Definitions for Discriminating quantum-optical beam-splitter channels with number-diagonal signal states: Applications to quantum reading and target detection

QAlgorithms.Papers.Independent.Intakethird100.P54.BlockReceiver

For any type $I$, natural number $M\ge0$, splitter pair $c$, vector $\psi\in H=\ell^2(I\times\mathcal N_M;\mathbb C)$, real weights $w_0,w_1$, and bounded complex-linear operator $E:H\to H$, where $\mathcal N_M=\{0,\ldots,M-1\}\to\mathbb N$, define $L_{b,k}$, for $b=0,1$ and $k\in\mathcal N_M$, by the coordinates

$$\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m},$$

where $R_b\in[0,1]$ and $\theta_b\in\mathbb R$ are splitter $c(b)$’s parameters, and use the zero vector if these coordinates are not square-summable. Put $V_k=\operatorname{span}_{\mathbb C}\{L_{0,k},L_{1,k}\}$ and $D_kv=w_0\langle L_{0,k},v\rangle L_{0,k}-w_1\langle L_{1,k},v\rangle L_{1,k}$. The defined proposition requires all of the following: for every $u,v\in H$, $\langle u,Ev\rangle=\overline{\langle v,Eu\rangle}$; for every $v\in H$, $0\le\operatorname{Re}\langle v,Ev\rangle\le\|v\|^2$; $E^2=E$; every $V_k$ is invariant under $E$; and for every $k$, every $v\in V_k$, and every real $a$ satisfying $D_kv=av$, one has $Ev=v$ if $a>0$ and $Ev=0$ if $a<0$. The last condition imposes no further requirement when $a=0$. The weights are unrestricted and $\psi$ need not be normalized. The case $M=0$ is included, with empty product $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.Channels

A channel pair is a function from the two Boolean labels to splitters. Writing its values as $c_0=c(\mathrm{false})$ and $c_1=c(\mathrm{true})$, each $c_b$ has a reflectance $R_b\in[0,1]$ and a real phase $\theta_b$. The two splitters may coincide.

QAlgorithms.Papers.Independent.Intakethird100.P54.CountsAs

For any type $I$, any natural number $M$, any bounded complex-linear operator $E$ on $\ell^2(I\times\mathbb N^M;\mathbb C)$, and any Boolean-valued function $d:\mathbb N^M\to\{\mathrm{false},\mathrm{true}\}$, this is the proposition that, for every vector $v$ in that Hilbert space and every coordinate $(i,n)\in I\times\mathbb N^M$,

$$(Ev)(i,n)= \begin{cases} 0,&d(n)=\mathrm{true},\\ v(i,n),&d(n)=\mathrm{false}. \end{cases}$$

There is no finiteness, countability, or nonemptiness assumption on $I$. If $I$ is empty, the coordinate condition is vacuous. If $M=0$, the occupation index set is a singleton, so $d$ has a single Boolean value.

QAlgorithms.Papers.Independent.Intakethird100.P54.HasSignalEnergy

For any type $I$, natural number $M$, square-summable complex family $\psi\in\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}})$, and real number $N$, define

$$P_\psi(n)= \sum_{q\in\mathbb N^{\{0,\ldots,M-1\}}} \begin{cases} \displaystyle\sum_{i\in I}|\psi(i,q)|^2, &\displaystyle\sum_{m=0}^{M-1}q_m=n,\\ 0,&\text{otherwise}. \end{cases}$$

The stated property means that $N\ge0$, the real series $\sum_{n=0}^{\infty}nP_\psi(n)$ is summable, and its sum equals $N$. No normalization of $\psi$ is required. The $n=0$ term contributes $0$; if $M=0$, if $I$ is empty, or if $\psi=0$, the property holds exactly when $N=0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.IsNDS

For any type $I$, natural number $M$, and square-summable complex family $\psi\in\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}})$, this property requires that every pair of distinct occupation vectors $n,q\in\mathbb N^{\{0,\ldots,M-1\}}$ satisfy

$$\sum_{i\in I}\psi(i,n)\,\overline{\psi(i,q)}=0.$$

There is no requirement for pairs with $n=q$, and no normalization or nonzero condition on $\psi$. If $M=0$, there are no distinct occupation vectors, so the property holds vacuously. If $I$ is empty or $\psi=0$, every displayed sum is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.IsOpticalEffect

For any type $X$ and continuous complex-linear operator $E:\ell^2(X)\to\ell^2(X)$, this property means that

$$\langle u,Ev\rangle=\overline{\langle v,Eu\rangle} \qquad\text{for every }u,v\in\ell^2(X),$$

and that

$$0\le\operatorname{Re}\langle v,Ev\rangle\le\|v\|^2 \qquad\text{for every }v\in\ell^2(X).$$

The vectors range over the entire Hilbert space, including $0$, without normalization. No idempotence requirement is included. If $X$ is empty, the unique operator on the zero Hilbert space satisfies the property.

QAlgorithms.Papers.Independent.Intakethird100.P54.PurifiesOpticalKernel

For any type $X$, arbitrary complex-valued kernel $\rho:X\times X\to\mathbb C$, and square-summable complex family $z\in\ell^2(X\times\mathbb N)$, this property means both that $\|z\|=1$ and that

$$\rho(x,y)=\sum_{e=0}^{\infty}z(x,e)\,\overline{z(y,e)} \qquad\text{for every }x,y\in X.$$

There are no separate assumptions on $\rho$; the displayed requirements are the entire property. If $X$ is empty, its Hilbert space is zero-dimensional and no such unit vector exists, so the property cannot hold.

