Definitions for On the role of coherent attacks in a type of strategic problem related to quantum key distribution

QAlgorithms.Papers.Independent.Intakethird100.P18.AncillaPOVM

For any normed additive commutative group $E$ equipped with a complex inner product space structure, and any type $O$ equipped with a measurable space structure, an ancilla measurement consists of a continuous complex linear operator $F_B:E\to E$ for every subset $B\subseteq O$, together with a measure $\mu_v$ on $O$ for every vector $v\in E$. These data must satisfy

$$\operatorname{Im}\langle v,F_Bv\rangle=0,\qquad 0\leq\operatorname{Re}\langle v,F_Bv\rangle,\qquad \mu_v(B)=\operatorname{ofReal}\!\left(\operatorname{Re}\langle v,F_Bv\rangle\right)$$

for every measurable $B$ and every $v$, and $F_O=\mathrm{id}_E$. Here $\operatorname{ofReal}(r)$ is the nonnegative extended real number obtained by replacing a negative real $r$ with zero. The operators are supplied on all subsets, but positivity and the displayed measure identity are required only on measurable subsets. Vectors need not be normalized; completeness of $E$, finiteness of $O$, and nonemptiness of $O$ are not assumed.

QAlgorithms.Papers.Independent.Intakethird100.P18.Attack

For any natural number $n$ and family of local protocols $P_i$, write $d_i>0$ and $c_i>0$ for their recorded signal dimensions and public alphabet sizes, and put $Z=\prod_i\{0,\ldots,d_i-1\}$ and $C=\prod_i\{0,\ldots,c_i-1\}$. An attack consists of a complete complex inner product space $E$, with its normed additive commutative group structure; an initial vector $e\in E$ satisfying $\|e\|=1$; a bijective complex linear isometry

$$U:\ell^2(Z;E)\longrightarrow\ell^2(Z;E);$$

an outcome type $O$ equipped with a measurable space structure; and, for every $c\in C$, a family of continuous complex linear operators $F_{c,B}:E\to E$, indexed by all subsets $B\subseteq O$, and measures $\mu_{c,v}$ on $O$, indexed by $v\in E$. For every measurable $B$ and every $v$, these satisfy $\operatorname{Im}\langle v,F_{c,B}v\rangle=0$, $\operatorname{Re}\langle v,F_{c,B}v\rangle\geq0$, and $\mu_{c,v}(B)=\operatorname{ofReal}(\operatorname{Re}\langle v,F_{c,B}v\rangle)$, while $F_{c,O}=\mathrm{id}_E$. Here $\ell^2(Z;E)$ is the space of $E$-valued functions on $Z$, with squared norm $\sum_z\|v(z)\|^2$. No finite dimensionality of $E$ or finiteness of $O$ is required. The case $n=0$ is included: both $Z$ and $C$ then contain one empty tuple, and $U$ acts on a space with one $E$-valued coordinate.

QAlgorithms.Papers.Independent.Intakethird100.P18.Inputs

For every natural number $n$, including $0$, and every family of local protocols $P_i$ indexed by $0\le i<n$, an input is a family $l=(l_i)_{i<n}$ with $l_i=(x_i,a_i,b_i)$, where $0\le x_i<P_i.\mathrm{rawSize}$, $0\le a_i<P_i.\mathrm{prepSize}$, and $0\le b_i<P_i.\mathrm{bobSize}$. There is no restriction requiring retained choices or positive weights. For $n=0$, there is exactly one such family, the empty family.

QAlgorithms.Papers.Independent.Intakethird100.P18.JointSpace

For any natural number $n$, any family $P_i$ of local protocols indexed by $i\in\{0,\ldots,n-1\}$, and any type $E$ equipped with a normed additive commutative group structure, let $d_i>0$ be the signal dimension recorded by $P_i$. The joint space is the space of functions from

$$Z=\prod_{i=0}^{n-1}\{0,\ldots,d_i-1\}$$

to $E$, equipped with the finite $\ell^2$ norm

$$\|v\|=\left(\sum_{z\in Z}\|v(z)\|^2\right)^{1/2}.$$

This definition requires neither a complex vector space structure nor completeness on $E$. It includes $n=0$: then $Z$ consists of the single empty tuple, so the joint space has one $E$-valued coordinate.

QAlgorithms.Papers.Independent.Intakethird100.P18.LocalProtocol

A local protocol consists of five strictly positive natural numbers $r,s,t,u,D$, specifying respectively the numbers of raw labels, preparation choices, Bob choices, public labels, and complex vector coordinates; three families of real weights $p_x$, $q_a$, and $v_b$, indexed respectively by $0\le x<r$, $0\le a<s$, and $0\le b<t$, with every weight nonnegative and $\sum_xp_x=\sum_aq_a=\sum_bv_b=1$; a vector $\psi_{x,a}\in\mathbb C^D$ for every $x,a$, satisfying $\|\psi_{x,a}\|=1$; and, for every $b$, matrices $M_{b,x}\in\mathbb C^{D\times D}$ indexed by raw labels $x$, each positive semidefinite and satisfying $\sum_xM_{b,x}=I_D$. It also includes a Boolean function $K(a,b)$ and a public-label function $C(a,b)\in\{0,\ldots,u-1\}$, subject to $\sum_{a,b:\,K(a,b)=\mathrm{true}}q_av_b>0$. Finally, for every $x,a,b$ with $K(a,b)=\mathrm{true}$, it requires $\operatorname{Re}\operatorname{tr}(M_{b,x}\rho_{x,a})=1$, where $(\rho_{x,a})_{j,k}=\psi_{x,a}(j)\overline{\psi_{x,a}(k)}$. This last requirement applies even when any of the corresponding weights is zero.

