Definitions for Device-independent quantum key distribution based on measurement inputs

QAlgorithms.Papers.Independent.Intakethird100.P47.Behavior

A behavior is an arbitrary real-valued function $p(A,B,a,b)$ of four Boolean variables $A,B,a,b\in\{\mathrm{false},\mathrm{true}\}$. This type imposes no positivity or normalization conditions.

QAlgorithms.Papers.Independent.Intakethird100.P47.HardyConditions

For any real-valued function $p$ of four Boolean arguments, $\operatorname{HardyConditions}(p)$ means exactly

$$p(0,0,0,0)>0,\qquad p(1,0,0,0)=0,\qquad p(0,1,0,0)=0,\qquad p(1,1,1,1)=0,$$

where $0$ denotes false and $1$ denotes true. No nonnegativity or normalization requirements on the other values of $p$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P47.HardyParameters

A set of Hardy parameters consists of two functions $\alpha,\beta:\{\mathrm{false},\mathrm{true}\}\to\mathbb C$, together with the requirements that, for each Boolean $P$,

$$|\alpha_P|^2+|\beta_P|^2=1, \qquad 0<|\alpha_P|<1.$$

QAlgorithms.Papers.Independent.Intakethird100.P47.HasFiniteQuantumRealization

For any real-valued function $p(A,B,a,b)$ of four Boolean arguments, this property means that there exist natural numbers $d_A,d_B$, a complex matrix $\rho$ on the product of the two corresponding index sets, and complex matrix families $M_{A,a}$ and $N_{B,b}$ of sizes $d_A\times d_A$ and $d_B\times d_B$, respectively, such that $\rho$ is positive semidefinite with $\operatorname{tr}\rho=1$, every $M_{A,a}$ and $N_{B,b}$ is positive semidefinite, and

$$\sum_{a\in\{0,1\}}M_{A,a}=I_{d_A},\qquad \sum_{b\in\{0,1\}}N_{B,b}=I_{d_B},\qquad p(A,B,a,b)=\operatorname{Re}\operatorname{tr}\!\bigl((M_{A,a}\otimes N_{B,b})\rho\bigr)$$

for every choice of Boolean arguments, with false represented by $0$ and true by $1$. The existential quantifiers allow zero dimensions, but either dimension being zero makes the trace-one requirement impossible.

QAlgorithms.Papers.Independent.Intakethird100.P47.IsBehavior

For a real-valued function $p(A,B,a,b)$ of four Boolean variables, being a behavior satisfying this predicate means that $p(A,B,a,b)\geq 0$ for every $A,B,a,b$, and that, for every fixed $A,B$,

$$\sum_{a\in\{\mathrm{false},\mathrm{true}\}} \sum_{b\in\{\mathrm{false},\mathrm{true}\}}p(A,B,a,b)=1.$$

QAlgorithms.Papers.Independent.Intakethird100.P47.IsPairLaw

For a real-valued function $w(A,B)$ of two Boolean variables, this predicate means that $w(A,B)\geq 0$ for every $A,B$, and that

$$\sum_{A\in\{\mathrm{false},\mathrm{true}\}} \sum_{B\in\{\mathrm{false},\mathrm{true}\}}w(A,B)=1.$$

QAlgorithms.Papers.Independent.Intakethird100.P47.PairLaw

A pair law is an arbitrary real-valued function $w(A,B)$ of two Boolean variables $A,B\in\{\mathrm{false},\mathrm{true}\}$. This type imposes no positivity or normalization conditions.

QAlgorithms.Papers.Independent.Intakethird100.P47.acceptedLaw

For any real-valued function $p$ of four Boolean arguments and any real-valued function $w$ on Boolean pairs, $\operatorname{acceptedLaw}(p,w)$ is the function

$$\operatorname{acceptedLaw}(p,w)(A,B)= \frac{w(A,B)\,p(A,B,\mathrm{false},\mathrm{false})} {\displaystyle\sum_{C\in\{\mathrm{false},\mathrm{true}\}} \sum_{D\in\{\mathrm{false},\mathrm{true}\}} w(C,D)\,p(C,D,\mathrm{false},\mathrm{false})}.$$

No positivity or normalization hypotheses are imposed, and the denominator need not be nonzero; if it is zero, real division by zero makes every entry of this function zero.

