[Paper Review] Bell-type inequality in quantum coherence theory as an entanglement witness

Kaifeng Bu, Asutosh Kumar+1 more|arXiv (Cornell University)|Mar 21, 2016

Quantum Mechanics and Applications3 references9 citations

TL;DR

This paper introduces Bell-type inequalities in quantum coherence theory that serve as entanglement witnesses for two-qubit pure states. By analyzing relative entropy and $l_1$-norm of coherence across specific measurement bases, it demonstrates that all two-qubit pure entangled states violate the relative entropy-based inequality, confirming its role as a universal entanglement witness.

ABSTRACT

Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative entropy of coherence is violated by all two-qubit pure entangled states, serving as an entanglement witness.

Motivation & Objective

• To establish a connection between quantum coherence measures and entanglement detection via Bell-type inequalities.
• To investigate whether coherence-based Bell-type inequalities can serve as entanglement witnesses for pure bipartite states.
• To determine the conditions under which different coherence measures (relative entropy and $l_1$-norm) lead to violation of these inequalities.
• To explore the robustness and universality of coherence-based inequalities in detecting entanglement across different quantum states.

Proposed method

• Construct Bell-type inequalities using four observables: $Q$, $R$, $S$, and $T$, defined in specific computational bases.
• Define coherence measures: relative entropy of coherence $\mathcal{C}_r$ and $l_1$-norm of coherence $\mathcal{C}_{l_1}$, applied to bipartite states.
• Evaluate coherence values for a two-qubit pure state $\ket{\Psi} = \cos(\theta/2)\ket{00} + e^{i\phi}\sin(\theta/2)\ket{11}$ under different measurement bases.
• Derive expressions for coherence sums and differences: $\mathcal{C}_r(Q,S) + \mathcal{C}_r(R,S) + \mathcal{C}_r(R,T) - \mathcal{C}_r(Q,T)$ and similar for $l_1$-norm.
• Analyze the inequality violation condition: whether the expression exceeds a threshold (e.g., 4 or 8) depending on the coherence measure.
• Use Shannon entropy $h(p,1-p) = -p\log_2 p - (1-p)\log_2(1-p)$ to compute coherence values in terms of $\theta$.

Experimental results

Research questions

• RQ1 Can Bell-type inequalities constructed from quantum coherence measures detect entanglement in two-qubit pure states?
• RQ2 Does the relative entropy of coherence-based Bell inequality violate for all two-qubit pure entangled states?
• RQ3 Are there limitations in using $l_1$-norm of coherence for entanglement detection via Bell-type inequalities?
• RQ4 What is the maximal violation of these coherence-based Bell inequalities, and how does it vary with entanglement degree?
• RQ5 Do these coherence-based inequalities also detect entanglement in two-qubit mixed states?

Key findings

• The relative entropy of coherence-based Bell inequality is violated by all two-qubit pure entangled states, with the expression exceeding 4 for $\theta \in (0, \pi)$.
• The $l_1$-norm of coherence-based inequality is violated only when $\theta \in (\arcsin(1/3), \pi - \arcsin(1/3))$, indicating partial detection capability.
• For maximally entangled states ($\theta = \pi/2$), both coherence measures lead to violation of their respective Bell-type inequalities.
• The relative entropy-based inequality yields a violation of $4 + h(\cos^2(\theta/2), \sin^2(\theta/2))$, which is always greater than 4 for entangled states.
• The $l_1$-norm-based expression reaches a maximum violation of $7 + 3\sin\theta$, exceeding 8 when $\sin\theta > 1/3$.
• The inequality for relative entropy of coherence is violated for all $\theta \in (0, \pi)$, confirming its universality as an entanglement witness.