Definitions for One-sided Device-Independent Quantum Key Distribution: Security, feasibility, and the connection with steering

QAlgorithms.Papers.Independent.Intakethird100.P94.BobBases

A value of this type is a family $U=(U_j)_{j\in B}$, where $B=\{\mathrm{false},\mathrm{true}\}$ and each $U_j$ is a complex unitary matrix whose rows and columns are indexed by $B$. Thus it specifies two $2\times2$ matrices satisfying $U_j^\ast U_j=U_jU_j^\ast=I$.

QAlgorithms.Papers.Independent.Intakethird100.P94.HiddenModel

Given any type $\Lambda$ equipped with a measurable-space structure and any family $U=(U_j)_{j\in B}$ of complex $2\times2$ unitary matrices, where $B=\{\mathrm{false},\mathrm{true}\}$, a value of this structure consists of a measure $\mu$ on $\Lambda$ with $\mu(\Lambda)=1$; a function $\rho:\Lambda\to\mathbb C^{B\times B}$ such that, for every $\ell\in\Lambda$, $\rho(\ell)$ is positive semidefinite and has trace $1$, and every entry function $\ell\mapsto\rho(\ell)_{x,y}$ is measurable for all $x,y\in B$; a function $r:B\times\Lambda\times B\times B\to\mathbb R$ such that $r(i,\ell,a,s)\ge0$ for every $i,a,s\in B$ and $\ell\in\Lambda$, $\sum_{a\in B}\sum_{s\in B}r(i,\ell,a,s)=1$ for every $i,\ell$, and $\ell\mapsto r(i,\ell,a,s)$ is measurable for every $i,a,s$; and a function $c:B\to B$ such that $a\ne c(i)$ implies $r(i,\ell,a,\mathrm{false})=0$ for every $i,\ell,a$. Finally, for every $i,j,a,s,b\in B$, the real function

$$\ell\longmapsto r(i,\ell,a,s)\, \operatorname{Re}\operatorname{tr}\!\bigl(E_{j,b}\rho(\ell)\bigr), \qquad (E_{j,b})_{x,y}=(U_j)_{x,b}\overline{(U_j)_{y,b}},$$

must be integrable with respect to $\mu$. The requirements on $\rho$ and $r$ hold at every $\ell$, including points of measure zero; such data cannot exist if $\Lambda$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P94.MutuallyUnbiased

For a family $U=(U_j)_{j\in B}$ of complex $2\times2$ unitary matrices, where $B=\{\mathrm{false},\mathrm{true}\}$, this is the proposition that every pair of column indices $a,b\in B$ satisfies

$$\left| \sum_{x\in B} \overline{(U_{\mathrm{false}})_{x,a}}\, (U_{\mathrm{true}})_{x,b} \right|^2 =\frac12.$$

QAlgorithms.Papers.Independent.Intakethird100.P94.basisEffect

For a family $U=(U_j)_{j\in B}$ of complex $2\times2$ unitary matrices, with $B=\{\mathrm{false},\mathrm{true}\}$, and indices $j,b\in B$, this defines the complex matrix $E_{j,b}$ with entries

$$(E_{j,b})_{x,y}=(U_j)_{x,b}\,\overline{(U_j)_{y,b}} \qquad(x,y\in B).$$

It is the outer product of column $b$ of $U_j$ with its conjugate transpose.

QAlgorithms.Papers.Independent.Intakethird100.P94.binaryError

For any function $P:B\times B\to\mathbb R$, where $B=\{\mathrm{false},\mathrm{true}\}$, this defines

$$\sum_{a\in B}\sum_{b\in B} \begin{cases} P(a,b),&a\ne b,\\ 0,&a=b, \end{cases} = P(\mathrm{false},\mathrm{true}) + P(\mathrm{true},\mathrm{false}).$$

No nonnegativity or normalization assumption is imposed on $P$.