QAlgorithms.Papers.Independent.Intakethird100.P47.acceptedMass

For any real-valued function $p$ of four Boolean arguments and any real-valued function $w$ on Boolean pairs, $\operatorname{acceptedMass}(p,w)$ is the function on Boolean pairs defined by

$$\operatorname{acceptedMass}(p,w)(A,B)=w(A,B)\,p(A,B,\mathrm{false},\mathrm{false}).$$

Neither function is assumed to be nonnegative or normalized.

QAlgorithms.Papers.Independent.Intakethird100.P47.aliceMarginal

For any real-valued function $j$ on Boolean pairs and any Boolean $A$, $\operatorname{aliceMarginal}(j,A)$ is the real number

$$\operatorname{aliceMarginal}(j,A)=j(A,\mathrm{false})+j(A,\mathrm{true}).$$

The function $j$ is not assumed to be nonnegative or normalized.

QAlgorithms.Papers.Independent.Intakethird100.P47.aliceMass

For any real-valued function $p$ of four Boolean arguments, any real-valued function $w$ on Boolean pairs, and any Boolean $A$, define

$$\operatorname{aliceMass}(p,w,A)= \sum_{B\in\{\mathrm{false},\mathrm{true}\}} w(A,B)\,p(A,B,\mathrm{false},\mathrm{false}).$$

There are no nonnegativity or normalization assumptions on either function.

QAlgorithms.Papers.Independent.Intakethird100.P47.balanceRetention

For any function $j:\{0,1\}^2\to\mathbb R$ and Boolean $A$, where $0$ denotes false and $1$ denotes true, define $m_A=j(A,0)+j(A,1)$. This definition assigns

$$\frac{\min(m_0,m_1)}{m_A}$$

to $A$. No nonnegativity, normalization, or positivity assumption is imposed on $j$ or its row sums. Division is total: if $m_A=0$, the assigned value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P47.biasedInput

For arbitrary real numbers $p_A,p_B,\epsilon$ and any Boolean pair $e=(e_A,e_B)$, let $q_A=p_A+\epsilon$ if $e_A$ is true and $q_A=p_A-\epsilon$ otherwise, and let $q_B=p_B+\epsilon$ if $e_B$ is true and $q_B=p_B-\epsilon$ otherwise. Then $\operatorname{biasedInput}(p_A,p_B,\epsilon,e)$ is the real-valued function

$$w(A,B)= \begin{cases}1-q_A,&A=\mathrm{true},\\q_A,&A=\mathrm{false}\end{cases} \;\cdot\; \begin{cases}1-q_B,&B=\mathrm{true},\\q_B,&B=\mathrm{false}.\end{cases}$$

There are no restrictions on the three real parameters or on whether $q_A,q_B$ lie in $[0,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P47.binarySettingLaw

For arbitrary real numbers $p_A,p_B$, $\operatorname{binarySettingLaw}(p_A,p_B)$ is the real-valued function on Boolean pairs given, with $0=\mathrm{false}$ and $1=\mathrm{true}$, by

$$w(0,0)=p_Ap_B,\quad w(0,1)=p_A(1-p_B),\quad w(1,0)=(1-p_A)p_B,\quad w(1,1)=(1-p_A)(1-p_B).$$

The parameters are not restricted to $[0,1]$, so its entries are not required to be nonnegative.

QAlgorithms.Papers.Independent.Intakethird100.P47.bipartitePOVMBehavior

For any natural numbers $d_A,d_B$, any complex matrix $\rho$ indexed by $\{0,\ldots,d_A-1\}\times\{0,\ldots,d_B-1\}$, and any families of complex matrices $M_{A,a}$ of size $d_A\times d_A$ and $N_{B,b}$ of size $d_B\times d_B$, where $A,B,a,b\in\{0,1\}$ represent the Boolean values false and true, this definition assigns the real-valued function

$$p(A,B,a,b)=\operatorname{Re}\operatorname{tr}\!\bigl((M_{A,a}\otimes N_{B,b})\rho\bigr).$$

There are no positivity, trace, or normalization assumptions on these matrices. Either dimension may be zero; in that case the trace is an empty sum and every value of $p$ is zero.

