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Quantum Entanglement and Conditional Information Transmission

Robert R. Tucci

TL;DR

The paper introduces a quantum entanglement measure based on conditional information transmission in Quantum Bayesian Nets and proves its equivalence to the Entanglement of Formation for bipartite pure states. It generalizes the measure to n-partite systems and develops a rich framework of definitions, dualities, and inequalities that connect tanglement with conditional mutual information and data processing concepts. For pure states, optimizing the measure over local unitaries recovers the familiar $E_F$, while mixed-state results show zero tanglement for conditionally separable states and structured expressions for Bell-diagonal mixtures. The work presents a set of properties, decomposition rules, and conditional data processing inequalities that underpin a general, information-theoretic approach to entanglement quantification, with potential implications for classical mutual information bounds. Overall, it offers a novel, scalable lens on entanglement that unifies quantum and classical information notions within Quantum Bayesian Net formalism.

Abstract

We propose a new measure of quantum entanglement. Our measure is defined in terms of conditional information transmission for a Quantum Bayesian Net. We show that our measure is identically equal to the Entanglement of Formation in the case of a bipartite (two listener) system occupying a pure state. In the case of mixed states, the relationship between these two measures is not known yet. We discuss some properties of our measure. Our measure can be easily and naturally generalized to handle n-partite (n-listener) systems. It is non-negative for any n. It vanishes for conditionally separable states with n listeners. It is symmetric under permutations of the n listeners. It decreases if listeners are merged, pruned or removed. Most promising of all, it is intimately connected with the Data Processing Inequalities. We also find a new upper bound for classical mutual information which is of interest in its own right.

Quantum Entanglement and Conditional Information Transmission

TL;DR

The paper introduces a quantum entanglement measure based on conditional information transmission in Quantum Bayesian Nets and proves its equivalence to the Entanglement of Formation for bipartite pure states. It generalizes the measure to n-partite systems and develops a rich framework of definitions, dualities, and inequalities that connect tanglement with conditional mutual information and data processing concepts. For pure states, optimizing the measure over local unitaries recovers the familiar , while mixed-state results show zero tanglement for conditionally separable states and structured expressions for Bell-diagonal mixtures. The work presents a set of properties, decomposition rules, and conditional data processing inequalities that underpin a general, information-theoretic approach to entanglement quantification, with potential implications for classical mutual information bounds. Overall, it offers a novel, scalable lens on entanglement that unifies quantum and classical information notions within Quantum Bayesian Net formalism.

Abstract

We propose a new measure of quantum entanglement. Our measure is defined in terms of conditional information transmission for a Quantum Bayesian Net. We show that our measure is identically equal to the Entanglement of Formation in the case of a bipartite (two listener) system occupying a pure state. In the case of mixed states, the relationship between these two measures is not known yet. We discuss some properties of our measure. Our measure can be easily and naturally generalized to handle n-partite (n-listener) systems. It is non-negative for any n. It vanishes for conditionally separable states with n listeners. It is symmetric under permutations of the n listeners. It decreases if listeners are merged, pruned or removed. Most promising of all, it is intimately connected with the Data Processing Inequalities. We also find a new upper bound for classical mutual information which is of interest in its own right.

Paper Structure

This paper contains 9 sections, 198 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Entanglement of Formation
  3. Some Definitions
  4. $ST$ for 2 Single-node Listeners and a Pure State
  5. $ST$ for 2 Single-Node Listeners and a Mixed State
  6. Properties of Tanglement and C.M.I.
  7. Proof that $E_F(|\psi_{xy}|)\leq E_F(\psi_{xy})$
  8. Proof that $H({\ul{x}} : {\ul{y}}) \leq E_F(\sqrt{P_{xy}})$
  9. Entropy of Density Matrix with Repeated Index Pairs

Figures (8)

  • Figure 1: CB net in which ${\ul{a}}$ and ${\ul{b}}$ are conditionally independent.
  • Figure 2: Plot of functions $p_0(t)$ and $h(p_0)$.
  • Figure 3: Net for 2 single-node listeners and a pure state.
  • Figure 4: Net that implements a 2 choice conditionally separable density matrix.
  • Figure 5: Net for 2 single-node listeners and a mixed state.
  • ...and 3 more figures