QAlgorithms.Papers.Independent.Intakethird100.P94.conditionalEntropyBits

For any finite type $X$, with its supplied finite enumeration, and any function $P:X\times B\to\mathbb R$, where $B=\{\mathrm{false},\mathrm{true}\}$, set $m_x=P(x,\mathrm{false})+P(x,\mathrm{true})$. This defines

$$\sum_{x\in X} \begin{cases} 0,&m_x=0,\\[2pt] m_x\,h_2\!\left(\dfrac{P(x,\mathrm{true})}{m_x}\right),&m_x\ne0, \end{cases} \qquad h_2(p)=\frac{-p\log p-(1-p)\log(1-p)}{\log2}.$$

Here the logarithms use the total real-logarithm convention: $\log0=0$ and $\log t=\log|t|$ for negative $t$. There is no assumption that $P$ is nonnegative or normalized, so the entropy argument need not belong to $[0,1]$; a zero row sum contributes zero even when it results from cancellation of nonzero entries. If $X$ is empty, the sum is zero.

QAlgorithms.Papers.Independent.Intakethird100.P94.detectedErrorMass

Let $\mathbb B=\{0,1\}$, with $0$ denoting false and $1$ true. For any measurable space $\Lambda$, two unitary complex matrices $U_0,U_1$ indexed by $\mathbb B$, and a model $L$, the model consists of a probability measure $\mu$ on $\Lambda$; matrices $\rho_\lambda$ that are positive semidefinite with trace $1$ and have measurable entries; nonnegative responses $r_i(\lambda,a,s)$, measurable in $\lambda$, satisfying $\sum_{a,s\in\mathbb B}r_i(\lambda,a,s)=1$ for every $i,\lambda$; and designated bits $c_i$ such that $r_i(\lambda,a,0)=0$ whenever $a\ne c_i$. Set $E_{j,b}(x,y)=U_j(x,b)\overline{U_j(y,b)}$ and $q_{j,b}(\lambda)=\operatorname{Re}\operatorname{tr}(E_{j,b}\rho_\lambda)$; the model also requires every function $r_i(\lambda,a,s)q_{j,b}(\lambda)$ to be integrable with respect to $\mu$. For every $i\in\mathbb B$, the definition is the real number

$$\operatorname{detectedErrorMass}(L,i) =\sum_{\substack{a,b\in\mathbb B\\a\ne b}} \int_\Lambda r_i(\lambda,a,1)q_{i,b}(\lambda)\,d\mu(\lambda).$$

This is an error mass without division by the detection probability, including when that probability is zero.

QAlgorithms.Papers.Independent.Intakethird100.P94.detectionProbability

Let $\Lambda$ be any measurable space, let $B=\{\mathrm{false},\mathrm{true}\}$, and let $U_j$ be a complex $2\times2$ unitary matrix for each $j\in B$. The supplied model contains a probability measure $\mu$; a matrix-valued function $\rho$ with measurable entries, positive semidefinite values, and trace $1$ at every $\ell\in\Lambda$; nonnegative responses $r(i,\ell,a,s)$ measurable in $\ell$ and satisfying $\sum_{a,s\in B}r(i,\ell,a,s)=1$ for every $i,\ell$; and a function $c:B\to B$ such that $a\ne c(i)$ implies $r(i,\ell,a,\mathrm{false})=0$. Define $(E_{j,b})_{x,y}=(U_j)_{x,b}\overline{(U_j)_{y,b}}$; the model also requires integrability of $\ell\mapsto r(i,\ell,a,s)\operatorname{Re}\operatorname{tr}(E_{j,b}\rho(\ell))$ for every $i,j,a,s,b\in B$. For each $i\in B$, this defines

$$\sum_{a\in B}\sum_{b\in B} \int_\Lambda r(i,\ell,a,\mathrm{true})\, \operatorname{Re}\operatorname{tr}\!\bigl(E_{i,b}\rho(\ell)\bigr) \,d\mu(\ell).$$

Both setting indices are fixed to $i$, the flag is fixed to $\mathrm{true}$, and both outcome indices are summed.

