Definitions for Tight bounds for private communication over bosonic Gaussian channels based on teleportation simulation with optimal finite resources

QAlgorithms.Papers.Independent.Intakethird100.P17.GaussianInConvention

For any natural number $n$, including $0$, put $J=\mathbb N^{\{0,\ldots,n-1\}}$, $H=\ell^2(J;\mathbb C)$, and $Q=\{0,\ldots,n-1\}\times\{0,1\}$, and let $e_j\in H$ have coordinate $1$ at $j$ and $0$ elsewhere. The argument $W$ is a family of bounded unitary operators $W_\xi:H\to H$, indexed by $\xi\in\mathbb R^Q$, satisfying

$$(W_\xi e_j)_i=\prod_{m=0}^{n-1}D\!\left(-\xi_{(m,1)}+\mathrm i\xi_{(m,0)};i_m,j_m\right),$$

where, for $z\in\mathbb C$ and $r,s\in\mathbb N$,

$$D(z;r,s)=e^{-|z|^2/2}\sqrt{r!\,s!} \sum_{h=0}^{\min(r,s)} \frac{z^{r-h}(-\overline z)^{s-h}}{h!\,(r-h)!\,(s-h)!}.$$

For any real $\kappa$, bounded complex-linear operator $\rho:H\to H$, vector $u\in\mathbb R^Q$, and real matrix $\sigma$ indexed by $Q$, this predicate means all of the following: there exist sequences $v_r,w_r\in H$ with $\sum_r\|v_r\|\|w_r\|<\infty$ such that $\rho x=\sum_r\langle w_r,x\rangle v_r$ for every $x\in H$; $\rho^*=\rho$ and $\operatorname{Re}\langle x,\rho x\rangle\ge0$ for every $x$; the family $(\langle e_j,\rho e_j\rangle)_{j\in J}$ is summable and has sum $1$; the real matrix $V=2\kappa\sigma$ is symmetric and positive definite, and $V+\mathrm i\Omega$ is positive semidefinite, where

$$\Omega_{(m,b),(\ell,c)} =\begin{cases} 1&m=\ell,\ b=0,\ c=1,\\ -1&m=\ell,\ b=1,\ c=0,\\ 0&\text{otherwise}; \end{cases}$$

for every $\xi\in\mathbb R^Q$, the family $(\langle e_j,\rho W_\xi e_j\rangle)_{j\in J}$ is summable and

$$\sum_{j\in J}\langle e_j,\rho W_\xi e_j\rangle = \exp\!\left( -\frac12\sum_{a,b\in Q}\xi_a(2\kappa\sigma_{ab})\xi_b +\mathrm i\sum_{a\in Q}\sqrt{2\kappa}\,u_a\xi_a \right);$$

and the real family

$$\left(\left(\sum_{m=0}^{n-1}j_m\right) \operatorname{Re}\langle e_j,\rho e_j\rangle\right)_{j\in J}$$

is summable. There is no explicit sign restriction on $\kappa$: the real square root is $0$ when $2\kappa\le0$; when $n>0$ and $\kappa=0$, the required positive definiteness of $V$ cannot hold. When $n=0$, $J$ is a singleton, $Q$ is empty, and the last summability condition concerns a single zero term.

QAlgorithms.Papers.Independent.Intakethird100.P17.PolynomialPairingSummable

For any natural number $n$, let $J=\mathbb N^{\{0,\ldots,n-1\}}$, $H=\ell^2(J;\mathbb C)$, and $e_j$ be the coordinate unit vector at $j$. For any bounded complex-linear operator $\rho:H\to H$ and arbitrary complex matrix $A:J\times J\to\mathbb C$, this predicate means that the single family indexed by pairs

$$\bigl((\rho e_j)_i A_{ji}\bigr)_{(i,j)\in J\times J}$$

is summable in $\mathbb C$, equivalently

$$\sum_{(i,j)\in J\times J}|(\rho e_j)_i A_{ji}|<\infty.$$

No positivity or normalization of $\rho$, or symmetry of $A$, is required. When $n=0$, the indexing set is a singleton and this predicate holds automatically.

