Monogamy of highly symmetric states

TL;DR

This work develops a unified framework, called G-extendibility, to study monogamy of entanglement for highly symmetric bipartite states on graphs by casting the problem as semidefinite programs and solving them analytically via representation theory. Central to the approach are Schur–Weyl dualities for unitary and orthogonal groups, Brauer algebras, and Jucys–Murphy elements, which allow exact computations of maximal overlaps with entangled subspaces for Werner, isotropic, and Brauer state families on the complete graph $K_n$. The authors derive closed-form expressions for the key maxima $q_W(n,d)$, $p_B(n,d)$, and $p_I(n,d)$, identify asymptotic limits, and map out the $K_n$-extendibility polytopes (at least in the qubit Brauer case), with implications for quantum marginal problems, quantum cloning, and network-based quantum information tasks. They further connect these static extendibility results to dynamic quantum-information tasks, showing how cycle-graph limits relate to Bell-state discrimination under time-constrained local operations and classical/quantum communication, yielding a bound of $ ext{ln}(2)$ for certain success probabilities and informing quantum position verification bounds.

Abstract

We investigate the extent to which two particles can be maximally entangled when they are also similarly entangled with other particles on a complete graph, focusing on Werner, isotropic, and Brauer states. To address this, we formulate and solve optimization problems that draw on concepts from many-body physics, computational complexity, and quantum cryptography. We approach the problem by formalizing it as a semi-definite program (SDP), which we solve analytically using tools from representation theory. Notably, we determine the exact maximum values for the projection onto the maximally entangled state and the antisymmetric Werner state, thereby resolving long-standing open problems in the field of quantum extendibility. Our results are achieved by leveraging SDP duality, the representation theory of symmetric, unitary and orthogonal groups, and the Brauer algebra.

Figures (12)

• Figure 1: The complete graph $K_5$ with $\mathrm{Aut}(K_5) \simeq S_5$, the star graph $K_{1,5}$ with $\mathrm{Aut}(K_{1,5}) \simeq S_5$, and the path graph $P_5$ with $\mathrm{Aut}(P_5) \simeq \mathbb{Z}_2$.
• Figure 2: The first values of $q_W(n,d)$. The values of $q_W(n,d)$, for which the Werner states $\rho_e$ are separable (i.e. $p \leq 1/2$), are in grey.
• Figure 3: The first values of $p'_{I}(n,d)$ and $p_{I}(n,d)$. The values for which the isotropic states $\rho_e$ are separable (i.e. $p \leq 1/d$), are in grey. Note that they decrease with respect to $n$, but are not monotonic with respect to $d$.
• Figure 4: Perfect matching for complete graphs $K_n$, with even $n$. For $K_n$ with odd $n$, some vertices are not matched .
• Figure 5: A typical behavior of the spectrum $f_{\mu,\lambda}(x)$ (thin red lines; $f(x)$ is in bold red) for all $(\lambda,\mu) \in { \Omega_{n,d}}$ when $d$ is odd, $n$ is odd and $d \leq n \leq 2d+1$. The coordinate $x=x^*$ corresponds to the optimal value $f(x^*) = p'_I(n,d)$. The plot corresponds to the parameters $n=5$ and $d=3$. The partitions $\lambda$ corresponding to the points $(\tilde{x},g(\lambda))$ are $\lambda_1 = (1^3)$, $\lambda_2 = (1)$, $\lambda_3 = (2,1)$, $\lambda_4 = (3)$, $\lambda_5 = (4,1)$, $\lambda_6 = (5)$. The partitions $\mu$ characterising the offsets $a(\mu)$ for the functions $f_{\mu,\lambda}(x)$ are $\mu_1 = (5)$, $\mu_2 = (4,1)$, $\mu_3 = (3,2)$, $\mu_4 = (3,1,1)$, $\mu_5 = (2,2,1)$. Some other values in that case are $\tilde{x} = 1/20$, $g(\lambda_1) = 1/4$, $x^* = -3/62$, $p'(n,d) = 7/31$.

Theorems & Definitions (45)

• Theorem : Summary of Theorems \ref{['thm:WernerStates']}, \ref{['thm:isotropicStates']}, \ref{['thm:BrauerStates']} and \ref{['lem:qB=qW']}
• Theorem : Theorem \ref{['thm:d=2_brauer_region']}
• Theorem 2.1: Schur--Weyl duality
• Theorem 2.2: Brauer
• Lemma 2.3: doty2019canonical
• Lemma 2.4: doty2019canonical
• Remark
• Remark 2.5
• Conjecture 3.1
• Theorem 4.1