QAlgorithms.Papers.Independent.Intakethird100.P94.flaggedConditionalEntropy

Let $\mathbb B=\{0,1\}$, representing false and true respectively. For any measurable space $\Lambda$, two unitary complex matrices $U_0,U_1$ indexed by $\mathbb B$, and a model $L$, the supplied data are a probability measure $\mu$; positive semidefinite matrices $\rho_\lambda$ of trace $1$, with measurable entries; nonnegative responses $r_i(\lambda,a,s)$, measurable in $\lambda$, whose sum over $a,s$ is $1$ for every $i,\lambda$; and bits $c_i$ such that $r_i(\lambda,a,0)=0$ whenever $a\ne c_i$. Write $E_{j,b}(x,y)=U_j(x,b)\overline{U_j(y,b)}$ and $q_{j,b}(\lambda)=\operatorname{Re}\operatorname{tr}(E_{j,b}\rho_\lambda)$; every product $r_i(\lambda,a,s)q_{j,b}(\lambda)$ is assumed integrable. For every $i\in\mathbb B$, define $p_i(a,s,b)=\int_\Lambda r_i(\lambda,a,s)q_{i,b}(\lambda)\,d\mu(\lambda)$ and $m_i(a,s)=\sum_b p_i(a,s,b)$. Then

$$\operatorname{flaggedConditionalEntropy}(L,i) =\sum_{a,s\in\mathbb B} \begin{cases} 0,&m_i(a,s)=0,\\ m_i(a,s)\,h_2\!\left(\dfrac{p_i(a,s,1)}{m_i(a,s)}\right),&m_i(a,s)\ne0, \end{cases}$$

where $h_2(p)=-[p\ln p+(1-p)\ln(1-p)]/\ln2$, with $0\ln0=0$. Both values of the flag $s$ are included, and each zero-mass pair $(a,s)$ contributes zero.

QAlgorithms.Papers.Independent.Intakethird100.P94.fullError

Let $\Lambda$ be any measurable space, let $B=\{\mathrm{false},\mathrm{true}\}$, and let $U_j$ be a complex $2\times2$ unitary matrix for each $j\in B$. The supplied model contains a probability measure $\mu$; a matrix-valued function $\rho$ with measurable entries, positive semidefinite values, and trace $1$ at every $\ell\in\Lambda$; nonnegative responses $r(i,\ell,a,s)$ measurable in $\ell$ and satisfying $\sum_{a,s\in B}r(i,\ell,a,s)=1$ for every $i,\ell$; and a function $c:B\to B$ such that $a\ne c(i)$ implies $r(i,\ell,a,\mathrm{false})=0$. Define $(E_{j,b})_{x,y}=(U_j)_{x,b}\overline{(U_j)_{y,b}}$; the model also requires integrability of $\ell\mapsto r(i,\ell,a,s)\operatorname{Re}\operatorname{tr}(E_{j,b}\rho(\ell))$ for every $i,j,a,s,b\in B$. For each $i\in B$, this defines

$$\sum_{\substack{a,b\in B\\a\ne b}} \ \sum_{s\in B} \int_\Lambda r(i,\ell,a,s)\, \operatorname{Re}\operatorname{tr}\!\bigl(E_{i,b}\rho(\ell)\bigr) \,d\mu(\ell).$$

Thus both setting indices are fixed to $i$, only unequal outcome indices contribute, and both values of the flag $s$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P94.maximumOverlap

For a family $U=(U_j)_{j\in B}$ of complex $2\times2$ unitary matrices, where $B=\{\mathrm{false},\mathrm{true}\}$, this defines

$$\max\!\left\{ \left| \sum_{x\in B} \overline{(U_{\mathrm{false}})_{x,a}}\, (U_{\mathrm{true}})_{x,b} \right|^2 :\ a,b\in B \right\}.$$

Thus the maximum is taken over all four choices of the two column indices.