QAlgorithms.Papers.Independent.Intakethird100.P17.PolynomialProductSummable

For any natural number $n$, with $J=\mathbb N^{\{0,\ldots,n-1\}}$, and arbitrary complex matrices $A,B:J\times J\to\mathbb C$, this predicate means that for every $i,j\in J$, the family

$$\bigl(A_{ik}B_{kj}\bigr)_{k\in J}$$

is summable in $\mathbb C$, equivalently $\sum_{k\in J}|A_{ik}B_{kj}|<\infty$. This is an entrywise requirement and imposes no uniform bound over $i,j$. When $n=0$, $J$ is a singleton and the predicate holds for every $A,B$.

QAlgorithms.Papers.Independent.Intakethird100.P17.StrictGibbsCovariance

For any natural number $n$, real $\kappa$, and real matrix $\sigma$ indexed by $Q=\{0,\ldots,n-1\}\times\{0,1\}$, this predicate means that $\sigma^{\mathsf T}=\sigma$, that $\sigma$ is positive definite over $\mathbb R$, and that the complex matrix

$$\kappa\sigma_{\mathbb C}+\frac{\mathrm i}{2}\Omega_{\mathbb C}$$

is positive definite over $\mathbb C$. Here the subscript $\mathbb C$ means entrywise embedding of a real matrix into $\mathbb C$, and

$$\Omega_{(m,b),(\ell,c)} =\begin{cases} 1&m=\ell,\ b=0,\ c=1,\\ -1&m=\ell,\ b=1,\ c=0,\\ 0&\text{otherwise}. \end{cases}$$

Positive definiteness requires the respective Hermitian quadratic forms to be strictly positive on every nonzero vector. There is no separate sign hypothesis on $\kappa$. When $n=0$, both index sets are empty and these conditions hold vacuously for the unique empty matrix, for every $\kappa$.

QAlgorithms.Papers.Independent.Intakethird100.P17.complexCovariance

For any natural number $n$, including $0$, and arbitrary real matrix $\sigma$ indexed by $Q=\{0,\ldots,n-1\}\times\{0,1\}$, this definition gives the complex matrix with entries

$$(\sigma_{\mathbb C})_{ab}=\sigma_{ab}+\mathrm i\,0 \qquad(a,b\in Q).$$

It imposes no symmetry, positivity, or covariance condition. When $n=0$, it is the unique matrix with empty row and column index sets.

QAlgorithms.Papers.Independent.Intakethird100.P17.crossEntropyCutoff

For every natural number $n$, bounded complex linear operators $\rho,\sigma$ on $\mathcal H_n=\ell^2(\{0,\ldots,n-1\}\to\mathbb N;\mathbb C)$, and natural number $k$, this defines

$$-\operatorname{Re}\sum_{j:\{0,\ldots,n-1\}\to\mathbb N} \left\langle e_j,\rho\bigl(f_k(\sigma)e_j\bigr)\right\rangle, \qquad f_k(x)=\frac{\log(\max\{x,1/(k+1)\})}{\log 2},$$

where $e_j$ is the standard unit vector at $j$, and $f_k(\sigma)$ is the real continuous functional calculus for self-adjoint operators, with value zero if $\sigma$ is not self-adjoint. No density, trace-class, or summability hypotheses are required. The displayed infinite sum is total and has value zero if its complex-valued summands are not summable. The cases $k=0$ and $n=0$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P17.densityCoefficients

For any natural number $n$, let $J=\mathbb N^{\{0,\ldots,n-1\}}$, let $H=\ell^2(J;\mathbb C)$, and let $e_j$ denote the vector with coordinate $1$ at $j$ and $0$ elsewhere. For any bounded complex-linear operator $\rho:H\to H$, this definition gives the complex matrix

$$\rho_{ij}=(\rho e_j)_i \qquad(i,j\in J).$$

No positivity, trace-class, or normalization assumption on $\rho$ is imposed. When $n=0$, $J$ is a singleton, so this matrix has one entry.