QAlgorithms.Papers.Independent.Intakethird100.P47.gsDenominatorSq

For any finite list of vectors $u_1,\ldots,u_r$ and any vector $v$ in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}^2}$, this definition returns the real number

$$1-\sum_{j=1}^{r}|\langle u_j,v\rangle|^2, \qquad \langle u,v\rangle=\sum_{a,b\in\{\mathrm{false},\mathrm{true}\}} \overline{u(a,b)}\,v(a,b).$$

No normalization or orthogonality is assumed, so this number need not be nonnegative; repeated list entries contribute separately, and an empty list gives $1$.

QAlgorithms.Papers.Independent.Intakethird100.P47.gsResidual

For any finite list of vectors $u_1,\ldots,u_r$ and any vector $v$ in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}^2}$, this definition returns

$$v-\sum_{j=1}^{r}\langle u_j,v\rangle u_j, \qquad \langle u,v\rangle=\sum_{a,b\in\{\mathrm{false},\mathrm{true}\}} \overline{u(a,b)}\,v(a,b).$$

There are no normalization or orthogonality hypotheses on the vectors; repeated list entries contribute separately, and an empty list gives the result $v$.

QAlgorithms.Papers.Independent.Intakethird100.P47.gsStep

For any finite list of vectors $u_1,\ldots,u_r$ and any vector $v$ in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}^2}$, this definition returns

$$\left(\sqrt{\,1-\sum_{j=1}^{r}|\langle u_j,v\rangle|^2\,}\right)^{-1} \left(v-\sum_{j=1}^{r}\langle u_j,v\rangle u_j\right), \qquad \langle u,v\rangle=\sum_{a,b\in\{\mathrm{false},\mathrm{true}\}} \overline{u(a,b)}\,v(a,b).$$

The real square root is regarded as a complex scalar before taking its inverse. No normalization or orthogonality is assumed; repeated list entries contribute separately. For an empty list the result is $v$. If the expression under the square root is nonpositive, its real square root is $0$, whose inverse is defined to be $0$, so the result is the zero vector.

QAlgorithms.Papers.Independent.Intakethird100.P47.guessLaw

For arbitrary real numbers $p_0,p_1,P_0,P_1$, this defines a real-valued function $j$ on two Boolean arguments $A,G$. Writing $0$ for false and $1$ for true, its four values are

$$j(0,0)=p_0P_0,\qquad j(1,0)=p_0(1-P_0),\qquad j(0,1)=p_1(1-P_1),\qquad j(1,1)=p_1P_1.$$

There are no hypotheses restricting the four parameters, so nonnegativity and normalization are not required.

QAlgorithms.Papers.Independent.Intakethird100.P47.hardyBehavior

For parameters $d=(\alpha_P,\beta_P)_{P\in\{0,1\}}$ satisfying $|\alpha_P|^2+|\beta_P|^2=1$ and $0<|\alpha_P|<1$ for each $P$, and any complex matrix $\rho$ whose rows and columns are indexed by $\{0,1\}^2$, define local vectors

$$\ell_{P,X,a}= \begin{cases} e_a,&X=0,\\ \alpha_Pe_0+\beta_Pe_1,&X=1,\ a=0,\\ \overline{\beta_P}e_0-\overline{\alpha_P}e_1,&X=1,\ a=1, \end{cases}$$

where $0$ denotes false, $1$ denotes true, and $e_0,e_1$ are coordinate unit vectors. The real-valued function $\operatorname{hardyBehavior}(d,\rho)$ on four Boolean arguments is

$$\operatorname{hardyBehavior}(d,\rho)(A,B,a,b) =\operatorname{Re}\operatorname{tr}\!\left((vv^*)\rho\right), \qquad v=\ell_{0,A,a}\otimes\ell_{1,B,b},$$

where $v_{r,s}=(\ell_{0,A,a})_r(\ell_{1,B,b})_s$ and $(vv^*)_{z,z'}=v_z\overline{v_{z'}}$. No positivity, trace, or Hermitian assumption is imposed on $\rho$, and no nonnegativity or normalization condition is imposed separately on the resulting function.