QAlgorithms.Papers.Independent.Intakethird100.P18.OptimalAt

For a local protocol $P$, any real number $d$, and a single attack $A$ on $P$, this predicate means that $D(A)=d$ and that every single attack $B$ on $P$ satisfying $D(B)=d$ also satisfies $I(B)\le I(A)$. Here single attacks may have their own complete complex inner-product ancilla spaces $E$, unit initial vectors $e$, surjective complex-linear isometries $U$ on the one-site signal–ancilla space, measurable outcome types $O$, and public-dependent measurements with positive normalized operator set maps $F_c$ and measures $\lambda_{c,v}$ satisfying $\lambda_{c,v}(B)=\max(\operatorname{Re}\langle v,F_c(B)v\rangle,0)$ for measurable $B$. To specify the two real quantities, write $\ell=(r,a,b)$, let $u,p,q$ be the protocol’s raw, preparation and Bob-setting weights, let $K=\sum_{a,b}\mathbf1_{\mathrm{keep}(a,b)}p(a)q(b)>0$, and put $w(\ell)=u(r)p(a)q(b)/K$ when retained and $0$ otherwise. For each attack $C$, identify the one-site signal tuples with the signal index set $S$, and set $v_\ell^C=U_C(s\mapsto\mathrm{prepare}(r,a)(s)e_C)$. Its disturbance is

$$D(C)=\sum_{\ell=(r,a,b)}w(\ell) \left[1-\operatorname{Re}\sum_{s,t\in S} M_b(r)_{s,t}\langle v_\ell^C(s),v_\ell^C(t)\rangle\right],$$

where $M_b(r)$ is Bob’s raw-outcome effect. For public value $c$ and raw value $x$, set $m_{c,x}=\sum_{\ell:\,\mathrm{announce}(a,b)=c,\ r=x}w(\ell)$, $m_c=\sum_xm_{c,x}$, $\nu_{c,x}^C=\sum_{\ell:\,\mathrm{announce}(a,b)=c,\ r=x}\max(w(\ell),0)\sum_s\lambda^C_{c,v_\ell^C(s)}$, and $\nu_c^C=\sum_x\nu_{c,x}^C$. With $H(p)=-\sum_xp(x)\log(p(x))/\log2$, its information is

$$I(C)=\sum_c\left[ m_cH\!\left(x\mapsto\frac{m_{c,x}}{m_c}\right) -\int H\!\left(x\mapsto \left(\frac{d\nu_{c,x}^C}{d\nu_c^C}(y)\right)_{\mathbb R}\right)d\nu_c^C(y) \right].$$

The real conversion sends $+\infty$ to $0$, division by $0$ and $\log0$ are $0$, and a nonintegrable real-valued integral is assigned $0$. There is no separate restriction on $d$, no uniqueness requirement, and competitors are compared only when their disturbance is exactly $d$.

QAlgorithms.Papers.Independent.Intakethird100.P18.ProductAttack

For every natural number $n$, including $0$, and every family of local protocols $P_i$, $i<n$, a product attack means a choice, for each $i<n$, of an attack on the singleton family containing $P_i$. Each choice supplies its own complete complex inner-product ancilla space $E_i$, unit initial vector, surjective complex-linear isometry on its one-site signal–ancilla $\ell^2$ space, measurable outcome type $O_i$, and measurement for every one-site public tuple. A measurement assigns continuous complex-linear operators $F_c(B)$ to outcome subsets, has real nonnegative quadratic forms on measurable $B$, satisfies $F_c(O_i)=I$, and supplies measures $\lambda_{c,v}$ with $\lambda_{c,v}(B)=\max(\operatorname{Re}\langle v,F_c(B)v\rangle,0)$ on measurable sets. The component spaces and outcome types may vary with $i$; the abbreviation itself specifies a family of single attacks. For $n=0$, it is the empty family.

QAlgorithms.Papers.Independent.Intakethird100.P18.Public

For every natural number $n$, including $0$, and every family of local protocols $P_i$ indexed by $0\le i<n$, a public value is a family $c=(c_i)_{i<n}$ with $c_i\in\{0,\ldots,P_i.\mathrm{publicSize}-1\}$. These values need not arise from the protocols’ announcement functions. For $n=0$, this set contains exactly the empty family.

QAlgorithms.Papers.Independent.Intakethird100.P18.QkdFeasible

For every natural number $n$, including $0$, every family of local protocols $P_i$, $i<n$, and every real-valued function $d$ on those indices, this predicate asserts the existence of an attack whose disturbance equals $d$ at every coordinate. Such an attack consists of a complete complex inner-product ancilla space $E$, a unit initial vector $e$, a surjective complex-linear isometry $U$ on the $\ell^2$ space of $E$-valued functions on the signal-tuple set $S$, a measurable outcome type, and public-dependent measurements with positive normalized operator set maps and measures satisfying the Born quadratic-form formula. For a label $\ell=((r_i,a_i,b_i))_i$, let $v_\ell=U(z\mapsto[\prod_i\mathrm{prepare}_i(r_i,a_i)(z_i)]e)$, let $K_i=\sum_{a,b}\mathbf1_{\mathrm{keep}_i(a,b)}p_i(a)q_i(b)>0$, and let $w(\ell)=\prod_i u_i(r_i)p_i(a_i)q_i(b_i)/K_i$ if every local pair is retained and $0$ otherwise, using the local raw, preparation and Bob-setting weights. Writing $M_{i,b_i}(r_i)$ for Bob’s local raw-outcome effect, the required coordinate equations are

$$d_i=\sum_\ell w(\ell)\left[ 1-\operatorname{Re} \sum_{\substack{z,t\in S\\z_j=t_j\ \text{for all }j\ne i}} M_{i,b_i}(r_i)_{z_i,t_i}\, \langle v_\ell(z),v_\ell(t)\rangle \right] \qquad(i<n).$$

The measurements are part of the existentially quantified attack even though they do not occur in these equations. No range restriction on $d$ is separately imposed. At $n=0$, there are no coordinate equations, while existence of an attack remains part of the definition.

QAlgorithms.Papers.Independent.Intakethird100.P18.QkdLabel

For a local protocol $P$, a label is a triple $(x,a,b)$, where $x\in\{0,\ldots,P.\mathrm{rawSize}-1\}$, $a\in\{0,\ldots,P.\mathrm{prepSize}-1\}$, and $b\in\{0,\ldots,P.\mathrm{bobSize}-1\}$. All such triples are included, regardless of their weights or whether $P.\mathrm{keep}(a,b)$ is true.

