[Paper Review] A Condition for the Nullity of Quantum Discord

Animesh Datta|arXiv (Cornell University)|Mar 27, 2010

Quantum Computing Algorithms and Architecture2 references26 citations

TL;DR

This paper establishes that zero quantum discord is equivalent to saturation of strong subadditivity of von Neumann entropy, providing a necessary and sufficient condition for zero discord states. It characterizes these states as pointer states and links their structure to the existence of non-demolition measurements, offering a fundamental information-theoretic boundary between classical and quantum correlations.

ABSTRACT

The positivity of quantum discord is shown to be equivalent to the strong subadditivity of von-Neumann entropy. This leads us to a necessary and sufficient condition characterizing the set of states with zero quantum discord. This also gives us a mathematical definition of pointer states, as they are the states with zero discord. Finally, we suggest that strong subadditivity of entropy might delineate the boundaries of the set of quantum correlations.

Motivation & Objective

• To identify a necessary and sufficient condition for a quantum state to have zero quantum discord.
• To establish a formal mathematical characterization of pointer states based on quantum discord nullity.
• To explore the role of strong subadditivity of von Neumann entropy in distinguishing quantum from classical correlations.
• To propose a criterion based on entropy inequalities to differentiate quantum theory from more general no-signalling theories.
• To provide a constructive framework for identifying zero-discord states in finite- and infinite-dimensional systems.

Proposed method

• Derives quantum discord as the difference between total correlations (quantum mutual information) and classical correlations (maximized over POVMs).
• Uses the Neumark extension to embed a measurement on system B into a unitary evolution on an enlarged system ABC.
• Applies the strong subadditivity inequality to the extended state $\rho'_{ABC}$, showing that equality holds if and only if the state is a short Markov chain.
• Demonstrates that equality in strong subadditivity corresponds to zero quantum discord, leading to a product structure in the eigenbasis of $\rho_B$.
• Characterizes zero-discord states as those where $\rho_{AB} = \sum_j p_j \rho_{A|j} \otimes |\lambda_j\rangle\!\langle\lambda_j|$, diagonal in the eigenbasis of $\rho_B$.
• Extends the characterization to infinite-dimensional systems under finiteness conditions on entropy.

Experimental results

Research questions

• RQ1 What is the necessary and sufficient condition for a quantum state to have zero quantum discord?
• RQ2 How is the saturation of strong subadditivity of von Neumann entropy related to the structure of zero-discord states?
• RQ3 Can pointer states be mathematically defined via the nullity of quantum discord?
• RQ4 Does strong subadditivity of entropy serve as a boundary condition distinguishing quantum from classical or more general no-signalling theories?
• RQ5 What is the structural form of states with zero quantum discord in terms of the eigenbasis of the reduced density matrix $\rho_B$?

Key findings

• Quantum discord is non-negative if and only if the strong subadditivity of von Neumann entropy holds, providing a new proof of this fact.
• Zero quantum discord occurs precisely when the extended state $\rho'_{ABC}$ satisfies equality in strong subadditivity, i.e., forms a short Markov chain.
• States with zero quantum discord have the form $\rho_{AB} = \sum_j p_j \rho_{A|j} \otimes |\lambda_j\rangle\!\langle\lambda_j|$, diagonal in the eigenbasis of $\rho_B$, confirming a product structure.
• Pointer states are mathematically defined as those with zero quantum discord, arising from the ability to extract information via non-demolition measurements.
• The set of zero-discord states is of measure zero in finite-dimensional Hilbert spaces, indicating their rarity.
• The quantity $H(AB) + H(BC) - H(ABC) - H(B)$ is positive for quantum states, zero for classical states, and negative for more general no-signalling theories, suggesting strong subadditivity as a boundary of quantum correlations.