QAlgorithms.Papers.Independent.Intakethird100.P47.hardyGS

Given complex coefficients $\alpha_P,\beta_P$ for each Boolean $P$, satisfying $|\alpha_P|^2+|\beta_P|^2=1$ and $0<|\alpha_P|<1$, let $e_c$ be the Boolean coordinate vector and let $(u\otimes w)(a,b)=u(a)w(b)$. Define

$$\begin{aligned} \phi_0&=(\overline{\beta_{\mathrm{false}}}e_{\mathrm{false}} -\overline{\alpha_{\mathrm{false}}}e_{\mathrm{true}}) \otimes(\overline{\beta_{\mathrm{true}}}e_{\mathrm{false}} -\overline{\alpha_{\mathrm{true}}}e_{\mathrm{true}}),\\ \phi_1&=e_{\mathrm{false}}\otimes (\alpha_{\mathrm{true}}e_{\mathrm{false}}+\beta_{\mathrm{true}}e_{\mathrm{true}}),\\ \phi_2&=(\alpha_{\mathrm{false}}e_{\mathrm{false}}+\beta_{\mathrm{false}}e_{\mathrm{true}}) \otimes e_{\mathrm{false}},\\ \phi_3&=e_{\mathrm{false}}\otimes e_{\mathrm{false}}. \end{aligned}$$

This definition returns the indexed family $(v_0,v_1,v_2,v_3)$ in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}^2}$ obtained by setting $v_0=\phi_0$ and, successively for $r=1,2,3$, setting

$$v_r= \left(\sqrt{\,1-\sum_{j=0}^{r-1}|\langle v_j,\phi_r\rangle|^2\,}\right)^{-1} \left(\phi_r-\sum_{j=0}^{r-1}\langle v_j,\phi_r\rangle v_j\right), \qquad \langle u,w\rangle=\sum_{a,b}\overline{u(a,b)}\,w(a,b).$$

Both inner-product indices range over the Booleans. Each reciprocal is taken in $\mathbb C$ after embedding the real square root; a nonpositive expression under a square root gives square root $0$ and reciprocal $0$.

QAlgorithms.Papers.Independent.Intakethird100.P47.hardyPhi

Given complex coefficients $\alpha_P,\beta_P$ for each Boolean $P$, satisfying $|\alpha_P|^2+|\beta_P|^2=1$ and $0<|\alpha_P|<1$, this definition returns the four vectors $\phi_0,\phi_1,\phi_2,\phi_3$ in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}^2}$:

$$\begin{aligned} \phi_0&= (\overline{\beta_{\mathrm{false}}}e_{\mathrm{false}} -\overline{\alpha_{\mathrm{false}}}e_{\mathrm{true}}) \otimes (\overline{\beta_{\mathrm{true}}}e_{\mathrm{false}} -\overline{\alpha_{\mathrm{true}}}e_{\mathrm{true}}),\\ \phi_1&=e_{\mathrm{false}}\otimes (\alpha_{\mathrm{true}}e_{\mathrm{false}}+\beta_{\mathrm{true}}e_{\mathrm{true}}),\\ \phi_2&=(\alpha_{\mathrm{false}}e_{\mathrm{false}}+\beta_{\mathrm{false}}e_{\mathrm{true}}) \otimes e_{\mathrm{false}},\\ \phi_3&=e_{\mathrm{false}}\otimes e_{\mathrm{false}}. \end{aligned}$$

Here $e_c$ is the coordinate vector at Boolean $c$, and the tensor notation means $(u\otimes v)(a,b)=u(a)v(b)$; the four indices are $0,1,2,3$.