QAlgorithms.Papers.Independent.Intakethird100.P18.Raw

For every natural number $n$, including $0$, and every family of local protocols $P_i$ indexed by $0\le i<n$, a raw value is a family $x=(x_i)_{i<n}$ with $x_i\in\{0,\ldots,P_i.\mathrm{rawSize}-1\}$. For $n=0$, this set contains exactly the empty family.

QAlgorithms.Papers.Independent.Intakethird100.P18.Signal

For every natural number $n$, including $0$, and every family of local protocols $P_i$ indexed by $0\le i<n$, a signal index is a family $z=(z_i)_{i<n}$ with $z_i\in\{0,\ldots,P_i.\mathrm{dimension}-1\}$. For $n=0$, this set contains exactly the empty family.

QAlgorithms.Papers.Independent.Intakethird100.P18.SingleAttack

For every local protocol $P$, a single attack means an attack on the singleton family whose only protocol is $P$. Explicitly, it consists of a complete complex inner-product space $E$, a unit vector $e\in E$, a surjective complex-linear isometry on the $\ell^2$ space of $E$-valued functions indexed by the one-site signal tuples, a measurable outcome type $O$, and a measurement for each one-site public tuple. Each measurement assigns a continuous complex-linear operator $F_c(B)$ to every subset $B\subseteq O$, has $\operatorname{Im}\langle v,F_c(B)v\rangle=0$ and $\operatorname{Re}\langle v,F_c(B)v\rangle\ge0$ for every vector $v$ and measurable $B$, satisfies $F_c(O)=I$, and supplies a measure $\lambda_{c,v}$ for every $v$, with $\lambda_{c,v}(B)=\max(\operatorname{Re}\langle v,F_c(B)v\rangle,0)$ on measurable sets. Neither the ancilla space nor the outcome type is required to be finite.

QAlgorithms.Papers.Independent.Intakethird100.P18.bobCorrect

For any natural number $n$, family of local protocols $P_i$, attack on that family, input label tuple $l_i=(x_i,a_i,b_i)$, and site $i\in\{0,\ldots,n-1\}$, let $Z=\prod_j\{0,\ldots,d_j-1\}$. Each protocol provides a unit preparation vector $v_j(x,a)\in\mathbb C^{d_j}$ and matrices $M_j(b,x)$, which are positive semidefinite and sum to the identity over the raw outcome label $x$ for each Bob label $b$. Let $E$, $e$, and $U$ be the attack’s complete complex inner product ancilla space, unit initial vector, and bijective complex linear isometry on $\ell^2(Z;E)$, and set $\Psi_l=U(z\mapsto(\prod_jv_j(x_j,a_j)(z_j))e)$. The defined real number is

$$\operatorname{Re}\left( \sum_{z\in Z}\sum_{w\in Z} \begin{cases} M_i(b_i,x_i)_{z_i,w_i}\, \langle\Psi_l(z),\Psi_l(w)\rangle, & z_j=w_j\ \text{for every }j\ne i,\\ 0,&\text{otherwise}. \end{cases} \right).$$

The inner product has $\Psi_l(z)$ as its first argument and $\Psi_l(w)$ as its second. No retention condition is imposed on $l$. If $n=0$, there is no site $i$ at which this definition can be evaluated.

QAlgorithms.Papers.Independent.Intakethird100.P18.experimentInformation

For any natural number $n$, including $0$, any family of local protocols $P_i$ indexed by $i<n$, any measurable outcome type $O$, and any assignment $L_{\ell,c}$ of a measure on $O$ to an input label $\ell=((r_i,a_i,b_i))_i$ and a public tuple $c$, the definition gives the following real number. Let $R_i,A_i,B_i,C_i$ be the protocol’s raw, preparation, Bob-setting and public index sets, let $u_i,p_i,q_i$ be its respective normalized nonnegative raw, preparation and Bob-setting weights, and put $K_i=\sum_{a,b}\mathbf1_{\mathrm{keep}_i(a,b)}p_i(a)q_i(b)>0$. Give a label weight $w(\ell)=\prod_i\bigl(u_i(r_i)p_i(a_i)q_i(b_i)/K_i\bigr)$ if every local pair is retained, and weight $0$ otherwise. Its raw tuple is $r(\ell)=(r_i)_i$ and its public tuple is $c(\ell)=(\mathrm{announce}_i(a_i,b_i))_i$. Define $m_{c,x}=\sum_{\ell:\,c(\ell)=c,\ r(\ell)=x}w(\ell)$, $m_c=\sum_xm_{c,x}$, $\nu_{c,x}=\sum_{\ell:\,c(\ell)=c,\ r(\ell)=x}\max(w(\ell),0)L_{\ell,c}$, and $\nu_c=\sum_x\nu_{c,x}$. For any real-valued function $p$ on the finite raw-tuple set, put $H(p)=-\sum_xp(x)\log(p(x))/\log2$, and put $h_{c,y}(x)=(d\nu_{c,x}/d\nu_c(y))_{\mathbb R}$, using the extended-nonnegative Radon–Nikodym derivative and real conversion that sends $+\infty$ to $0$. Then

$$\operatorname{experimentInformation}(P,L) =\sum_{c\in\prod_iC_i} \left[ m_cH\!\left(x\mapsto\frac{m_{c,x}}{m_c}\right) -\int_O H(h_{c,y})\,d\nu_c(y) \right].$$

The assignment $L$ is arbitrary; it is not assumed to consist of probability measures, and no integrability hypothesis is required. Division by $0$ and $\log0$ use their total real-function values, both $0$, and the real-valued integral is $0$ when its integrand is not integrable. For $n=0$, all tuple sets are singletons and empty products have value $1$.

