Extremal Marginal States of Maximal Rank in $(d, d+m)$

Abstract: We study the extreme points of the convex set $\mathcal{C}(ρ_1,ρ_2)$ of bipartite quantum states with fixed marginals $ρ_1$ and $ρ_2$. We construct extreme points in $(d,\,d+m)$ dimension, of rank $d+m$, matching the highest possible value, for all $d\geq 3$, $m > \frac{d^2-2d-2}{2}$ (when $d=2$, $m\geq 1$). 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 $\mathcal{C}(ρ_1,ρ_2)$, it is sufficient to study the special case $\mathcal{C}(\mathcal{D}_1,\mathcal{D}_2)$, where the marginals are diagonal. Additionally, we observe that it is sufficient to consider $d_1\leq d_2$. Thus, our results show that apart from possibly a few finite cases, for each $d_1$, the maximal rank is achieved almost all times.

• The paper presents a novel construction of extreme points with maximal rank in (d, d+m) bipartite quantum states, showing that the Parthasarathy bound is attained for most cases.
• It leverages the Choi–Jamiołkowski isomorphism to connect extreme points with completely positive maps and reduces the problem to full-rank diagonal marginals.
• The study demonstrates that these extreme states are separable and entanglement breaking, offering significant insights for quantum channel certification and noisy quantum process analysis.

Extremal Marginal States of Maximal Rank in $(d, d+m)$

Problem Overview and Context

This work addresses the extremal structure of the convex set $\mathcal{C}(\rho_1, \rho_2)$, which consists of bipartite quantum states in $H_1 \otimes H_2$ with fixed marginals $\rho_1$ and $\rho_2$. The principal question is to determine the maximal rank achievable by an extreme point in this set—$\mathcal{MR}(\rho_1, \rho_2)$—and whether the upper bound established by Parthasarathy, $\lfloor \sqrt{d_1^2 + d_2^2 - 1} \rfloor$, can be attained for various dimension pairs $(d_1, d_2)$ and choices of $\rho_1$, $\rho_2$.

Prior work established sharpness of this bound for certain special cases, but there remained an incomplete understanding of which dimension pairs and marginals permit extreme points achieving this maximal rank, particularly in the case of asymmetric or high-dimensional systems.

Main Results and Methodology

Construction of Extremal States in $\mathcal{C}(\rho_1, \rho_2)$0 Systems

The central achievement of the paper is the explicit construction of an infinite family of extreme points in $\mathcal{C}(\rho_1, \rho_2)$1 for systems with Hilbert space dimensions $\mathcal{C}(\rho_1, \rho_2)$2, with $\mathcal{C}(\rho_1, \rho_2)$3, $\mathcal{C}(\rho_1, \rho_2)$4. These states are of rank $\mathcal{C}(\rho_1, \rho_2)$5, which coincides with Parthasarathy's upper bound for the maximal rank under the provided conditions.

• Such extremal states have marginals $\mathcal{C}(\rho_1, \rho_2)$6 and $\mathcal{C}(\rho_1, \rho_2)$7, where $\mathcal{C}(\rho_1, \rho_2)$8 and $\mathcal{C}(\rho_1, \rho_2)$9, $H_1 \otimes H_2$0 is the $H_1 \otimes H_2$1 all-ones matrix.
• The approach leverages the Choi–Jamiołkowski isomorphism to connect the structure of extreme points in $H_1 \otimes H_2$2 with completely positive (CP) maps between matrix algebras whose Kraus operators fulfill explicit algebraic constraints.

The construction shows that, except for finitely many exceptional cases, the maximal rank of extreme points is realized almost universally for these parameter regimes.

Reduction to Canonical Form

A significant theoretical development is a reduction of the general problem—studying extreme points for arbitrary marginals—to the special case when both marginals are full-rank diagonal (i.e., maximally mixed up to permutation). This reduction is enabled by applying unitary transformations, which preserve extremality due to their isometric nature, together with the observation that the ordering $H_1 \otimes H_2$3 is sufficient to cover all cases by symmetry.

Separability and Entanglement Breaking Properties

It is proven that the constructed extremal states in this infinite family are, in fact, separable, and the corresponding CP maps are entanglement breaking (EB) with EB-rank exactly $H_1 \otimes H_2$4. This is established via a check of the PPT condition and results connecting the Kraus/Choi rank to separability for low-rank states.

Extension and Coverage of Prior Results

The general construction recovers and subsumes most previously known isolated cases of maximal rank attainment in past literature. The proof strategy and explicit maps constructed extend directly to higher-dimensional analogues by appropriate tensoring of lower-dimensional extremal maps.

Numerical Results and Claims

• Existence of Extreme Points with Maximal Possible Rank: For all $H_1 \otimes H_2$5 with $H_1 \otimes H_2$6, $H_1 \otimes H_2$7, there exist extreme points with rank $H_1 \otimes H_2$8, which achieves the Parthasarathy bound.
• Complete Characterization for These Families: Apart from exceptions in small dimensions, $H_1 \otimes H_2$9 equals the theoretical upper bound in almost all $\rho_1$0 cases treated.
• Separability: All constructed extremal states are proven to be separable, coinciding with their Kraus/Choi rank being equal to the larger of the two subsystem dimensions.

Theoretical and Practical Implications

This work refines the understanding of the geometric and convex-analytic structure of bipartite quantum states with prescribed marginals, particularly in revealing the role of dimension as the critical constraint for the attainability of maximal-rank extreme points.

For quantum information theory, this impacts the study of marginal problems, the structure of quantum channels (through the Choi isomorphism), and provides explicit constructions of extreme entanglement breaking channels with maximal rank, which are relevant for the characterization of noisy quantum processes and resilience in quantum system identification.

On the mathematical side, the reduction to diagonal marginals provides a pathway for algebraic and combinatorial approaches to the extremality problem, potentially facilitating algorithmic verification and automated search in high dimensions.

Future Directions

There remain open questions concerning the finite exceptional cases where the bound may not be achieved, particularly in low-dimensional asymmetric systems. The extension to multipartite systems with more than two parties, and the study of extremality for non-diagonal (highly nonclassical) marginals, would be natural continuations. Additionally, a systematic characterization of when extreme points can be entangled versus separable in this maximal-rank regime would clarify the interplay between rank, extremality, and entanglement.

Algorithmic generation and classification of extremal channels in arbitrary dimensions may also find further application in quantum device certification and benchmarking.

Conclusion

The paper introduces a constructive method for realizing extreme bipartite states with fixed marginals and maximal rank in $\rho_1$1 quantum systems, achieving the theoretical upper bound in a broad family of cases and showing that these states are necessarily separable. The results unify and extend existing examples, streamline the study of the marginal problem to focus primarily on diagonal marginals, and showcase the utility of CP map techniques in quantum convex analysis. These advances refine the conceptual and practical understanding of extremal quantum correlations subject to marginal constraints (2605.26920).