QAlgorithms.Papers.Independent.Intakethird100.P94.observed

Let $\Lambda$ be any measurable space, let $B=\{\mathrm{false},\mathrm{true}\}$, and let $U_j$ be a complex $2\times2$ unitary matrix for each $j\in B$. Suppose the supplied model consists of a probability measure $\mu$ on $\Lambda$, matrices $\rho(\ell)$ that are positive semidefinite with trace $1$ for every $\ell$ and have measurable entry functions, nonnegative responses $r(i,\ell,a,s)$ measurable in $\ell$ and satisfying $\sum_{a,s\in B}r(i,\ell,a,s)=1$ for every $i,\ell$, and a function $c:B\to B$ satisfying $r(i,\ell,a,\mathrm{false})=0$ whenever $a\ne c(i)$. Write $(E_{j,b})_{x,y}=(U_j)_{x,b}\overline{(U_j)_{y,b}}$, and assume, as part of the model, that every function $\ell\mapsto r(i,\ell,a,s)\operatorname{Re}\operatorname{tr}(E_{j,b}\rho(\ell))$ is $\mu$-integrable for all Boolean indices $i,j,a,s,b$. For any such indices, this defines the real number

$$\int_\Lambda r(i,\ell,a,s)\, \operatorname{Re}\operatorname{tr}\!\bigl(E_{j,b}\rho(\ell)\bigr) \,d\mu(\ell).$$

QAlgorithms.Papers.Independent.Intakethird100.P94.projectiveOverlapIncompatibility

For a family $U=(U_j)_{j\in B}$ of complex $2\times2$ unitary matrices, where $B=\{\mathrm{false},\mathrm{true}\}$, this defines the real number

$$-\log_2\!\left( \max_{a,b\in B} \left| \sum_{x\in B} \overline{(U_{\mathrm{false}})_{x,a}}\, (U_{\mathrm{true}})_{x,b} \right|^2 \right).$$

The logarithm here is the real base-$2$ logarithm.

QAlgorithms.Papers.Independent.Intakethird100.P94.squaredOverlap

For a family $U=(U_j)_{j\in B}$ of complex $2\times2$ unitary matrices, where $B=\{\mathrm{false},\mathrm{true}\}$, and column indices $a,b\in B$, this defines the real number

$$c_{a,b} = \left| \sum_{x\in B} \overline{(U_{\mathrm{false}})_{x,a}}\, (U_{\mathrm{true}})_{x,b} \right|^2.$$

Here the squared complex modulus of a number $z$ means $(\operatorname{Re}z)^2+(\operatorname{Im}z)^2$.

QAlgorithms.Papers.Independent.Intakethird100.P94.weightedDetectedEntropy

Let $\mathbb B=\{0,1\}$, with false represented by $0$ and true by $1$. For any measurable space $\Lambda$, two unitary complex matrices $U_0,U_1$ indexed by $\mathbb B$, and a model $L$, assume that $L$ supplies a probability measure $\mu$; positive semidefinite trace-one matrices $\rho_\lambda$ with measurable entries; and responses $r_i(\lambda,a,s)$ that are nonnegative, measurable in $\lambda$, and satisfy $\sum_{a,s}r_i(\lambda,a,s)=1$ for every $i,\lambda$. It also supplies bits $c_i$ with $r_i(\lambda,a,0)=0$ for $a\ne c_i$. Define $E_{j,b}(x,y)=U_j(x,b)\overline{U_j(y,b)}$, $q_{j,b}(\lambda)=\operatorname{Re}\operatorname{tr}(E_{j,b}\rho_\lambda)$, and $p_i(a,s,b)=\int_\Lambda r_i(\lambda,a,s)q_{i,b}(\lambda)\,d\mu(\lambda)$; all products $r_i(\lambda,a,s)q_{j,b}(\lambda)$, for every choice of indices, are required to be integrable. For every $i\in\mathbb B$, put $\eta_i=\sum_{a,b}p_i(a,1,b)$ and $e_i=\sum_{a\ne b}p_i(a,1,b)$. With $h_2(p)=-[p\ln p+(1-p)\ln(1-p)]/\ln2$, using $0\ln0=0$, the definition is

$$\operatorname{weightedDetectedEntropy}(L,i)= \begin{cases} 0,&\eta_i=0,\\ \eta_i h_2(e_i/\eta_i),&\eta_i\ne0. \end{cases}$$

In particular, the zero-efficiency branch returns zero without using a conditional error ratio.