[Paper Review] A paradox regarding monogamy of entanglement

Anna Karlsson|arXiv (Cornell University)|Nov 21, 2019

Quantum Mechanics and Applications29 references4 citations

TL;DR

This paper challenges the foundational assumption of monogamy of entanglement in quantum mechanics by presenting a counterexample in an alternative qubit entanglement model that does not rely on density matrices. The model allows for three qubits to be mutually fully entangled, violating the monogamy constraint inherent in standard density matrix theory, which arises from classical correlation assumptions in the partial trace and purification procedures. The key contribution is showing that monogamy is a theoretical assumption, not a physical necessity, and that entanglement entropy in the new model behaves classically.

ABSTRACT

In density matrix theory, entanglement is monogamous. However, we show that qubits can be arbitrarily entangled in a different, recently constructed model of qubit entanglement [arXiv:1907.11805]. We illustrate the differences between these two models, analyse how the density matrix property of monogamy of entanglement originates in assumptions of classical correlations in the construction of that model, and explain the counterexample to monogamy in the alternative model. We conclude that monogamy of entanglement is a theoretical assumption, not necessarily a physical property, and discuss how contemporary theory relies on that assumption. The properties of entanglement entropy are very different in the two models; a priori, the entropy in the alternative model is classical.

Motivation & Objective

• To challenge the assumption that entanglement must be monogamous in quantum theory.
• To demonstrate that monogamy of entanglement is not a physical necessity but a consequence of specific modeling choices in density matrix theory.
• To construct a counterexample where three qubits are mutually fully entangled in an alternative entanglement model.
• To clarify the role of classical correlations in the partial trace and purification procedures that enforce monogamy in standard quantum mechanics.
• To examine how entanglement entropy differs in the alternative model, where it behaves classically rather than quantum mechanically.

Proposed method

• The paper introduces an alternative model of qubit entanglement based on orthogonal information theory, where correlations are defined via vector-based spin directions and relative angles rather than density matrices.
• It constructs a three-qubit state where all pairs are fully entangled by defining correlations through relative orientations (θ, α) of spin quantization axes.
• The model uses a generalized overlap expression ⟨S₁(a)S₂(b)⟩ = ⟨s₁s₂⟩ × f(θ, α, θ_ab) to compute correlations, with averaging over uncertainty intervals (Δ, δ).
• The key equation (C.10) computes the average correlation as a function of angle deviations, showing non-zero entanglement even when one-to-one correlations are not perfectly matched.
• It contrasts this with the standard density matrix approach, where monogamy arises from the partial trace and classical correlation mediation via local hidden variables (λ).
• The paper analyzes the entropy in the new model, showing it behaves like classical entropy (e.g., not von Neumann), and argues this undermines the assumption that trace-based entropy is fundamental.

Experimental results

Research questions

• RQ1 Is monogamy of entanglement a necessary physical property, or is it a consequence of the mathematical structure of density matrix theory?
• RQ2 Can a consistent model of qubit entanglement be constructed that allows for non-monogamous, mutual full entanglement among three qubits?
• RQ3 How do the assumptions of classical correlations in the partial trace procedure lead to monogamy in standard quantum mechanics?
• RQ4 What are the implications for entanglement entropy when the von Neumann trace-based definition is replaced by a classical-like entropy in an alternative model?
• RQ5 Does the existence of a non-monogamous model imply that current quantum information theory, built on density matrices, may be incomplete or based on unverified assumptions?

Key findings

• A counterexample is constructed where three qubits are mutually fully entangled, violating the monogamy of entanglement as defined in standard density matrix theory.
• The model allows for non-monogamous entanglement by defining correlations through relative spin axis orientations (θ, α) rather than through density matrices.
• The average correlation in the model is given by equation (C.10), which shows non-zero entanglement even when one-to-one spin correlations are absent, provided angular deviations are bounded.
• Monogamy in the density matrix model arises from the classical correlation structure in the partial trace and purification process, not from fundamental quantum principles.
• Entanglement entropy in the alternative model behaves like classical entropy, suggesting that the von Neumann entropy may not be a faithful representation of physical entanglement in all cases.
• The paper concludes that monogamy of entanglement is a theoretical assumption, not a physical law, and that current quantum theory may rest on unverified foundational choices.