QAlgorithms.Papers.Independent.Intakethird100.P17.gaussianSigma

For every natural number $n$, real number $\kappa$, real matrices $\sigma_1,\sigma_2$ indexed by $J=\{0,\ldots,n-1\}\times\{0,1\}$, and real vector $\delta$ indexed by $J$, this defines

$$\frac{ \log Z^2+\kappa\,\operatorname{tr}(\sigma_1G) +\kappa\sum_{a,b\in J}\delta_aG_{ab}\delta_b }{2\log 2}, \qquad Z^2=\operatorname{Re}\det\!\left(\kappa\sigma_2+\frac{i}{2}\Omega\right),$$

where $\Omega_{(r,0),(r,1)}=1$, $\Omega_{(r,1),(r,0)}=-1$, all its other entries are zero, and real matrices inside complex expressions are embedded entrywise. Here

$$B=2\kappa\,\sigma_2(i\Omega),\qquad G_{ab}=\operatorname{Re}\left[ 2i\Omega\sum_{k=0}^{\infty}\frac{(B^{-1})^{2k+1}}{2k+1} \right]_{ab}.$$

No covariance, positivity, invertibility, or summability conditions are imposed. Singular matrix inverses and nonsummable matrix series have value zero. The real logarithm is total, with $\log 0=0$ and $\log x=\log|x|$ for negative $x$, so $Z^2$ need not be positive. The case $n=0$ is included and gives zero.

QAlgorithms.Papers.Independent.Intakethird100.P17.gaussianVarianceExpression

For every natural number $n$, real number $\kappa$, real matrices $\sigma_1,\sigma_2$ indexed by $J=\{0,\ldots,n-1\}\times\{0,1\}$, and real vector $\delta$ indexed by $J$, let $\Omega_{(r,0),(r,1)}=1$, $\Omega_{(r,1),(r,0)}=-1$, with all other entries zero. Regarding real matrices as complex matrices entrywise, set

$$B_r=2\kappa\,\sigma_r(i\Omega),\qquad (G_r)_{ab}=\operatorname{Re}\left[ 2i\Omega\sum_{k=0}^{\infty}\frac{(B_r^{-1})^{2k+1}}{2k+1} \right]_{ab}\quad(r=1,2), \qquad D=G_1-G_2.$$

The definition returns the real number

$$\frac{ 4\kappa^2\operatorname{tr}(\sigma_1D\sigma_1D) +\operatorname{tr}(D\Omega D\Omega) +\displaystyle\sum_{a,b\in J}\delta_a \bigl(8\kappa^2G_2\sigma_1G_2\bigr)_{ab}\delta_b }{ 2(2\log 2)^2 }.$$

No positivity, symmetry, invertibility, convergence, or sign conditions on the result are required. Singular matrix inverses and nonsummable matrix series have value zero. Both $\kappa=0$ and $n=0$ are allowed; for $n=0$, the result is zero.

QAlgorithms.Papers.Independent.Intakethird100.P17.gibbsSeriesTerm

For any natural number $n$, real $\kappa$, arbitrary real matrix $\sigma$ indexed by $Q=\{0,\ldots,n-1\}\times\{0,1\}$, and natural number $k$, including $0$, embed $\sigma$ entrywise into the complex matrix $\sigma_{\mathbb C}$, and define $\Omega$ by

$$\Omega_{(m,b),(\ell,c)} =\begin{cases} 1&m=\ell,\ b=0,\ c=1,\\ -1&m=\ell,\ b=1,\ c=0,\\ 0&\text{otherwise}. \end{cases}$$

With $B=2\kappa\,\sigma_{\mathbb C}(\mathrm i\Omega_{\mathbb C})$, this definition gives the complex matrix

$$T_k=\frac{1}{2k+1}(B^{-1})^{2k+1}.$$

The exponent $2k+1$ is a positive natural number, and its reciprocal is taken in $\mathbb C$. No symmetry, positivity, strict Gibbs condition, or invertibility hypothesis is imposed. The matrix inverse is total: for a singular $B$, $B^{-1}=0$, hence $T_k=0$. When $n=0$, the result is the unique empty matrix.

