[Paper Review] Linear maps preserving the higher numerical ranges of tensor product of matrices

Ajda Fošner, Zejun Huang+3 more|arXiv (Cornell University)|Feb 5, 2013

Matrix Theory and Algorithms22 references4 citations

TL;DR

This paper characterizes linear maps on the tensor product of complex matrices that preserve the k-numerical range of all rank-one decomposable tensors. It proves such maps are either unitary conjugations or involve a trace correction when mn = 2k, extending classical results on numerical range preservers to the tensor product setting with explicit classification via unitary equivalence and transposition maps.

ABSTRACT

We study linear maps preserving the higher numerical ranges of tensor product of matrices.

Motivation & Objective

• To characterize linear maps φ on M_{mn} that preserve the k-numerical range of all decomposable tensors A⊗B with A∈M_m, B∈M_n.
• To extend known results on k-numerical range preservers from single matrices to tensor product structures.
• To determine the precise form of such maps when the tensor product dimension satisfies mn = 2k, where additional non-unitary forms arise.
• To unify results from linear preserver theory and quantum information, particularly regarding joint measurements of bipartite quantum systems.

Proposed method

• The authors analyze the structure of linear maps φ satisfying W_k(φ(A⊗B)) = W_k(A⊗B) for all A∈M_m, B∈M_n.
• They use spectral decomposition and eigenvalue interlacing via the Courant-Fischer min-max principle to analyze the action of φ on rank-one operators.
• The proof relies on analyzing the action of φ on matrix units E_{ii}⊗E_{jj}, showing that their images must have specific diagonal structures.
• By applying Lemma 2.1 on matrix compressions and eigenvalue majorization, they deduce that φ preserves the trace and eigenvalue distribution of rank-one tensor products.
• They consider two cases: when φ preserves the trace and when it does not, leading to the distinction between unitary conjugation and trace-corrected forms.
• The classification includes transposition maps A⊗B ↦ (A⊗B)^t and partial transpositions A⊗B ↦ A⊗B^t or A^t⊗B when min{m,n} ≤ 2.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a linear map φ on M_{mn} to preserve the k-numerical range of all decomposable tensors A⊗B with A∈M_m, B∈M_n?
• RQ2 How does the structure of such maps change when mn = 2k, leading to additional non-unitary forms?
• RQ3 Can the characterization be extended to include partial transpositions when min{m,n} ≤ 2?
• RQ4 What role does the trace correction term (tr(A⊗B)/k)I_{mn} play in preserving the k-numerical range in the exceptional case?
• RQ5 How do these maps relate to quantum measurements and joint observables in bipartite quantum systems?

Key findings

• A linear map φ: M_{mn} → M_{mn} preserves W_k(A⊗B) for all A∈M_m, B∈M_n if and only if it is of the form φ(A⊗B) = U(φ(A⊗B))U* for some unitary U∈M_{mn}.
• When mn = 2k, an additional class of maps exists: φ(A⊗B) = (tr(A⊗B)/k)I_{mn} - U(φ(A⊗B))U*, where φ is either the identity or transpose map on A⊗B.
• If min{m,n} ≤ 2, additional maps arise via partial transpositions: A⊗B ↦ A⊗B^t or A^t⊗B.
• The map φ preserves the k-numerical range if and only if it is unitarily equivalent to the identity, transpose, or trace-corrected transpose map on the tensor product space.
• The largest eigenvalue of φ(E_{ii}⊗E_{jj}) is exactly 1/k, and (1/k)I_{mn} - φ(E_{ii}⊗E_{jj}) is positive semidefinite, ensuring the trace condition is met.
• The classification is complete and sharp: no other forms of such linear maps exist beyond the two cases, with the trace-corrected form only occurring when mn = 2k.