_{AB}) - S(\rho_B). \]

An alternative expression for mutual information, based on a measurement on subsystem \(B\), is: \[ J(A:B) = S(\rho_A) - S(A|B), \] where \(S(A|B)\) is the conditional entropy after a measurement on \(B\).

Quantum discord \(D(A:B)\) is then defined as the difference between these two expressions: \[ D(A:B) = I(A:B) - J(A:B). \]

Properties of Quantum Discord

Quantum discord has several key properties that distinguish it from other measures of quantum correlations:

**Non-negativity**: Quantum discord is always non-negative, \(D(A:B) \geq 0\).
**Asymmetry**: Quantum discord is generally asymmetric, meaning \(D(A:B) \neq D(B:A)\).
**Zero Discord States**: States with zero discord are known as classical-quantum states. These states can be written as:

 \[ \rho_{AB} = \sum_i p_i \rho_A^i \otimes |i\rangle_B \langle i|, \]
 where \(\{ |i\rangle_B \}\) is an orthonormal basis for subsystem \(B\), \(\rho_A^i\) are density matrices for subsystem \(A\), and \(p_i\) are probabilities.

Quantum Discord in Quantum Information Processing

Quantum discord has been found to be a useful resource in various quantum information processing tasks. Unlike entanglement, which is fragile and can be easily destroyed by decoherence, quantum discord can persist in mixed states and noisy environments. This makes it a valuable resource for quantum computation and communication.

Quantum Algorithms

Quantum discord plays a role in the efficiency of certain quantum algorithms. For example, the DQC1 (Deterministic Quantum Computation with One Qubit) model, which uses only one qubit and a maximally mixed state, can solve certain problems more efficiently than classical algorithms. The presence of non-zero quantum discord in the DQC1 model suggests that quantum correlations other than entanglement can provide a computational advantage.

Quantum Communication

In quantum communication protocols, such as quantum key distribution and quantum teleportation, quantum discord can enhance the performance and security of these protocols. Even in the absence of entanglement, non-classical correlations quantified by quantum discord can be exploited to achieve secure communication.

Experimental Realizations

Quantum discord has been experimentally observed and measured in various physical systems, including:

**Nuclear Magnetic Resonance (NMR)**: NMR systems have been used to create and measure quantum discord in liquid-state molecules.
**Optical Systems**: Quantum discord has been demonstrated using entangled photon pairs and optical interferometers.
**Solid-State Systems**: Quantum discord has been observed in solid-state qubits, such as superconducting circuits and quantum dots.

These experimental realizations have provided insights into the behavior of quantum correlations in different physical contexts and have paved the way for practical applications of quantum discord.

Theoretical Developments

The study of quantum discord has led to several theoretical advancements in quantum information theory. Researchers have developed various methods to quantify and compute quantum discord, including analytical expressions for specific classes of states and numerical algorithms for general states.

Discord Monotones

Discord monotones are functions that quantify quantum discord and satisfy certain properties, such as non-negativity and invariance under local unitary operations. Examples of discord monotones include the geometric discord and the relative entropy of discord.

Discord and Quantum Phase Transitions

Quantum discord has been used to study quantum phase transitions in many-body systems. It has been shown that quantum discord can signal critical points in quantum systems, even in the absence of entanglement. This has provided a new perspective on the role of quantum correlations in phase transitions.

References

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