[Paper Review] Concurrence of Lorentz-positive maps

Roland Hildebrand|ArXiv.org|Dec 8, 2006

Advanced Topics in Algebra12 references3 citations

TL;DR

This paper introduces explicit formulae for the concurrence and I-fidelity of positive operators and bipartite density matrices by leveraging Lorentz-positive maps and generalized eigenvalue pencils. It establishes that for operators with input space 𝒟(2), the concurrence and I-fidelity reduce to convex and concave roofs over Lorentz cones, yielding closed-form expressions using the second largest and smallest generalized eigenvalues of a matrix pencil, respectively.

ABSTRACT

Let L_n be the n-dimensional Lorentz cone. A linear map M from R^m to R^n is called Lorentz-positive if M[L_m] is contained in L_n. We extend the notion of concurrence, which was initially introduced to quantify the entanglement of bipartite density matrices, to Lorentz-positive maps and provide an explicite formula for it. This allows us to obtain formulae for the concurrence of arbitrary positive operators taking 2 x 2 complex hermitian matrices as input and consequently of arbitrary bipartite density matrices of rank 2. Namely, let P: H(2) o H(d) be a positive operator, and let λ_1,...,λ_4 be the generalized eigenvalues of the pencil σ_2(P(X)) - λdet X, in decreasing order, where σ_2 is the second symmetric function of the spectrum. Then the concurrence is given by the expression C(P;X) = 2\sqrt{σ_2(P(X)) - λ_2 det X}. As an application, we compute the concurrences of the density matrices of all graphs with 2 edges. Similar results apply for a function which we call I-fidelity, with the second largest generalized eigenvalue λ_2 replaced by the smallest eigenvalue λ_4.

Motivation & Objective

• To generalize the concept of concurrence beyond quantum states to positive operators on Hermitian matrices.
• To define and compute the convex and concave roofs of the square root of the second symmetric function over Lorentz cones.
• To derive explicit expressions for concurrence and I-fidelity of positive maps with input space ℋ(2), particularly for rank-2 bipartite density matrices.
• To establish a connection between concurrence of positive operators and the partial trace map, enabling computation for bipartite systems.
• To provide a systematic method for computing optimal decompositions into pure states that realize the concurrence and I-fidelity.

Proposed method

• The paper defines Lorentz-positive maps as linear maps between real vector spaces that preserve the Lorentz cone, enabling the use of quadratic forms with signature (+---...).
• It introduces the generalized eigenvalue problem for the pencil σ₂^{d₂}(Φ(X)) - λ det X, where λ₁ ≥ λ₂ ≥ λ₃ ≥ λ₄ are the generalized eigenvalues.
• The concurrence is computed as C(Φ;X) = 2√[σ₂^{d₂}(Φ(X)) - λ₂ det X], with λ₂ being the second largest generalized eigenvalue.
• The I-fidelity is defined analogously using the smallest generalized eigenvalue λ₄, yielding F(Φ;X) = 2√[σ₂^{d₂}(Φ(X)) - λ₄ det X].
• The method relies on the fact that for ℋ(2), the positive operator Φ induces a Lorentz-positive map, allowing reduction to the Lorentz cone structure.
• The optimal decomposition achieving the concurrence or I-fidelity is derived from the eigenvector corresponding to λ₂ (for concurrence) or λ₄ (for I-fidelity).

Experimental results

Research questions

• RQ1 How can the concurrence of a positive operator Φ:ℋ(2)→ℋ(d₂) be computed explicitly using generalized eigenvalues of a matrix pencil?
• RQ2 What is the role of the Lorentz cone in generalizing the concept of concurrence beyond quantum states to positive maps?
• RQ3 Can the I-fidelity, defined as the concave roof of the same function, be computed with a similar formula to the concurrence?
• RQ4 For which classes of positive operators does the convex roof of the square root of the second symmetric function admit a closed-form expression?
• RQ5 What is the structure of the optimal pure-state decomposition that realizes the concurrence or I-fidelity for rank-2 density matrices?

Key findings

• The concurrence of a positive operator Φ:ℋ(2)→ℋ(d₂) is given by C(Φ;X) = 2√[σ₂^{d₂}(Φ(X)) - λ₂ det X], where λ₂ is the second largest generalized eigenvalue of the pencil σ₂^{d₂}(Φ(X)) - λ det X.
• The I-fidelity of the same operator is F(Φ;X) = 2√[σ₂^{d₂}(Φ(X)) - λ₄ det X], with λ₄ being the smallest generalized eigenvalue.
• For bipartite density matrices of rank 2, the concurrence and I-fidelity can be computed using the same formulae by identifying Φ with the partial trace map.
• The optimal decomposition achieving the concurrence or I-fidelity involves only two pure states, with the decomposition vector derived from the eigenvector of the matrix pencil corresponding to λ₂ or λ₄.
• The paper provides explicit concurrence values for all 20 graphs with two edges, showing that isomorphic graphs yield identical concurrence and I-fidelity due to shared spectrum and range.
• The results extend to all positive operators with input space ℋ(2), including those of arbitrary rank, by reducing the problem to the Lorentz cone structure via the generalized eigenvalue pencil.