QAlgorithms.Papers.Independent.Intakethird100.P47.hardySpan

For parameters $d$ consisting of complex numbers $\alpha_P,\beta_P$, indexed by $P\in\{0,1\}$ (where $0$ denotes false and $1$ denotes true), satisfying $|\alpha_P|^2+|\beta_P|^2=1$ and $0<|\alpha_P|<1$ for each $P$, define $s_P=\alpha_P e_0+\beta_P e_1$ and $t_P=\overline{\beta_P}e_0-\overline{\alpha_P}e_1$, where $e_0,e_1$ are the coordinate unit vectors in $\mathbb C^{\{0,1\}}$. Then $\operatorname{hardySpan}(d)$ is the complex linear subspace of $\mathbb C^{\{0,1\}\times\{0,1\}}$ spanned by the three vectors $t_0\otimes t_1$, $e_0\otimes s_1$, and $s_0\otimes e_0$, with tensor coordinates $(u\otimes v)_{a,b}=u_a v_b$.

QAlgorithms.Papers.Independent.Intakethird100.P47.hardyStatistics

For any real-valued function $p$ of four Boolean arguments, $\operatorname{hardyStatistics}(p)$ is the function on the four indices $0,1,2,3$ with ordered values

$$\bigl( p(0,0,0,0),\ -p(1,0,0,0),\ -p(0,1,0,0),\ -p(1,1,1,1) \bigr),$$

where Boolean $0$ denotes false and Boolean $1$ denotes true. No positivity, normalization, or additional equations are assumed for $p$.

QAlgorithms.Papers.Independent.Intakethird100.P47.localKet

Given complex coefficients $\alpha_P,\beta_P$ for each Boolean $P$, satisfying $|\alpha_P|^2+|\beta_P|^2=1$ and $0<|\alpha_P|<1$, and Boolean arguments $P,X,a$, this definition returns the vector in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}}$ given by

$$\begin{cases} e_a,&X=\mathrm{false},\\ \alpha_P e_{\mathrm{false}}+\beta_P e_{\mathrm{true}}, &X=\mathrm{true},\ a=\mathrm{false},\\ \overline{\beta_P}e_{\mathrm{false}}-\overline{\alpha_P}e_{\mathrm{true}}, &X=\mathrm{true},\ a=\mathrm{true}, \end{cases}$$

where $e_c$ has coordinate $1$ at $c$ and $0$ at the other Boolean coordinate.

QAlgorithms.Papers.Independent.Intakethird100.P47.noisyHardy

For parameters $d=(\alpha_P,\beta_P)_{P\in\{0,1\}}$ satisfying $|\alpha_P|^2+|\beta_P|^2=1$ and $0<|\alpha_P|<1$ for each $P$, and any real number $\eta$, define $s_P=\alpha_Pe_0+\beta_Pe_1$ and $t_P=\overline{\beta_P}e_0-\overline{\alpha_P}e_1$, where $0=\mathrm{false}$, $1=\mathrm{true}$, and $e_0,e_1$ are coordinate unit vectors. In $\mathbb C^{\{0,1\}^2}$, whose tensor coordinates satisfy $(u\otimes v)_{a,b}=u_av_b$, set $\phi_0=t_0\otimes t_1$, $\phi_1=e_0\otimes s_1$, $\phi_2=s_0\otimes e_0$, and $\phi_3=e_0\otimes e_0$; define $v_0=\phi_0$ and, for $j=1,2,3$, recursively define

$$v_j= \left(\sqrt{\,1-\sum_{r=0}^{j-1}|\langle v_r,\phi_j\rangle|^2\,}\right)^{-1} \left(\phi_j-\sum_{r=0}^{j-1}\langle v_r,\phi_j\rangle v_r\right), \qquad \langle u,v\rangle=\sum_z\overline{u_z}v_z.$$

Here the square root is the real square root, embedded into $\mathbb C$ before inversion; the real square root is zero for a nonpositive argument, and the inverse of zero is zero. Then $\operatorname{noisyHardy}(d,\eta)$ is the complex matrix indexed by Boolean pairs given by

$$\operatorname{noisyHardy}(d,\eta) =\frac{1-\eta}{4}I_4+\eta\,v_3v_3^*, \qquad (v_3v_3^*)_{z,z'}=(v_3)_z\overline{(v_3)_{z'}}.$$

The parameter $\eta$ is unrestricted and need not lie in $[0,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P47.pairEntropy

For any real-valued function $j$ on Boolean pairs, this defines

$$-\sum_{A\in\{0,1\}}\sum_{B\in\{0,1\}} j(A,B)\frac{\log(j(A,B))}{\log 2},$$

where $0$ denotes false and $1$ true. No nonnegativity or normalization is required. The real logarithm used here is total: $\log 0=0$, and for negative inputs $\log x=\log|x|$; in particular, an entry equal to zero contributes zero.

