On the rank of extremal marginal states

Let $ρ_1$ and $ρ_2$ be two states on $\mathbb{C}^{d_1}$ and $\mathbb{C}^{d_2}$ respectively. The marginal state space, denoted by $\mathcal{C}(ρ_1,ρ_2)$, is the set of all states $ρ$ on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ with partial traces $ρ_1, ρ_2$. K. R. Parthasarathy established that if $ρ$ is an extreme point of $\mathcal{C}(ρ_1,ρ_2)$, then the rank of $ρ$ does not exceed $\sqrt{d_1^2+d_2^2-1}$. Rudolph posed a question regarding the tightness of this bound. In 2010, Ohno gave an affirmative answer by providing examples in low-dimensional matrix algebras $\mathbb{M}_3$ and $\mathbb{M}_4$. This article aims to provide a positive answer to the Rudolph question in various matrix algebras. Our approaches, to obtain the extremal marginal states with tight upper bound, are based on Choi-Jamiołkowski isomorphism and tensor product of extreme points.

💡 Research Summary

The paper investigates the maximal possible rank of extreme points in the set of bipartite quantum states with prescribed marginals, denoted 𝒞(ρ₁, ρ₂). Parthasarathy proved that any extreme point ρ ∈ 𝒞(ρ₁, ρ₂) satisfies rank ρ ≤ ⌊√(d₁² + d₂² − 1)⌋, where d₁ and d₂ are the local dimensions. Rudolph asked whether this bound is tight. Ohno answered affirmatively for the low‑dimensional cases d₁ = d₂ = 3 and 4, but the question remained open for larger dimensions and for the unequal‑dimension case.

The authors address this gap by exploiting the Choi–Jamiołkowski isomorphism, which identifies a bipartite state with a completely positive (CP) linear map Φ: M_{d₁}→M_{d₂}. Under this correspondence, the rank of the state equals the Choi‑rank of Φ, i.e., the minimal number of Kraus operators needed to represent Φ. An extreme point of 𝒞(ρ₁, ρ₂) translates into an extreme point of the convex set of CP maps with fixed output Φ(I_{d₁}) = ρ₂ and fixed dual output Φ* (I_{d₂}) = ρ₁ᵀ. Theorem 2.8 (originally due to Parthasarathy and later refined) gives a simple algebraic criterion: a CP map Φ with minimal Kraus decomposition Φ = ∑{j=1}^n Ad{V_j} is extreme iff the family {V_i* V_j}{i,j} is linearly independent in M{d₂}.

The central technical contribution is a systematic method for constructing high‑dimensional extreme points from known low‑dimensional ones by taking tensor products of CP maps. If Φ₁ and Φ₂ are extreme CP maps with Kraus sets {V_j^{(1)}} and {W_k^{(2)}}, then the tensor product map Φ₁⊗Φ₂ has Kraus operators {V_j^{(1)}⊗W_k^{(2)}}. The authors prove that the linear independence of the products {V_i* V_j⊗W_k* W_l} is preserved under tensoring, guaranteeing that Φ₁⊗Φ₂ remains extreme. Consequently, the Choi‑rank multiplies: rank J(Φ₁⊗Φ₂) = rank J(Φ₁)·rank J(Φ₂). By starting from Ohno’s explicit extreme points in M₃ and M₄ (which achieve the bound for d=3,4), and repeatedly tensoring, the authors generate families of extreme states that attain the Parthasarathy bound for a wide range of dimensions.

Specifically, they prove the bound is sharp for:

• Equal dimensions d₁ = d₂ = 5, 9, 12.
• All dimensions of the form d₁ = d₂ = 5k with 3 ≤ k ≤ 14 (i.e., 15, 20, …, 70).
• The mixed‑dimension pair (d₁,d₂) = (3,4).
• The family (d₁,d₂) = (2,d) for any d ≥ 4.

In each case they exhibit explicit Kraus operators, verify the linear independence condition, and thus confirm that the constructed state’s rank equals ⌊√(d₁² + d₂² − 1)⌋.

The paper also explores situations where the bound is not attained. For the pair (d₁,d₂) = (d, d+1) with d ≥ 4, they construct an extreme point with rank d + 1, which is strictly smaller than the general bound ⌊√(d² + (d+1)² − 1)⌋. This demonstrates that while the bound is sharp in many regimes, it is not universally tight.

The authors discuss the reduction to invertible marginals, showing that any extreme point can be restricted to the support of the marginals without loss of generality. They also provide a detailed analysis of the Choi‑rank versus state rank relationship, and include an appendix listing all Kraus matrices used in the constructions and the linear independence checks.

Overall, the work settles Rudolph’s question for a large class of matrix algebras, confirming that Parthasarathy’s rank bound is indeed sharp for most equal‑dimension cases beyond d = 4, and for several unequal‑dimension scenarios, while also clarifying the existence of lower‑rank extreme points in certain off‑diagonal cases. The methods combine algebraic criteria for extremality with constructive tensor‑product techniques, offering a versatile toolkit for future investigations of marginal problems in quantum information theory.