QAlgorithms.Papers.Independent.Intakethird100.P17.kappaGibbsMatrix

For every natural number $n$, every real number $\kappa$, and every real matrix $\sigma$ indexed by $J=\{0,\ldots,n-1\}\times\{0,1\}$, this defines a real matrix $G_\kappa(\sigma)$. Let $\Omega_{(r,0),(r,1)}=1$, $\Omega_{(r,1),(r,0)}=-1$, and let all other entries of $\Omega$ be zero; regard real matrices as complex matrices entrywise. Then

$$B=2\kappa\,\sigma(i\Omega),\qquad G_\kappa(\sigma)_{ab} =\operatorname{Re}\left[ 2i\Omega\sum_{k=0}^{\infty} \frac{(B^{-1})^{2k+1}}{2k+1} \right]_{ab}.$$

There are no positivity, symmetry, invertibility, or convergence assumptions. Matrix inversion is total and gives the zero matrix for a singular matrix; the infinite matrix sum is the total mathematical sum, which is zero when the series is not summable. In particular, $\kappa=0$ is allowed. When $n=0$, the result is the unique matrix on the empty index set.

QAlgorithms.Papers.Independent.Intakethird100.P17.oscillatorCrossEntropy

For every natural number $n$ and bounded complex linear operators $\rho,\sigma$ on $\mathcal H_n=\ell^2(\{0,\ldots,n-1\}\to\mathbb N;\mathbb C)$, this defines the real value selected by the total filter-limit operation, as $k\to\infty$ through natural numbers, for the sequence

$$-\operatorname{Re}\sum_j \left\langle e_j,\rho\bigl(f_k(\sigma)e_j\bigr)\right\rangle, \qquad f_k(x)=\frac{\log(\max\{x,1/(k+1)\})}{\log 2}.$$

Here $e_j$ is the standard number basis, indexed by all functions $j:\{0,\ldots,n-1\}\to\mathbb N$, and $f_k(\sigma)$ uses the real continuous functional calculus for self-adjoint operators, giving zero outside that condition. Nonsummable complex sums have value zero. No convergence, self-adjointness, density, or trace-class hypothesis is imposed: if the real sequence converges, the selected value is its limit, but the definition itself does not assert convergence. The case $n=0$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P17.oscillatorLogCutoff

For every natural number $n$, every bounded complex linear operator $\rho$ on $\mathcal H_n=\ell^2(\{0,\ldots,n-1\}\to\mathbb N;\mathbb C)$, and every natural number $k$, this defines the operator obtained by applying the real continuous functional calculus for self-adjoint operators to $\rho$ with the scalar function

$$f_k(x)=\frac{\log\!\left(\max\left\{x,\frac1{k+1}\right\}\right)}{\log 2}.$$

No self-adjointness, positivity, normalization, or trace-class assumption on $\rho$ is supplied; the total functional-calculus operation gives zero when its self-adjointness condition fails. The scalar logarithm always has a positive argument here. Both $k=0$, where the lower cutoff is $1$, and $n=0$, where $\mathcal H_0$ is one-dimensional, are included.

QAlgorithms.Papers.Independent.Intakethird100.P17.oscillatorRelativeEntropy

For every natural number $n$ and bounded complex linear operators $\rho,\sigma$ on $\mathcal H_n=\ell^2(\{0,\ldots,n-1\}\to\mathbb N;\mathbb C)$, supplied with proofs that each admits a representation

$$Tx=\sum_{r=0}^{\infty}\langle v_r,x\rangle u_r \quad\text{for every }x,\qquad \sum_{r=0}^{\infty}\|u_r\|\,\|v_r\|<\infty$$

for some sequences $u_r,v_r\in\mathcal H_n$, this defines an element of $[0,\infty]$. For every natural number $m$ and every orthonormal family $v_0,\ldots,v_{m-1}$ in $\mathcal H_n$, form

$$X_{ab}=\langle v_a,\rho v_b\rangle,\qquad Y_{ab}=\langle v_a,\sigma v_b\rangle.$$

Assign this pair the value

$$R(X,Y)= \begin{cases} \max\!\left\{0,\operatorname{Re}\operatorname{tr} \bigl(h(X)-X\log(Y)-X+Y\bigr)\right\}, &\ker Y\subseteq\ker X,\\ +\infty,&\text{otherwise}, \end{cases} \qquad h(x)=x\log x,$$

where $h(X)$ and $\log(Y)$ use the real continuous functional calculus for self-adjoint matrices, including its total default outside the required conditions. The defined value is

$$\frac{\displaystyle\sup_{m,v}R(X,Y)}{\log 2},$$

with the denominator interpreted in $[0,\infty]$. No positivity or trace-one assumption is imposed on either operator. Empty orthonormal families $m=0$, and the case $n=0$, are included.

