We find the spectrum in maximal lexicographic order for quantum states ρ A B ∈ H A ⊗ H B \rho _{AB}\in \mathcal {H}_A\otimes \mathcal {H}_B with margins ρ A = 1 n I n \rho _A=\frac {1}{n}I_n and ρ B = 1 m I m \rho _B=\frac {1}{m}I_m and discuss the construction of ρ A B \rho _{AB} . By nonzero rectangular Kronecker coefficients, we give counterexamples for Klyachko’s conjecture which says that a quantum state with maximal lexicographical spectrum has minimal rank among all states with given margins. Moreover, we show that quantum states with the maximal lexicographical spectrum are extreme points.
We find the spectrum in maximal lexicographic order for quantum states ρ A B ∈ H A ⊗ H B \rho _{AB}\in \mathcal {H}_A\otimes \mathcal {H}_B with margins ρ A = 1 n I n \rho _A=\frac {1}{n}I_n and ρ B = 1 m I m \rho _B=\frac {1}{m}I_m and discuss the construction of ρ A B \rho _{AB} . By nonzero rectangular Kronecker coefficients, we give counterexamples for Klyachko’s conjecture which says that a quantum state with maximal lexicographical spectrum has minimal rank among all states with given margins. Moreover, we show that quantum states with the maximal lexicographical spectrum are extreme points.
Extremal Marginal States of Maximal Rank in (d,d+m)
We study the extreme points of the convex set C(ρ1,ρ2) of bipartite quantum states with fixed marginals ρ1 and ρ2. We construct extreme points in (d,d+m) dimension, of rank d+m, matching the highest possible value, for all d≥3, m>2d2−2d−2 (when d=2, m≥1). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of C(ρ1,ρ2), it is sufficient to study the special case C(D1,D2), where the marginals are diagonal. Additionally, we observe that it is sufficient to consider d1≤d2. Thus, our results show that apart from possibly a few finite cases, for each d1, the maximal rank is achieved almost all times.
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.
We study the convex set C(ρ1,ρ2) of all bipartite quantum states with fixed marginal states ρ1 and ρ2. The extremal states in this set have recently been characterized by Parthasarathy [Ann. Henri Poincaré (to appear), quant-ph/0307182]. Here we present an alternative necessary and sufficient condition for a state in C(ρ1,ρ2) to be extremal. Our approach is based on a canonical duality between bipartite states and a certain class of completely positive maps and has the advantage that it is easier to check and to construct explicit examples of extremal states. In dimension 2×2 we give a simple new proof for the fact that all extremal states in C(121,121) are precisely the projectors onto maximally entangled wave functions. We also prove that in higher dimension this does not hold and construct an explicit example of an extremal state in C(131,131) that is not maximally entangled. Generalizations of this result to higher dimensions are also discussed.
Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem A, the marginal maximal-mixing defect is defined by DA=2∣A∣Tr(ρA2)−1=2∣A∣ρA−2∣A∣IA22, and Qex,εD(n,k) is defined as the maximum number of k-body marginals satisfying DA≤ε in an n-qubit pure state. This counting problem differs from approximate k-uniformity, which requires all k-body marginals to obey a common error bound.
For pure states on 4m qubits, the following local stability inequality is established: ∑i∈TDT∖{i}≥1(∣T∣=2m+1). It follows that, whenever ε<1/(2m+1), the hypergraph of ε-good 2m-subsets is K2m+1(2m)-free. Combined with the known exact eight-qubit construction, this yields the stability plateau Qex,εD(8,4)=56,0≤ε<51. For odd systems of 2k+1 qubits, the exact forbidden hypergraph Hk is used to derive explicit finite-error stability radii. In particular, Qex,εD(9,4)≤120 for 0≤ε<1/17. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.