Coherent attacks give no raw-information advantage at fixed local disturbances

For every natural number $n\ge1$, every family $P=(P_i)_{i<n}$ of local protocols, and every real vector $d=(d_i)_{i<n}$ that is the disturbance vector of some joint attack, the following existence statement holds. Here each protocol has positive finite raw, preparation, Bob-choice, public, and signal index sets $X_i,S_i,B_i,C_i,Z_i$; nonnegative, separately normalized weights $r_i,p_i,q_i$ on the first three sets; unit preparation vectors $\psi^i_{x,a}\in\mathbb C^{Z_i}$; and positive semidefinite matrices $M^i_{b,x}$ satisfying $\sum_xM^i_{b,x}=I$ for every $b$. It has retention and announcement functions $k_i(a,b)$ and $c_i(a,b)$, positive retained mass $s_i=\sum_{a,b:k_i(a,b)=\mathrm{true}}p_i(a)q_i(b)$, and $\operatorname{Re}\operatorname{tr}(M^i_{b,x}\psi^i_{x,a}(\psi^i_{x,a})^*)=1$ for every retained $(a,b)$ and every $x$. Give a local label $\ell_i=(x_i,a_i,b_i)$ weight $w_i(\ell_i)=r_i(x_i)p_i(a_i)q_i(b_i)/s_i$ when retained and $0$ otherwise; for joint labels put $W(\ell)=\prod_iw_i(\ell_i)$, $x(\ell)=(x_i)_i$, and $c(\ell)=(c_i(a_i,b_i))_i$. An attack $C$ on this family consists of a complete complex inner-product space $E$, a unit vector $e\in E$, a complex-linear isometric equivalence $U$ of the Euclidean space of $E$-valued functions on $Z=\prod_iZ_i$, an arbitrary measurable outcome space $O$, and a measurement for each public string $c\in\prod_iC_i$. A measurement supplies continuous complex-linear operators $F_c(D)$ for measurable $D\subseteq O$, whose quadratic forms are real and nonnegative, with $F_c(O)=I$, and measures $\lambda_{c,v}$ satisfying $\lambda_{c,v}(D)=\max\{\operatorname{Re}\langle v,F_c(D)v\rangle,0\}$. Put $\Phi^C_\ell=U(z\mapsto(\prod_i\psi^i_{x_i,a_i}(z_i))e)$ and $\mu^C_{\ell,c}=\sum_z\lambda_{c,\Phi^C_\ell(z)}$. Its disturbance at site $i$ is exactly

For any outcome laws $\mu_{\ell,c}$, define $\pi_{c,x}=\sum_{\ell:c(\ell)=c,\ x(\ell)=x}W(\ell)$, $p_c=\sum_x\pi_{c,x}$, $\nu_{c,x}=\sum_{\ell:c(\ell)=c,\ x(\ell)=x}W(\ell)\mu_{\ell,c}$, and $\nu_c=\sum_x\nu_{c,x}$. With $H(t)=-\sum_xt(x)\log(t(x))/\log2$ and $h_{c,y}(x)=(d\nu_{c,x}/d\nu_c)(y)$ converted to a real number, define $J(\mu)=\sum_c[p_cH(x\mapsto\pi_{c,x}/p_c)-\int H(h_{c,y})\,d\nu_c(y)]$; the information $I(C)$ is $J(\mu^C)$. Division by zero and $\log0$ have value $0$, an infinite Radon–Nikodym derivative converts to $0$, and a nonintegrable real integrand has integral $0$. A single-site attack uses these same definitions on the one-element family containing $P_i$; denote its disturbance and information by $\Delta(B)$ and $I(B)$. The hypothesis on $d$ is precisely that some attack $C_0$ satisfies $\Delta_i(C_0)=d_i$ for every $i$. The conclusion is that there exists a tuple $A=(A_i)_{i<n}$ of single-site attacks such that each $\Delta(A_i)=d_i$, and every single-site attack $B$ on $P_i$ with $\Delta(B)=d_i$ satisfies $I(B)\le I(A_i)$; moreover, writing $I_{\mathrm{prod}}(A)=J(\mu^A)$ for the product laws $\mu^A_{\ell,c}=\bigotimes_i\mu^{A_i}_{\ell_i,c_i}$, one has

The asserted product disturbance vector is exactly $(\Delta(A_i))_{i<n}=d$; neither the chosen attacks nor the individual maximizers are asserted to be unique.