QAlgorithms.Papers.Independent.Intakethird100.P17.oscillatorRelativeVariance

For every natural number $n$, bounded complex linear operators $\rho,\sigma$ on $\mathcal H_n=\ell^2(\{0,\ldots,n-1\}\to\mathbb N;\mathbb C)$, and real number $m$, this defines the real value selected by the total filter-limit operation, as $k\to\infty$, for

$$\operatorname{Re}\sum_j\langle e_j,\rho(L_k(L_ke_j))\rangle, \qquad L_k=f_k(\rho)-f_k(\sigma)-mI,\qquad f_k(x)=\frac{\log(\max\{x,1/(k+1)\})}{\log 2}.$$

Here $e_j$ is the standard number basis indexed by $j:\{0,\ldots,n-1\}\to\mathbb N$; the real functional calculus gives zero outside self-adjoint operators, and a nonsummable complex trace series has value zero. No convergence, density, trace-class, or positivity hypotheses are imposed, and $m$ is arbitrary. If the displayed real sequence converges, the definition gives its limit; no such convergence is asserted by the definition. The case $n=0$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P17.polynomialExpectation

For any natural number $n$, let $J=\mathbb N^{\{0,\ldots,n-1\}}$, $H=\ell^2(J;\mathbb C)$, and $e_j$ be the coordinate unit vector at $j$. For any bounded complex-linear operator $\rho:H\to H$ and arbitrary complex matrix $A:J\times J\to\mathbb C$, this definition assigns the real number

$$\operatorname{Re}\left( \sum_{(i,j)\in J\times J}(\rho e_j)_i A_{ji} \right).$$

The sum is over pairs, rather than an iterated sum. There is no summability, positivity, normalization, or symmetry hypothesis; a nonsummable complex family has total sum $0$, so the defined value is then $0$. When $n=0$, the expression is the real part of the product of the sole operator coefficient and the sole entry of $A$.

QAlgorithms.Papers.Independent.Intakethird100.P17.polynomialKernelProduct

For any natural number $n$, let $J=\mathbb N^{\{0,\ldots,n-1\}}$. For arbitrary complex matrices $A,B:J\times J\to\mathbb C$, their defined kernel product is the matrix

$$(A\mathbin{\star}B)_{ij}=\sum_{k\in J}A_{ik}B_{kj} \qquad(i,j\in J).$$

No summability hypothesis is required: each sum is the total infinite-sum operation, which returns $0$ if its complex-valued family is not summable. When $n=0$, $J$ is a singleton and this is ordinary multiplication of the two sole entries.

QAlgorithms.Papers.Independent.Intakethird100.P17.quadraticKernel

For any natural number $n$, real $\kappa$, and arbitrary real matrix $A$ indexed by $Q=\{0,\ldots,n-1\}\times\{0,1\}$, this definition gives a complex matrix on $J=\mathbb N^{\{0,\ldots,n-1\}}$ by

$$Q_{\kappa,A}(i,j) =\sum_{a\in Q}\sum_{b\in Q}A_{ab} \left(\sum_{k\in J}K_{\kappa,a}(i,k)K_{\kappa,b}(k,j)\right).$$

Here, writing $a=(m,c)$, let $R_m(i,j)$ be $\sqrt{j_m+1}$ if $i$ is obtained from $j$ by increasing coordinate $m$ by $1$, and $0$ otherwise; then

$$K_{\kappa,(m,0)}(i,j) =\frac{R_m(i,j)+R_m(j,i)}{\sqrt{2\kappa}}, \qquad K_{\kappa,(m,1)}(i,j) =\frac{\mathrm i(R_m(i,j)-R_m(j,i))}{\sqrt{2\kappa}}.$$

All real factors are embedded in $\mathbb C$. No symmetry or positivity of $A$, or positivity of $\kappa$, is assumed. Infinite sums use the total sum operation, returning $0$ for a nonsummable family. For $\kappa\le0$, division by the zero real square root makes every $K_{\kappa,a}$, and hence $Q_{\kappa,A}$, zero. For $n=0$, the outer sums are empty and the resulting matrix is zero on the singleton $J$.