QAlgorithms.Papers.Independent.Intakethird100.P18.inputAmplitude

For every natural number $n$, every family of local protocols $P_i$ indexed by $i<n$, every input family $l_i=(x_i,a_i,b_i)$ in the respective finite ranges, and every signal index $z=(z_i)_{i<n}$ with $0\le z_i<P_i.\mathrm{dimension}$, the input amplitude is the complex number $\prod_{i<n}\psi^{(i)}_{x_i,a_i}(z_i)$, where $\psi^{(i)}_{x_i,a_i}=P_i.\mathrm{prepare}(x_i,a_i)\in\mathbb C^{P_i.\mathrm{dimension}}$ is the protocol’s unit vector. This formula uses neither the Bob choices nor the weights or retention conditions. For $n=0$, its value is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P18.posterior

For any natural number $n$, including $0$, let $P_i$ be a local protocol for each $i<n$, and write $R_i,A_i,B_i,C_i$ for its finite raw-value, preparation-setting, Bob-setting and public-announcement sets. An input label is a tuple $\ell=((r_i,a_i,b_i))_{i<n}$; a raw tuple is $x\in\prod_iR_i$, and a public tuple is $c\in\prod_iC_i$. Define the retained mass $K_i=\sum_{a\in A_i}\sum_{b\in B_i}\mathbf1_{\mathrm{keep}_i(a,b)}p_i(a)q_i(b)>0$, where $p_i,q_i$ are the protocol’s preparation and Bob-setting weights, and define $w(\ell)=\prod_i w_i(r_i,a_i,b_i)$, where $w_i(r,a,b)=u_i(r)p_i(a)q_i(b)/K_i$ when the pair $(a,b)$ is retained and $0$ otherwise, with $u_i$ the raw-value weight. Write $c(\ell)=(\mathrm{announce}_i(a_i,b_i))_i$ and $r(\ell)=(r_i)_i$. For any type $O$ equipped with a measurable space and any assignment of a measure $L_{\ell,c}$ on $O$ to every label and public tuple, set $\nu_{c,x}=\sum_{\ell:\,c(\ell)=c,\ r(\ell)=x}\max(w(\ell),0)L_{\ell,c}$ and $\nu_c=\sum_x\nu_{c,x}$. For every such $c$, every $y\in O$, and every raw tuple $x$, the defined posterior is

$$\operatorname{post}(c,y,x)=\left(\frac{d\nu_{c,x}}{d\nu_c}(y)\right)_{\mathbb R},$$

where the derivative is the measure library’s Radon–Nikodym derivative, valued in the extended nonnegative reals, and conversion to a real sends $+\infty$ to $0$. No probability, finiteness, absolute-continuity or sigma-finiteness hypothesis is imposed on the supplied measures, and this is a pointwise definition for every $y$. When $n=0$, the label, raw and public tuple sets are singletons and the empty product defining $w$ is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P18.priorMass

For any natural number $n$ and family of local protocols $P_i$, let $X_i,A_i,B_i,C_i$ be their finite raw, preparation, Bob, and public alphabets. Each protocol supplies nonnegative unit-sum weights $r_i,p_i,q_i$ on its first three alphabets, a Boolean retention rule $k_i:A_i\times B_i\to\{\mathrm{false},\mathrm{true}\}$, and an announcement map $t_i:A_i\times B_i\to C_i$, with positive retained mass $s_i=\sum_{a,b}\mathbf1_{\{k_i(a,b)=\mathrm{true}\}}p_i(a)q_i(b)$. Define

$$\lambda_i(u,a,b)= \begin{cases} r_i(u)p_i(a)q_i(b)/s_i,&k_i(a,b)=\mathrm{true},\\ 0,&\text{otherwise}. \end{cases}$$

For every public tuple $c\in\prod_i C_i$ and raw tuple $x\in\prod_iX_i$, the prior mass is

$$\sum_{l\in\prod_i(X_i\times A_i\times B_i)} \begin{cases} \displaystyle\prod_i\lambda_i(u_i,a_i,b_i), & t_i(a_i,b_i)=c_i\ \text{and }u_i=x_i\ \text{for every }i,\\ 0,&\text{otherwise}, \end{cases}$$

where $l_i=(u_i,a_i,b_i)$. No positivity assumption is made on the resulting mass for the chosen $c,x$. For $n=0$, all tuple spaces contain one empty tuple, the conditions are vacuous, and the value is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P18.productDisturbance

For every natural number $n$, including $0$, every family of local protocols $P_i$, $i<n$, and every family $A_i$ of single attacks on them, this defines a real-valued function on the site indices by taking the sole-site disturbance of $A_i$ at coordinate $i$. Explicitly, let $u_i(r),p_i(a),q_i(b)$ be the protocol’s raw, preparation and Bob-setting weights, let $K_i=\sum_{a,b}\mathbf1_{\mathrm{keep}_i(a,b)}p_i(a)q_i(b)>0$, and put $w_i(r,a,b)=u_i(r)p_i(a)q_i(b)/K_i$ for retained pairs and $0$ otherwise. Each single attack supplies a complete complex inner-product ancilla space $E_i$, a unit vector $e_i$, a surjective complex-linear isometry $U_i$ on its one-site signal–ancilla $\ell^2$ space, and public-dependent measurements on a measurable outcome type. Identifying one-site signal tuples with the local signal set $S_i$, let $v^{(i)}_{r,a,b}=U_i(s\mapsto\mathrm{prepare}_i(r,a)(s)e_i)$. With $M_{i,b}(r)$ denoting Bob’s local effect for raw outcome $r$ at setting $b$, the defined function is

$$\operatorname{productDisturbance}(A)_i =\sum_{r,a,b}w_i(r,a,b) \left[ 1-\operatorname{Re}\sum_{s,t\in S_i} M_{i,b}(r)_{s,t} \langle v^{(i)}_{r,a,b}(s),v^{(i)}_{r,a,b}(t)\rangle \right].$$

The component measurements do not enter this formula. When $n=0$, the result is the unique real-valued function on the empty index set.

