[Paper Review] Exact interferometers for the concurrence and residual 3-tangle

Hilary A. Carteret|arXiv (Cornell University)|Sep 29, 2003

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper presents exact quantum circuits to measure the concurrence of two-qubit mixed states and the residual 3-tangle for pure three-qubit states without requiring state tomography or the Structural Physical Approximation (SPA). The method uses generalized interferometers with controlled unitary operations and visibility proportional to entanglement invariants, enabling direct measurement of concurrence and 3-tangle via interference patterns, with improved robustness against state-dependent errors compared to SPA-based approaches.

ABSTRACT

In this paper we describe a set of circuits that can measure the concurrence of a two qubit density matrix without requiring the deliberate addition of noise. We then extend these methods to obtain a circuit to measure one type of three qubit entanglement for pure states, namely the 3-tangle.

Motivation & Objective

• To develop a method for measuring bipartite entanglement (concurrence) in two-qubit mixed states without reconstructing the density matrix.
• To extend the interferometric approach to measure the residual 3-tangle for pure three-qubit states.
• To avoid the state-dependent errors inherent in the Structural Physical Approximation (SPA) by using direct interferometric measurement of entanglement invariants.
• To reduce the number of state copies required compared to full state tomography, particularly for three-qubit entanglement.

Proposed method

• Generalizes the Mach-Zehnder interferometer to use controlled completely positive maps via convex combinations of unitaries, enabling measurement of trace functionals of density matrices.
• Employs a visibility proportional to the real part of the trace of a unitary-evolved density matrix, allowing direct measurement of entanglement invariants like concurrence.
• Uses the Wootters spin-flip operation implicitly via a controlled circuit structure that measures |Tr(Uρ)|, which is related to the concurrence via eigenvalue decomposition.
• Constructs a circuit for the 3-tangle using controlled operations on three qubits, with visibility proportional to |τ_ABC|², derived from the sum of squared eigenvalues of ρ_AB~ρ_AB and ρ_AC~ρ_AC.
• Implements the 3-tangle measurement via a superposition of circuits with and without matrix multiplication, using a |−⟩ ancilla to induce interference with a phase shift.
• Combines one-party and two-party circuits to measure Tr(ρ_i²) and τ_AB, enabling reconstruction of the 3-tangle using the identity τ_ABC = 4det(ρ_A) − τ_AB − τ_AC.

Experimental results

Research questions

• RQ1 Can the concurrence of a two-qubit mixed state be measured directly without state reconstruction or the SPA, using only interferometric techniques?
• RQ2 Is it possible to design a circuit that measures the residual 3-tangle for pure three-qubit states using only a small number of state copies and standard quantum operations?
• RQ3 How can the visibility of interferometric circuits be optimized to measure entanglement invariants like concurrence and 3-tangle with high sensitivity?
• RQ4 What is the minimal number of quantum circuits required to measure the 3-tangle, and how does this compare to full state tomography?
• RQ5 Can the 3-tangle be measured via a single circuit with visibility proportional to |τ_ABC|², and what is the required control structure?

Key findings

• The concurrence can be measured using a generalized interferometer with visibility proportional to |Tr(Uρ)|, avoiding the need for the SPA and its state-dependent errors.
• The method for measuring concurrence is applicable to any two-qubit mixed state and does not require prior knowledge of the density matrix.
• For pure three-qubit states, the residual 3-tangle can be measured using five circuits: two for concurrence, three for single-party purity, with a total of 14 circuits possible via optimization.
• The 3-tangle can be measured via a visibility proportional to |τ_ABC|² = 16λ₁²λ₂², derived from Tr((ρ_AB~ρ_AB)²) and Tr(ρ_AB~ρ_AB)², enabling direct interferometric detection.
• The proposed circuit for |τ_ABC|² requires only four copies of the state per run, significantly fewer than the 22 copies needed by the first method, improving resource efficiency.
• The method avoids the need for full state tomography, reducing the number of required measurements from 16 parameters to just 14 circuits for pure three-qubit states.