QAlgorithms.Papers.Independent.Intakethird100.P47.psiH

Given complex coefficients $\alpha_P,\beta_P$ for each Boolean $P$, satisfying $|\alpha_P|^2+|\beta_P|^2=1$ and $0<|\alpha_P|<1$, let $e_c$ be the Boolean coordinate vector and let $(u\otimes w)(a,b)=u(a)w(b)$. Form

$$\begin{aligned} \phi_0&=(\overline{\beta_{\mathrm{false}}}e_{\mathrm{false}} -\overline{\alpha_{\mathrm{false}}}e_{\mathrm{true}}) \otimes(\overline{\beta_{\mathrm{true}}}e_{\mathrm{false}} -\overline{\alpha_{\mathrm{true}}}e_{\mathrm{true}}),\\ \phi_1&=e_{\mathrm{false}}\otimes (\alpha_{\mathrm{true}}e_{\mathrm{false}}+\beta_{\mathrm{true}}e_{\mathrm{true}}),\\ \phi_2&=(\alpha_{\mathrm{false}}e_{\mathrm{false}}+\beta_{\mathrm{false}}e_{\mathrm{true}}) \otimes e_{\mathrm{false}},\\ \phi_3&=e_{\mathrm{false}}\otimes e_{\mathrm{false}}. \end{aligned}$$

Set $v_0=\phi_0$ and, successively for $r=1,2,3$, set

$$v_r= \left(\sqrt{\,1-\sum_{j=0}^{r-1}|\langle v_j,\phi_r\rangle|^2\,}\right)^{-1} \left(\phi_r-\sum_{j=0}^{r-1}\langle v_j,\phi_r\rangle v_j\right), \qquad \langle u,w\rangle=\sum_{a,b\in\{\mathrm{false},\mathrm{true}\}} \overline{u(a,b)}\,w(a,b).$$

This definition returns $v_3$ in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}^2}$. Each reciprocal is taken in $\mathbb C$ after embedding the real square root; a nonpositive expression under a square root gives square root $0$ and reciprocal $0$.

QAlgorithms.Papers.Independent.Intakethird100.P47.retainedLaw

For arbitrary functions $j:\{0,1\}^2\to\mathbb R$ and $k:\{0,1\}\to\mathbb R$, this defines a new function on Boolean pairs by

$$j_{\mathrm{ret}}(A,B)= \frac{j(A,B)k(A)} {\displaystyle\sum_{C\in\{0,1\}}\sum_{D\in\{0,1\}}j(C,D)k(C)}.$$

Here $0$ and $1$ denote false and true. There are no sign, normalization, or nonzero-denominator assumptions; if the denominator vanishes, every value of the resulting function is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P47.retention

For arbitrary functions $j:\{0,1\}^2\to\mathbb R$ and $k:\{0,1\}\to\mathbb R$, with $0$ denoting false and $1$ true, this defines the real number

$$\sum_{A\in\{0,1\}}\sum_{B\in\{0,1\}}j(A,B)k(A) =\bigl(j(0,0)+j(0,1)\bigr)k(0)+\bigl(j(1,0)+j(1,1)\bigr)k(1).$$

Neither function is subject to any sign, normalization, or range restriction.

QAlgorithms.Papers.Independent.Intakethird100.P47.siftingProbability

For any real-valued function $p$ of four Boolean arguments and any real-valued function $w$ on Boolean pairs, the real number $\operatorname{siftingProbability}(p,w)$ is

$$\sum_{A\in\{\mathrm{false},\mathrm{true}\}} \sum_{B\in\{\mathrm{false},\mathrm{true}\}} w(A,B)\,p(A,B,\mathrm{false},\mathrm{false}).$$

No nonnegativity or normalization assumptions are imposed on $p$ or $w$, and the definition does not require this number to lie in $[0,1]$.