QAlgorithms.Papers.Independent.Intakethird100.P18.productInformation

For any natural number $n$, any family $P=(P_i)_{i<n}$ of local protocols, and any tuple $A=(A_i)_{i<n}$ of single-site attacks, this defines a real number as follows. Each protocol $P_i$ has positive finite raw, preparation, Bob-choice, public, and signal index sets $X_i,S_i,B_i,C_i,Z_i$; nonnegative weights $r_i,p_i,q_i$ on $X_i,S_i,B_i$, respectively, each summing to $1$; unit vectors $\psi^i_{x,a}\in\mathbb C^{Z_i}$; and, for each $b\in B_i$, positive semidefinite matrices $M^i_{b,x}$ indexed by $x\in X_i$, with $\sum_xM^i_{b,x}=I$. It also has a retention function $k_i:S_i\times B_i\to\{\mathrm{false},\mathrm{true}\}$ and announcement function $c_i:S_i\times B_i\to C_i$, with retained mass $s_i=\sum_{a,b:k_i(a,b)=\mathrm{true}}p_i(a)q_i(b)>0$, and requires $\operatorname{Re}\operatorname{tr}(M^i_{b,x}\psi^i_{x,a}(\psi^i_{x,a})^*)=1$ whenever $k_i(a,b)=\mathrm{true}$. For a label $\ell_i=(x_i,a_i,b_i)$, put $w_i(\ell_i)=r_i(x_i)p_i(a_i)q_i(b_i)/s_i$ if it is retained, and $0$ otherwise. A single-site attack $A_i$ consists of a complete complex inner-product space $E_i$, a unit vector $e_i\in E_i$, a complex-linear isometric equivalence $U_i$ of the space of $E_i$-valued functions on $Z_i$ with its Euclidean norm, an arbitrary measurable outcome space $O_i$, and a measurement for each public value $c\in C_i$. Such a measurement assigns each measurable set $D\subseteq O_i$ a continuous complex-linear operator $F^i_c(D)$, with $\langle v,F^i_c(D)v\rangle$ real and nonnegative for every $v$, and $F^i_c(O_i)=I$, together with a measure $\lambda^i_{c,v}$ satisfying $\lambda^i_{c,v}(D)=\max\{\operatorname{Re}\langle v,F^i_c(D)v\rangle,0\}$. Define $\Phi^i_{\ell_i}=U_i(z\mapsto\psi^i_{x_i,a_i}(z)e_i)$ and $\mu^i_{\ell_i,c}=\sum_{z\in Z_i}\lambda^i_{c,\Phi^i_{\ell_i}(z)}$. For joint labels $\ell=(\ell_i)_i$, set $W(\ell)=\prod_iw_i(\ell_i)$, $x(\ell)=(x_i)_i$, $c(\ell)=(c_i(a_i,b_i))_i$, and let $\mu_{\ell,c}=\bigotimes_i\mu^i_{\ell_i,c_i}$ be the product outcome measure on $\prod_iO_i$. For $c\in\prod_iC_i$ and $x\in\prod_iX_i$, 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 $\nu_c=\sum_x\nu_{c,x}$. Write $h_{c,y}(x)$ for the real value of the Radon–Nikodym derivative $d\nu_{c,x}/d\nu_c$ at $y$, with an infinite derivative converted to $0$, and write $H(t)=-\sum_x t(x)\log(t(x))/\log 2$. The defined value is

$$\operatorname{productInformation}(A) =\sum_c\left[ p_c\,H\!\left(x\mapsto\frac{\pi_{c,x}}{p_c}\right) -\int H(h_{c,y})\,d\nu_c(y) \right].$$

Zero public masses are included, with division by zero evaluated as $0$; the real logarithm at $0$ is $0$, and the integral is the total real Lebesgue integral, which is $0$ for a nonintegrable integrand. The definition includes $n=0$, with empty products equal to $1$; in that case its value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P18.publicInput

For every natural number $n$, every family of local protocols $P_i$ indexed by $i<n$, and every input family $l_i=(x_i,a_i,b_i)$ in the respective raw, preparation, and Bob ranges, the associated public value is the family $c_i=P_i.\mathrm{announce}(a_i,b_i)$, with $0\le c_i<P_i.\mathrm{publicSize}$. This is defined for every input, including unretained choices and choices of weight zero. For $n=0$, it is the empty family.

QAlgorithms.Papers.Independent.Intakethird100.P18.publicMass

For any natural number $n$ and family of local protocols $P_i$, let $X_i,A_i,B_i,C_i$ be their finite raw, preparation, Bob, and public alphabets. Write $r_i,p_i,q_i$ for their nonnegative unit-sum weights, $k_i$ for their Boolean retention rules, and $t_i:A_i\times B_i\to C_i$ for their announcement maps. With the required positive masses $s_i=\sum_{a,b}\mathbf1_{\{k_i(a,b)=\mathrm{true}\}}p_i(a)q_i(b)$, set $\lambda_i(u,a,b)=r_i(u)p_i(a)q_i(b)/s_i$ when $k_i(a,b)=\mathrm{true}$, and set it to zero otherwise. For every public tuple $c\in\prod_iC_i$, the public mass is

$$\sum_{x\in\prod_iX_i} \ \sum_{l\in\prod_i(X_i\times A_i\times B_i)} \begin{cases} \displaystyle\prod_i\lambda_i(u_i,a_i,b_i), & t_i(a_i,b_i)=c_i\ \text{and }u_i=x_i\ \text{for every }i,\\ 0,&\text{otherwise}, \end{cases}$$

where $l_i=(u_i,a_i,b_i)$. The chosen public tuple need not have positive mass. For $n=0$, each sum has one term, its conditions are vacuous, and the value is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P18.publicOutcome