QAlgorithms.Papers.Independent.Intakethird100.P17.quadraticVariance

For any natural number $n$, real $\kappa$, bounded complex-linear operator $\rho$ on $H=\ell^2(J;\mathbb C)$, where $J=\mathbb N^{\{0,\ldots,n-1\}}$, and arbitrary real matrix $A$ indexed by $Q=\{0,\ldots,n-1\}\times\{0,1\}$, let $e_j$ be the coordinate unit vector at $j$. Define $R_m(i,j)=\sqrt{j_m+1}$ when $i$ is obtained by increasing coordinate $m$ of $j$ by $1$, and $R_m(i,j)=0$ otherwise, and set

$$K_{\kappa,(m,0)}(i,j)=\frac{R_m(i,j)+R_m(j,i)}{\sqrt{2\kappa}}, \qquad K_{\kappa,(m,1)}(i,j)= \frac{\mathrm i(R_m(i,j)-R_m(j,i))}{\sqrt{2\kappa}},$$ $$Q_{ij}=\sum_{a,b\in Q}A_{ab} \sum_{k\in J}K_{\kappa,a}(i,k)K_{\kappa,b}(k,j).$$

The defined real number is

$$\operatorname{Re}\left( \sum_{(i,j)\in J\times J}(\rho e_j)_i \left(\sum_{k\in J}Q_{jk}Q_{ki}\right) \right) - \left[ \operatorname{Re}\left( \sum_{(i,j)\in J\times J}(\rho e_j)_iQ_{ji} \right) \right]^2.$$

No summability, normalization, positivity, or symmetry assumptions are imposed, and no nonnegativity of this value is asserted. Each infinite sum returns $0$ if its family is not summable. Real square roots of nonpositive numbers and division by zero have value $0$; consequently the expression is $0$ when $\kappa\le0$. It is also $0$ when $n=0$, because then $Q$ is empty and the matrix $Q_{ij}$ is zero.

QAlgorithms.Papers.Independent.Intakethird100.P17.quadratureKernel

For any natural number $n$, real $\kappa$, quadrature index $a=(m,b)\in\{0,\ldots,n-1\}\times\{0,1\}$, and occupation indices $i,j\in\mathbb N^{\{0,\ldots,n-1\}}$, define

$$R_m(i,j)= \begin{cases} \sqrt{j_m+1}&i\text{ equals }j\text{ with coordinate }m\text{ replaced by }j_m+1,\\ 0&\text{otherwise}. \end{cases}$$

The complex-valued kernel is

$$K_{\kappa,(m,b)}(i,j)= \begin{cases} \dfrac{R_m(i,j)+R_m(j,i)}{\sqrt{2\kappa}}&b=0,\\[6pt] \dfrac{\mathrm i\bigl(R_m(i,j)-R_m(j,i)\bigr)}{\sqrt{2\kappa}}&b=1, \end{cases}$$

with real quantities embedded in $\mathbb C$. No positivity assumption on $\kappa$ is imposed. If $\kappa\le0$, its real square-root denominator is $0$, and complex division by $0$ makes the kernel $0$. For $n=0$, there is no quadrature index $a$, so there are no such evaluations.

QAlgorithms.Papers.Independent.Intakethird100.P17.relativeVarianceCutoff

For every natural number $n$, bounded complex linear operators $\rho,\sigma$ on $\mathcal H_n=\ell^2(\{0,\ldots,n-1\}\to\mathbb N;\mathbb C)$, real number $m$, and natural number $k$, this defines

$$\operatorname{Re}\sum_j\langle e_j,\rho(L_k(L_ke_j))\rangle, \qquad L_k=f_k(\rho)-f_k(\sigma)-mI,\qquad f_k(x)=\frac{\log(\max\{x,1/(k+1)\})}{\log 2}.$$

The vectors $e_j$ are the standard unit vectors indexed by all $j:\{0,\ldots,n-1\}\to\mathbb N$. Each functional calculus is the real continuous functional calculus for self-adjoint operators and gives zero when its self-adjointness condition fails; a nonsummable complex trace series has value zero. There are no density, trace-class, positivity, summability, or restrictions on $m$, and $m$ is not required to be an expectation. The product is $L_k\circ L_k$. The cases $k=0$ and $n=0$ are included.