[Paper Review] Violation of Bell inequalities and Quantum Tomography with Pure-states, Werner-states and Maximally Entangled Mixed States created by a Universal Quantum Entangler

Marco Barbieri, F. De Martini+2 more|ArXiv.org|Mar 4, 2003

Quantum Mechanics and Applications3 references3 citations

TL;DR

This paper presents a high-brilliance universal quantum entangler based on spontaneous parametric down-conversion (SPDC) that generates and characterizes pure entangled states, Werner states, and maximally entangled mixed states (MEMS) in a 2×2 Hilbert space. The source enables the first experimental violation of Bell inequalities with a pure-state (source quantum efficiency ≈1) and full quantum tomography of Werner and MEMS, demonstrating precise control over entanglement and mixedness.

ABSTRACT

Entangled pure-states, Werner-states and generalized mixed-states of any structure, spanning a 2x2 Hilbert space are created by a novel high-brilliance universal source of polarization-entangled photon pairs. The violation of a Bell inequality has been tested for the first time with a pure-state, indeed a conceptually relevant ideal condition, and with Werner-states. The generalized ''maximally entangled mixed states'' (MEMS) were also synthetized for the first time and their exotic properties investigated by means of a quantum tomographic technique.

Motivation & Objective

• To develop a universal, high-brilliance source for generating arbitrary entangled states in a 2×2 Hilbert space, overcoming limitations of conventional SPDC sources with low source quantum efficiency (SQE).
• To test Bell inequality violation with a pure entangled state, achieving SQE ≈ 1, thus eliminating the detection loophole and enabling direct measurement of joint detection probabilities.
• To experimentally synthesize and characterize Werner states with variable mixing parameters, investigating their entanglement properties via quantum tomography.
• To generate and fully characterize maximally entangled mixed states (MEMS) for the first time, validating their exotic entanglement structure under controlled decoherence.
• To provide a flexible platform for testing foundational quantum phenomena and advancing quantum information protocols using mixed states as practical resources.

Proposed method

• A thin Type I phase-matched β-barium-borate (BBO) crystal is pumped by a cw UV laser to generate polarization-entangled photon pairs via SPDC.
• A spatial filter with a movable plate (G) controls the overlap of the emission cones, enabling precise tuning of the mixing parameter p and thus the creation of Werner states and MEMS.
• The source achieves high brilliance and source quantum efficiency (SQE ≈ 1), allowing nearly all generated photon pairs to be detected, minimizing the detection loophole.
• Quantum tomography is performed by measuring the density matrix elements using a set of measurement bases, enabling full reconstruction of the density matrix ρ.
• The entanglement content is quantified via concurrence C(ρ) and tangle T = [C(ρ)]², while mixedness is measured via linear entropy SL = 1 − Trρ².
• Theoretical models for Werner states and MEMS are implemented using matrix forms with parameters A, B, C, D, and a piecewise function g(p) to define the MEMS structure.

Experimental results

Research questions

• RQ1 Can Bell inequality violation be demonstrated with a pure entangled state using a source with near-unity source quantum efficiency (SQE ≈ 1), thereby eliminating the detection loophole?
• RQ2 What is the behavior of Bell inequality violation as a function of mixedness in Werner states, and does it vanish at SL = 8/9 as predicted?
• RQ3 Can maximally entangled mixed states (MEMS) be experimentally synthesized and characterized with high fidelity using a universal source?
• RQ4 How accurately can quantum tomography reconstruct the density matrix of Werner states and MEMS, and what are the deviations from theoretical predictions?
• RQ5 What is the relationship between entanglement (tangle T) and mixedness (linear entropy SL) in experimentally generated mixed states?

Key findings

• The first experimental violation of a Bell inequality was achieved with a pure entangled state, with a Bell parameter S = 1.048 ± 0.011, confirming the violation in a conceptually ideal setup with SQE ≈ 1.
• For Werner states, Bell inequality violation was observed only when linear entropy SL < 1/2, consistent with theoretical predictions that entanglement becomes non-distillable and Bell non-violating for SL > 8/9.
• Quantum tomography of a Werner state with p ≈ 0.42 (SL ≈ 0.82) accurately reproduced the theoretical density matrix, with experimental data on tangle T(SL) showing good agreement with the theoretical curve T(SL) = ¼(1 − 3√(1 − SL))².
• The first experimental synthesis and tomographic characterization of maximally entangled mixed states (MEMS) was achieved, with a state having p = 0.56 and g(p) = 1/3, though agreement with theory was less accurate than for Werner states.
• The tangle T of the MEMS state was experimentally determined, confirming its status as the maximally entangled state for a given mixedness, validating the theoretical framework of MEMS.
• The source demonstrated high flexibility, enabling the generation of any bi-partite 2×2 state via control of the spatial overlap parameter p and the filter plate position Δx.