For any natural number $n$, family of local protocols $P_i$, and measurable space $O$, let $X_i,A_i,B_i,C_i$ be the protocols’ finite raw, preparation, Bob, and public alphabets. Write $r_i,p_i,q_i$ for their nonnegative unit-sum weights, $k_i$ for their Boolean retention rules, and $t_i:A_i\times B_i\to C_i$ for their announcement maps. Define the required positive retained masses $s_i=\sum_{a,b}\mathbf1_{\{k_i(a,b)=\mathrm{true}\}}p_i(a)q_i(b)$, and let $\lambda_i(u,a,b)=r_i(u)p_i(a)q_i(b)/s_i$ when $k_i(a,b)=\mathrm{true}$, and zero otherwise. Given any assignment of a measure $\nu_{l,c}$ on $O$ to every label tuple $l\in L=\prod_i(X_i\times A_i\times B_i)$ and public tuple $c\in\prod_iC_i$, the public outcome measure at $c$ is

$$\sum_{x\in\prod_iX_i}\ \sum_{l\in L} \begin{cases} \displaystyle \operatorname{ofReal}\!\left(\prod_i\lambda_i(u_i,a_i,b_i)\right)\nu_{l,c}, & t_i(a_i,b_i)=c_i\ \text{and }u_i=x_i\ \text{for every }i,\\ 0,&\text{otherwise}, \end{cases}$$

where $l_i=(u_i,a_i,b_i)$ and $\operatorname{ofReal}$ converts a real scalar to an extended nonnegative scalar by truncating negative values to zero. The supplied measures may be arbitrary, and no measurability of their assignment is required. The definition does not divide by the mass of the chosen public tuple. For $n=0$, both sums have one term and the result is the supplied measure at the unique empty input and public tuples.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdDisturbance

For any natural number $n$, family of local protocols $P_i$, and attack on that family, each protocol supplies finite raw, preparation, and Bob alphabets $X_i,A_i,B_i$; nonnegative weights $r_i(x),p_i(a),q_i(b)$, each summing to $1$ on its alphabet; a Boolean retention rule $k_i(a,b)$; unit preparation vectors $v_i(x,a)\in\mathbb C^{d_i}$; and positive semidefinite matrices $M_i(b,x)$ summing to the identity over $x$. Its retained mass

$$s_i=\sum_{a\in A_i}\sum_{b\in B_i} \mathbf1_{\{k_i(a,b)=\mathrm{true}\}}p_i(a)q_i(b)$$

is required to be positive. For $l=(x_i,a_i,b_i)_i\in L=\prod_i(X_i\times A_i\times B_i)$, put

$$w(l)=\prod_i \begin{cases} r_i(x_i)p_i(a_i)q_i(b_i)/s_i,&k_i(a_i,b_i)=\mathrm{true},\\ 0,&\text{otherwise}. \end{cases}$$

If the attack has complete complex inner product ancilla space $E$, unit initial vector $e$, and bijective complex linear isometry $U$ on $\ell^2(Z;E)$, where $Z=\prod_i\{0,\ldots,d_i-1\}$, set $\Psi_l=U(z\mapsto(\prod_jv_j(x_j,a_j)(z_j))e)$. The disturbance is the function assigning to each site $i$ the real number

$$\sum_{l\in L}w(l)\left[ 1- \operatorname{Re}\left( \sum_{\substack{z,w\in Z\\z_j=w_j\ \forall j\ne i}} M_i(b_i,x_i)_{z_i,w_i} \langle\Psi_l(z),\Psi_l(w)\rangle \right)\right].$$

All label tuples are summed, including those with zero weight. For $n=0$, the result is the unique function with empty domain.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdInformation

For every natural number $n$, including $0$, every family of local protocols $P_i$, $i<n$, and every attack $A$ on that family, this definition assigns a real information value as follows. The attack supplies a complete complex inner-product space $E$, a vector $e\in E$ with $\|e\|=1$, a surjective complex-linear isometry $U$ on the space of $E$-valued functions on the finite signal-tuple set $S=\prod_i\{0,\ldots,\mathrm{dimension}_i-1\}$, equipped with the $\ell^2$ norm, a measurable outcome type $O$, and a measurement for each public tuple $c$. Each measurement supplies continuous complex-linear operators $F_c(B)$ on $E$, with real nonnegative quadratic forms on measurable sets $B$, with $F_c(O)=I$, and measures $\lambda_{c,v}$ satisfying $\lambda_{c,v}(B)=\max(\operatorname{Re}\langle v,F_c(B)v\rangle,0)$. For a label $\ell=((r_i,a_i,b_i))_i$, define

$$v_\ell =U\!\left(z\mapsto \left[\prod_i\mathrm{prepare}_i(r_i,a_i)(z_i)\right]e\right), \qquad L_{\ell,c}=\sum_{z\in S}\lambda_{c,v_\ell(z)}.$$

Let $K_i=\sum_{a,b}\mathbf1_{\mathrm{keep}_i(a,b)}p_i(a)q_i(b)>0$, using the protocol’s preparation and Bob-setting weights, and let $w(\ell)=\prod_i u_i(r_i)p_i(a_i)q_i(b_i)/K_i$ when every pair is retained, and $0$ otherwise, where $u_i$ is its raw-value weight. Set $c(\ell)=(\mathrm{announce}_i(a_i,b_i))_i$, $r(\ell)=(r_i)_i$, $m_{c,x}=\sum_{\ell:\,c(\ell)=c,\ r(\ell)=x}w(\ell)$, $m_c=\sum_xm_{c,x}$, $\nu_{c,x}=\sum_{\ell:\,c(\ell)=c,\ r(\ell)=x}\max(w(\ell),0)L_{\ell,c}$, and $\nu_c=\sum_x\nu_{c,x}$. With $H(p)=-\sum_xp(x)\log(p(x))/\log2$, the assigned value is

$$\operatorname{qkdInformation}(A) =\sum_c\left[ m_cH\!\left(x\mapsto\frac{m_{c,x}}{m_c}\right) -\int_O H\!\left(x\mapsto \left(\frac{d\nu_{c,x}}{d\nu_c}(y)\right)_{\mathbb R}\right)d\nu_c(y) \right].$$

Real conversion of an infinite derivative is $0$; real division by $0$ and $\log0$ are $0$, and a nonintegrable real-valued integral is assigned $0$. No positive lower bound on $n$ is required; at $n=0$, the tuple sets are singletons and empty products equal $1$.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdInputWeight

