Many-copy Bell nonlocality of every entangled state
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Problem
Does every entangled bipartite state become Bell nonlocal under collective local measurements on some finite number of copies? Fix an integer \(d\geq2\) and a state \(\rho\) on \(\mathbb C^d\otimes\mathbb C^d\). At blocklength \(n\), Alice and Bob choose arbitrary collective positive operator-valued measurements \(\{M_{a|x}\}_a\) and \(\{N_{b|y}\}_b\) on their respective \(n\) subsystems. Settings \(x,y\) and outcomes \(a,b\) range over finite sets, and
A behavior is Bell local when it admits a local hidden-variable decomposition
where \(\mu\) is a probability measure independent of the settings and \(p_A,p_B\) are local response distributions. For \(s\geq2\), let \(\mathsf L^{(s)}\) be the set of states on \(\mathbb C^s\otimes\mathbb C^s\) all of whose behaviors, for all finite setting and outcome sets and all local positive operator-valued measurements, are Bell local. Let \(\mathsf L_\infty^{(d)}\) be the set of states \(\rho\) for which every behavior in Eq. (1) admits a decomposition as in Eq. (2) for every finite \(n\); equivalently, \(\rho^{\otimes n}\in\mathsf L^{(d^n)}\) for all \(n\geq1\), with Alice’s \(n\) subsystems grouped against Bob’s. Let \(\mathrm{SEP}_d\) denote the separable states on \(\mathbb C^d\otimes\mathbb C^d\). No communication, auxiliary entangled state, or postselection is allowed, and every measurement outcome is retained. The question is whether
Source
Cavalcanti, Acín, Brunner, and Vértesi explicitly ask, in the discussion of their results, whether every entangled state is a nonlocal resource when a sufficient number of copies is considered [CABV13]. Šupić, Skrzypczyk, and Cavalcanti state in Sec. I that whether every entangled state can be superactivated in this way is open [SSC17]. Eq. (3) is a precise formulation of that question.
Progress
Palazuelos proves that Bell nonlocality can be superactivated by tensor powers without local preprocessing: certain isotropic states that admit a local hidden-variable model for all local positive operator-valued measurements have tensor powers violating a Bell inequality based on the Khot–Vishnoi game [Pal12]. Thus
\begin{equation} \rho\in\mathsf L^{(d)}, \qquad \rho^{\otimes n}\notin\mathsf L^{(d^n)} \tag{4} \end{equation}is possible for some states \(\rho\) and finite \(n\). Existence of the effect in Eq. (4) is settled; its universality over entangled states is the question.
Cavalcanti, Acín, Brunner, and Vértesi prove the sufficient condition
\begin{equation} F_d(\rho)>\frac1d \quad\Longrightarrow\quad \rho\notin\mathsf L_\infty^{(d)}, \tag{5} \end{equation}where
\begin{equation} F_d(\rho):=\max_{U\in\mathrm U(d)} \langle\Phi_d|(I_d\otimes U)\rho(I_d\otimes U^\dagger)|\Phi_d\rangle, \qquad |\Phi_d\rangle:=\frac1{\sqrt d}\sum_{j=0}^{d-1}|jj\rangle, \tag{6} \end{equation}is the fully entangled fraction and \(\mathrm U(d)\) is the unitary group [CABV13]. Their proof twirls \(\rho\) into an isotropic state with the same value of Eq. (6) and applies Khot–Vishnoi Bell inequalities to collective local measurements on \(\rho^{\otimes k}\); the twirling unitaries are absorbed into the local measurements, so no filtering, auxiliary state, or postselection is used. Eq. (5) covers every entangled isotropic state, but not every entangled state: the authors note that it does not apply to some distillable states.
Šupić, Skrzypczyk, and Cavalcanti distinguish the many-copy scenario from quantum networks, hidden nonlocality with local preprocessing, and Bell tests with quantum inputs, in which every entangled state can be detected [SSC17]. The unresolved implication is
\begin{equation} \rho\notin\mathrm{SEP}_d \quad\Longrightarrow\quad \exists\,n\geq1:\ \rho^{\otimes n}\notin\mathsf L^{(d^n)}, \tag{7} \end{equation}not a scenario in which a referee supplies additional quantum inputs.
Renner, Lobo, Konderak, Augusiak, and Acín prove a stronger many-copy statement for pure states: for every pure entangled state \(|\psi\rangle\) there is an integer \(k\) such that suitable local measurements on \(|\psi\rangle^{\otimes k}\) give a behavior with zero local content (Result 2 in Sec. V of [RLKAA26]). The local content of a nonsignaling behavior \(p\) is
\begin{equation} \operatorname{LC}(p):=\max\left\{q\in[0,1]:\ p=q\,p_L+(1-q)\,p_{\mathrm{NS}},\ p_L\ \text{Bell local},\ p_{\mathrm{NS}}\ \text{nonsignaling}\right\}. \tag{8} \end{equation}Pure entangled states are already Bell nonlocal for a single copy, as recalled in [Pal12]; the conclusion \(\operatorname{LC}(p)=0\) in Eq. (8) strengthens the pure-state case and does not address mixed states.
Comment
No proof or counterexample to Eq. (3) is known. A counterexample must be an entangled state with a local hidden-variable model for every finite tensor power and all collective local measurements, not merely a state that passes finitely many Bell tests. This many-copy property is not a regularized rate, so activation by copies of the same state is meaningful here. Activation with auxiliary states, local filtering, networks, or quantum inputs concerns different scenarios. Literature checked through 15 September 2026.