[Paper Review] Quantitative complementarity relations in bipartite systems

Matthias Jakob, János A. Bergou|ArXiv.org|Feb 10, 2003

Advanced Algebra and Logic1 references10 citations

TL;DR

This paper introduces a triality relation in two-qubit systems that unifies wave-particle duality (via visibility and predictability) with bipartite entanglement (via concurrence), showing that these three quantum properties are mutually exclusive and quantitatively linked. The key contribution is a complete complementarity relation ${\cal{C}}^2 + {\cal{V}}_k^2 + {\cal{P}}_k^2 = 1$ for pure states, revealing that Bell inequality violations arise from the exclusion of single-particle realities due to maximal entanglement.

ABSTRACT

We introduce a complete set of complementary quantities in bipartite, two-dimensional systems. Complementarity then relates the quantitative entanglement measure concurrence which is a bipartite property to the single-particle quantum properties predictability and visibility, for the most general quantum state of two qubits. Consequently, from an interferometric point of view, the usual wave-particle duality relation must be extended to a ``triality'' relation containing, in addition, the quantitative entanglement measure concurrence, which has no classical counterpart and manifests a genuine quantum aspect of bipartite systems. A generalized duality relation, that also governs possible violations of the Bell's inequality, arises between single- and bipartite properties.

Motivation & Objective

• To extend wave-particle duality to bipartite systems by incorporating entanglement as a fundamental quantum property.
• To derive a quantitative complementarity relation involving concurrence, single-particle visibility, and predictability for two-qubit systems.
• To demonstrate that violations of Bell’s inequality are governed by the same complementarity principle that unifies local and nonlocal quantum properties.
• To show that entanglement, quantified by concurrence, is operationally measurable through interferometric visibility and predictability.

Proposed method

• Define the most general pure two-qubit state $|\Theta\rangle = a|00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$ with normalization $|a|^2 + |b|^2 + |c|^2 + |d|^2 = 1$.
• Introduce concurrence as ${\cal{C}} = 2|ad - bc|$, a measure of bipartite entanglement derived from the state and its spin-flipped version.
• Define single-particle visibility ${\cal{V}}_k = 2|\langle\Theta|\sigma_k^+|\Theta\rangle|$ and predictability ${\cal{P}}_k = |\langle\Theta|\sigma_{z,k}|\Theta\rangle|$ for each qubit, representing local wave and particle aspects.
• Derive the triality relation ${\cal{C}}^2 + {\cal{V}}_k^2 + {\cal{P}}_k^2 = 1$ for pure states, showing mutual exclusivity of the three quantum properties.
• Extend the relation to mixed states via the inequality ${\cal{C}}^2 + {\cal{V}}_k^2 + {\cal{P}}_k^2 \leq 1$, noting that two-particle visibility no longer directly relates to concurrence in mixed systems.
• Relate the complementarity relation to Bell inequality violations by expressing the maximum Bell violation ${\cal{B}} = 2\sqrt{2 - {\cal{S}}_k^2}$, where ${\cal{S}}_k^2 = {\cal{V}}_k^2 + {\cal{P}}_k^2$.

Experimental results

Research questions

• RQ1 How can wave-particle duality in single systems be generalized to include entanglement in bipartite systems?
• RQ2 What quantitative relation unifies entanglement (concurrence), single-particle visibility, and predictability in two-qubit systems?
• RQ3 How does the complementarity between local and nonlocal properties govern the violation of Bell’s inequality?
• RQ4 Why does the direct link between two-particle visibility and concurrence break down in mixed quantum states?
• RQ5 Can entanglement be operationally quantified using only measurable single-particle properties like visibility and predictability?

Key findings

• The paper derives a triality complementarity relation ${\cal{C}}^2 + {\cal{V}}_k^2 + {\cal{P}}_k^2 = 1$ for pure two-qubit states, showing that concurrence, visibility, and predictability are mutually exclusive quantum properties.
• The concurrence ${\cal{C}} = 2|ad - bc|$ is shown to be a direct measure of bipartite entanglement with no classical counterpart, emerging naturally in the complementarity framework.
• For pure states, the maximum violation of Bell’s inequality occurs when ${\cal{S}}_k^2 = {\cal{V}}_k^2 + {\cal{P}}_k^2 = 0$, meaning single-particle realities are fully excluded, leading to ${\cal{B}} = 2\sqrt{2}$.
• In mixed states, the inequality ${\cal{C}}^2 + {\cal{V}}_k^2 + {\cal{P}}_k^2 \leq 1$ holds, but the direct connection between concurrence and two-particle visibility is lost.
• The study reveals that the violation of Bell’s inequality is not independent but is governed by the same complementarity principle that unifies single-particle and bipartite quantum properties.
• Entanglement is operationally quantified via measurable single-particle observables, establishing concurrence as an operational concept grounded in interferometric measurements.