Nonexistence of maximally entangled mixed states for a fixed spectrum

The existence of a maximally entangled pure state is a cornerstone result of entanglement theory that has paramount consequences in quantum information theory. A natural generalization of this property is to consider whether a notion of maximal entanglement is possible among all states with the same spectrum (where the aforementioned case of pure states corresponds to the particular choice in which the spectrum is a delta distribution, i.e., rank-1 states). Despite positive evidence in the past that such a notion might exist at least in the case of two-qubit states, it was recently shown in [Phys. Rev. Lett. 133, 050202 (2024)] that the answer to the above question is negative. This reference proved this for particular choices of the spectrum in the case of rank-2 two-qubit density matrices. While this settles the problem in general, it still leaves open whether there are other choices of the spectrum outside the case of pure states where a maximally entangled state for a fixed spectrum might exist. In this work we extend this impossibility result to all rank-2 and rank-3 two-qubit states as well as for a large class of eigenvalue distributions in the case where the rank equals four.

💡 Research Summary

The paper investigates whether a universal “maximally entangled mixed state” (MEMS) can exist for two‑qubit density matrices when the eigenvalue spectrum is fixed, extending beyond the trivial pure‑state case (rank‑1). The authors build on recent work that showed, for the specific rank‑2 spectrum (λ, 1‑λ, 0, 0) with λ∈(2/3, 1), no state can dominate all others under LOCC‑preserving transformations, thereby ruling out a universal maximally entangled state for that family.

The present work generalizes this impossibility to all rank‑2 and rank‑3 spectra, and to a large class of full‑rank (rank‑4) spectra. Two complementary approaches are employed. First, the authors consider separable (SEP) maps, a superset of LOCC that still cannot generate entanglement. They compare the candidate MEMS introduced in earlier literature (ρ_λ = ∑_i λ_i ξ_i, where ξ_i are Bell‑state projectors or computational‑basis projectors) with any other state σ sharing the same spectrum. By exploiting PPT (positive partial transpose) criteria, majorization relations, and monotones such as the quantum relative entropy, they prove that for any spectrum with λ₄ = 0 but λ₂ ≠ 0 (i.e., rank‑2 or rank‑3 cases) there exists at least one σ that cannot be obtained from ρ_λ via any SEP map. Consequently, no state in S(λ) can be universally maximal under LOCC.

Second, the authors revisit the non‑entangling (NE) class, the most permissive set of operations that merely preserve separability of separable inputs. They formulate the transformation ρ_λ → σ as a linear program (LP): the variables are the probabilities associated with Kraus operators, while the constraints enforce spectrum preservation, the PPT condition on the output, and the tensor‑product structure of the Kraus operators. Solving the LP numerically for a dense grid of spectra, they find that for virtually all full‑rank spectra satisfying the entanglement condition λ₁ − λ₃ − 2√(λ₂λ₄) > 0, the LP is infeasible, meaning that no NE map can convert the MEMS into certain other states with the same eigenvalues. This extends the non‑existence result to a broad family of rank‑4 states.

The technical insight is that even when allowing the most generous transformation sets (SEP, NE), the fixed‑spectrum set S(λ) lacks a single state that can be converted into every other member. The obstruction is not merely a difference in a particular entanglement measure; it is a simultaneous violation of several LOCC‑invariant quantities (majorization order, PPT positivity, relative entropy monotones). Hence, the notion of a universal maximally entangled state is confined to the pure‑state case.

The paper concludes that, in realistic scenarios where noise forces the preparation of mixed states with a given spectrum, the optimal entangled resource must be chosen task‑specifically rather than universally. This has practical implications for quantum communication and computation protocols that rely on entanglement generation under spectral constraints. The authors also note that extending these results to higher‑dimensional or multipartite systems, and developing more efficient analytical tools beyond the LP approach, remain open challenges. Overall, the work provides a comprehensive and rigorous demonstration that “maximally entangled mixed states for a fixed spectrum do not exist” beyond the trivial pure‑state situation.