For every natural number $n$, every family of local protocols $P_i$ indexed by $i<n$, and every input family $l_i=(x_i,a_i,b_i)$ in the respective raw, preparation, and Bob ranges, the input weight is $\prod_{i<n}w_i$, where $w_i=P_i.\mathrm{rawWeight}(x_i)P_i.\mathrm{prepWeight}(a_i)P_i.\mathrm{bobWeight}(b_i)/R_i$ when $P_i.\mathrm{keep}(a_i,b_i)=\mathrm{true}$, and $w_i=0$ otherwise; here $R_i=\sum_{a,b:\,P_i.\mathrm{keep}(a,b)=\mathrm{true}}P_i.\mathrm{prepWeight}(a)P_i.\mathrm{bobWeight}(b)>0$. Thus an unretained local choice gives input weight zero, and for $n=0$ the empty product gives input weight $1$.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdJointOutcome

For any natural number $n$, family of local protocols $P_i$, and type $O$ equipped with a measurable space structure, let $X_i,A_i,B_i,C_i$ be the protocols’ finite raw, preparation, Bob, and public alphabets. Their nonnegative unit-sum weights $r_i,p_i,q_i$, Boolean retention rules $k_i$, and announcement maps $t_i:A_i\times B_i\to C_i$ determine positive retained masses $s_i=\sum_{a,b}\mathbf1_{\{k_i(a,b)=\mathrm{true}\}}p_i(a)q_i(b)$ and weights $\lambda_i(u,a,b)=r_i(u)p_i(a)q_i(b)/s_i$ for retained pairs, with $\lambda_i(u,a,b)=0$ otherwise. Let $L=\prod_i(X_i\times A_i\times B_i)$, $C=\prod_i C_i$, and suppose an arbitrary measure $\nu_{l,c}$ on $O$ is supplied for every $l\in L,c\in C$; these measures are not required to be probability measures or to arise from an attack. For every $c\in C$ and $x\in\prod_iX_i$, the joint outcome measure is

$$J_{c,x}= \sum_{l\in L} \begin{cases} \displaystyle \operatorname{ofReal}\!\left(\prod_i\lambda_i(u_i,a_i,b_i)\right)\nu_{l,c}, & t_i(a_i,b_i)=c_i\ \text{and }u_i=x_i\ \text{for every }i,\\ 0,&\text{otherwise}, \end{cases}$$

where $l_i=(u_i,a_i,b_i)$, $\operatorname{ofReal}$ replaces a negative real number by zero and regards the result as an extended nonnegative real, and multiplication denotes scalar multiplication of a measure. No measurability condition on the assignment $(l,c)\mapsto\nu_{l,c}$ is assumed. For $n=0$, this is the supplied measure at the unique empty input and public tuples.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdLabelWeight

For a local protocol $P$ and any triple $l=(x,a,b)$ of a raw label, a preparation choice, and a Bob choice in their respective finite ranges, the label weight is $P.\mathrm{rawWeight}(x)\,P.\mathrm{prepWeight}(a)\,P.\mathrm{bobWeight}(b)/R_P$ if $P.\mathrm{keep}(a,b)=\mathrm{true}$, and is $0$ otherwise, where $R_P=\sum_{a',b':\,P.\mathrm{keep}(a',b')=\mathrm{true}}P.\mathrm{prepWeight}(a')P.\mathrm{bobWeight}(b')$. This denominator is strictly positive by the requirements on $P$; a retained triple can still have weight zero.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdOneSite

For every local protocol $P$, this defines the family indexed by the singleton set $\{0\}$ whose value at its only index is $P$:

$$\operatorname{qkdOneSite}(P)_0=P.$$

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdOutcomeLaw

For any natural number $n$, family of local protocols $P_i$, attack on that family, input label tuple $l_i=(x_i,a_i,b_i)$, and public tuple $c=(c_i)$ in the product of the protocols’ finite public alphabets, let $E$, $e$, and $U$ be the attack’s complete complex inner product ancilla space, unit initial vector, and bijective complex linear isometry on $\ell^2(Z;E)$, where $Z=\prod_i\{0,\ldots,d_i-1\}$. Using the protocols’ unit preparation vectors $v_i(x_i,a_i)$, define

$$\Psi_l=U\left(z\longmapsto\left(\prod_i v_i(x_i,a_i)(z_i)\right)e\right).$$

The attack has a measurable outcome space $O$ and, for each public tuple $c$ and ancilla vector $v$, a measure $\mu_{c,v}$ on $O$, with continuous complex linear effects $F_{c,B}$ satisfying $F_{c,O}=\mathrm{id}$ and, for measurable $B$, real nonnegative $\langle v,F_{c,B}v\rangle$ and $\mu_{c,v}(B)=\operatorname{ofReal}(\operatorname{Re}\langle v,F_{c,B}v\rangle)$. The outcome law is the measure

$$\nu_{l,c}=\sum_{z\in Z}\mu_{c,\Psi_l(z)}.$$

The public tuple $c$ is an independent argument; it is not required to equal the announcement associated with $l$. When $n=0$, the sum has one term, corresponding to the empty signal tuple.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdOutputState

For any natural number $n$, family of local protocols $P_i$, and attack on that family, let $E$ be its complete complex inner product ancilla space, $e\in E$ its unit initial vector, and $U$ its bijective complex linear isometry on the space of $E$-valued functions on $Z=\prod_i\{0,\ldots,d_i-1\}$, with the finite $\ell^2$ norm. Each protocol supplies a unit preparation vector $v_i(x,a)\in\mathbb C^{d_i}$, indexed by a raw label $x$ and preparation label $a$. For any input label tuple $l$, whose $i$-th component is a triple $l_i=(x_i,a_i,b_i)$ of raw, preparation, and Bob labels in their respective finite alphabets, the output state is defined by

$$\Psi_l = U\left(z\longmapsto \left(\prod_{i=0}^{n-1}v_i(x_i,a_i)(z_i)\right)e\right).$$

The label tuple is unrestricted: no retention condition or positive label weight is required, and its Bob labels do not enter this formula. For $n=0$, the product is $1$, there is a single signal coordinate, and the input to $U$ is the function taking that coordinate to $e$.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdProductLaw

For every natural number $n$, including $0$, every family of local protocols $P_i$, every family $A_i$ of single attacks on those protocols, every input-label tuple $\ell=(\ell_i)_i$, and every public tuple $c=(c_i)_i$, this defines a measure on the dependent outcome-product space $\prod_{i<n}O_i$, with its product measurable structure, where $O_i$ is the measurable outcome type of $A_i$. To specify each component measure, write $\ell_i=(r_i,a_i,b_i)$, let $E_i$ be the attack’s complete complex inner-product ancilla space, $e_i$ its unit initial vector, and $U_i$ its surjective complex-linear isometry on the one-site signal–ancilla $\ell^2$ space. Identify the one-site signal tuples with the local signal set $S_i$, and put $v_i=U_i(s\mapsto\mathrm{prepare}_i(r_i,a_i)(s)e_i)$. The measurement selected by public value $c_i$ supplies measures $\lambda_{i,c_i,v}$ satisfying $\lambda_{i,c_i,v}(B)=\max(\operatorname{Re}\langle v,F_{i,c_i}(B)v\rangle,0)$ on measurable sets, with real nonnegative operator quadratic forms and $F_{i,c_i}(O_i)=I$. Then

$$\operatorname{qkdProductLaw}(A,\ell,c) =\bigotimes_{i<n}\left(\sum_{s\in S_i}\lambda_{i,c_i,v_i(s)}\right).$$

This is the measure-theoretic product of the component measures; it includes no label weights and imposes no retention or announcement-consistency condition on $\ell,c$. When $n=0$, it uses the empty product of measures on the singleton empty-tuple space.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdRetainedMass

For a local protocol $P$, its retained mass is the real number $R_P=\sum_{a=0}^{P.\mathrm{prepSize}-1}\sum_{b=0}^{P.\mathrm{bobSize}-1}\mathbf 1_{\{P.\mathrm{keep}(a,b)=\mathrm{true}\}}\,P.\mathrm{prepWeight}(a)\,P.\mathrm{bobWeight}(b)$, where the indicator is $1$ when its condition holds and $0$ otherwise. The requirements on $P$ ensure $R_P>0$.

QAlgorithms.Papers.Independent.Intakethird100.P18.qkdShannonBits

For any type $X$ equipped with a finite enumeration and any function $p:X\to\mathbb R$, the defined real number is

$$-\sum_{x\in X}p(x)\frac{\log(p(x))}{\log 2}.$$

There is no assumption that the values of $p$ are nonnegative or sum to $1$. The logarithm is the total real logarithm used by the code: $\log 0=0$, and for negative arguments it equals the logarithm of the absolute value. Thus zero-valued terms contribute zero, negative values are also covered by the definition, and an empty $X$ gives zero.

QAlgorithms.Papers.Independent.Intakethird100.P18.rawInput

For every natural number $n$, every family of local protocols $P_i$ indexed by $i<n$, and every input family $l_i=(x_i,a_i,b_i)$ in the respective raw, preparation, and Bob ranges, the associated raw value is the family $(x_i)_{i<n}$, obtained by taking the raw component at every index. This is defined regardless of retention or weights; for $n=0$, it is the empty family.

QAlgorithms.Papers.Independent.Intakethird100.P18.singleDisturbance

For every local protocol $P$ and every single attack $A$ on it, this defines a real disturbance at the sole site. Write $R,A_0,B_0,S$ for the protocol’s finite raw, preparation, Bob-setting and signal index sets, $u(r),p(a),q(b)$ for its normalized nonnegative weights, and $M_b(r)$ for Bob’s effect for raw outcome $r$ at setting $b$. Let $K=\sum_{a,b}\mathbf1_{\mathrm{keep}(a,b)}p(a)q(b)>0$, and let $w(r,a,b)=u(r)p(a)q(b)/K$ for retained pairs and $0$ otherwise. The single attack supplies a complete complex inner-product ancilla space $E$, a unit vector $e$, a surjective complex-linear isometry $U$ on the one-site signal–ancilla space, and public-dependent measurements on a measurable outcome type; those measurements do not enter this disturbance formula. Identifying the singleton-indexed signal tuples with $S$, put $v_{r,a,b}=U(s\mapsto\mathrm{prepare}(r,a)(s)e)$. Then

$$\operatorname{singleDisturbance}(A) =\sum_{r\in R}\sum_{a\in A_0}\sum_{b\in B_0} w(r,a,b) \left[ 1-\operatorname{Re}\sum_{s,t\in S} M_b(r)_{s,t}\, \langle v_{r,a,b}(s),v_{r,a,b}(t)\rangle \right].$$

This is exactly coordinate $0$ of the disturbance function for the singleton protocol family.

QAlgorithms.Papers.Independent.Intakethird100.P18.singleLaw

For every local protocol $P$, every single attack $A$ on $P$, every label $\ell=(r,a,b)$ consisting of a raw value, preparation setting and Bob setting, and every public value $c$, this defines a measure on the attack’s measurable outcome type $O$. The attack has a complete complex inner-product ancilla space $E$, a unit initial vector $e$, and a surjective complex-linear isometry $U$ on the one-site signal–ancilla $\ell^2$ space. Its measurement at the singleton public tuple with value $c$ provides measures $\lambda_{c,v}$ and continuous complex-linear operators $F_c(B)$, with real nonnegative quadratic forms on measurable $B$, $F_c(O)=I$, and $\lambda_{c,v}(B)=\max(\operatorname{Re}\langle v,F_c(B)v\rangle,0)$ for measurable $B$. Identifying the singleton-indexed signal tuples with the protocol’s signal set $S$, put $v_\ell=U(s\mapsto\mathrm{prepare}(r,a)(s)e)$. The defined measure is

$$\operatorname{singleLaw}(A,\ell,c)=\sum_{s\in S}\lambda_{c,v_\ell(s)}.$$

Both the input label and public value are passed as constant singleton-indexed tuples. The definition requires neither that $(a,b)$ is retained nor that $c=\mathrm{announce}(a,b)$, and it applies no label-weight factor.