QAlgorithms.Papers.Independent.Intakethird100.P54.Reading

For any channel pair $c$, consisting of splitters with reflectances $R_0,R_1\in[0,1]$ and real phases $\theta_0,\theta_1$, this property means exactly that $\theta_0=0$ and $\theta_1=0$, as equalities of real numbers. It imposes no ordering or distinctness condition on the reflectances.

QAlgorithms.Papers.Independent.Intakethird100.P54.Resolution

For any type $X$ and arbitrary complex-valued kernel $\rho:X\times X\to\mathbb C$, a resolution consists of a real sequence $(p_n)_{n\in\mathbb N}$ and vectors $u_n\in\ell^2(X)$ such that $p_n\ge0$ for every $n$, the sequence $(p_n)$ is summable with $\sum_{n=0}^{\infty}p_n=1$, every $u_n$ with $p_n>0$ has norm $1$, every distinct pair $n,m$ with $p_n>0$ and $p_m>0$ satisfies $\langle u_n,u_m\rangle=0$, and

$$\rho(x,y)=\sum_{n=0}^{\infty}p_n\,u_n(x)\,\overline{u_n(y)} \qquad\text{for every }x,y\in X.$$

Vectors at indices of zero weight have no normalization or orthogonality requirement. The sequence may have finitely or infinitely many positive weights. Existence of a resolution is not asserted for an arbitrary kernel; if $X$ is empty, no resolution exists because total weight $1$ forces a positive weight but the zero Hilbert space has no unit vector.

QAlgorithms.Papers.Independent.Intakethird100.P54.Splitter.field

For any splitter $a$, with reflectance $R\in[0,1]$ and real phase $\theta$, this defines the complex number $\sqrt R\,e^{i\theta}$. When $R=0$, this number is zero for every phase.

QAlgorithms.Papers.Independent.Intakethird100.P54.Splitter

A splitter consists of two real numbers, a reflectance $R$ and a phase $\theta$, together with proofs that $0\le R$ and $R\le1$. Both endpoints $R=0$ and $R=1$ are allowed, and $\theta$ is unrestricted.

QAlgorithms.Papers.Independent.Intakethird100.P54.Splitter.r

For any splitter $a$, with reflectance $R\in[0,1]$ and real phase $\theta$, this defines $r(a)=\sqrt R$, using the nonnegative real square root; in particular, $r(a)=0$ when $R=0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.Splitter.t

For any splitter $a$, with reflectance $R\in[0,1]$ and real phase $\theta$, this defines $t(a)=\sqrt{1-R}$, using the nonnegative real square root; in particular, $t(a)=0$ when $R=1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.ValidSplitters

For any channel pair $c$, whose splitter at label $b\in\{0,1\}$ has reflectance $R_b\in[0,1]$ and real phase $\theta_b$, this property means exactly that $R_0\le R_1$ and $\sqrt{R_0}\,e^{i\theta_0}\ne\sqrt{R_1}\,e^{i\theta_1}$. Equality of the reflectances is allowed provided the complex fields differ; if both reflectances are zero, the property fails.

QAlgorithms.Papers.Independent.Intakethird100.P54.attenuate

For any type $I$, any natural number $M$, a splitter $a$ with reflectance $R\in[0,1]$ and phase $\theta\in\mathbb R$, and any complex kernel $\rho$ on $I\times\mathbb N^{\{0,\ldots,M-1\}}$, this defines another kernel as follows. Put $r=\sqrt R$, $t=\sqrt{1-R}$, and, for occupation vectors $p,k$, put $A_a(p,k)=\prod_{m=0}^{M-1}\sqrt{\binom{p_m+k_m}{k_m}}\,e^{\,i(p_m+k_m)\theta}r^{p_m}t^{k_m}$. Then

$$\operatorname{attenuate}(a,\rho)\bigl((i,p),(j,q)\bigr) =\sum_{k\in\mathbb N^M}A_a(p,k)\,\overline{A_a(q,k)}\, \rho\bigl((i,p+k),(j,q+k)\bigr).$$

Addition of occupation vectors is componentwise. No positivity, normalization, or summability hypothesis is imposed on $\rho$; the infinite sum is the total complex summation operation, which equals zero if the summand is not summable. The definition includes $M=0$, when there is one empty occupation vector, all products are empty products equal to $1$, and the resulting kernel equals $\rho$. The endpoint reflectances $0$ and $1$ are included, with natural powers satisfying $0^0=1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.bayesError

For any type $X$, arbitrary kernels $\rho,\sigma:X\times X\to\mathbb C$, and arbitrary real numbers $\pi_0,\pi_1$, put $H=\ell^2(X;\mathbb C)$. A resolution $(w_n,u_n)$ of a kernel $K$ consists of summable nonnegative real weights with $\sum_nw_n=1$, vectors $u_n\in H$ normalized whenever $w_n>0$, orthogonality $\langle u_n,u_m\rangle=0$ whenever $n\ne m$ and both weights are positive, and the identities $K(x,y)=\sum_nw_nu_n(x)\overline{u_n(y)}$. The value $\operatorname{bayesError}(\rho,\sigma,\pi_0,\pi_1)$ is the real infimum, over all resolutions $(a_n,u_n)$ of $\rho$, resolutions $(b_n,v_n)$ of $\sigma$, and continuous complex-linear operators $E:H\to H$ satisfying

$$\langle u,Ev\rangle=\overline{\langle v,Eu\rangle}\quad(u,v\in H), \qquad 0\le\operatorname{Re}\langle v,Ev\rangle\le\|v\|^2\quad(v\in H),$$

of the numbers

$$\pi_0\sum_{n=0}^{\infty}a_n\bigl(1-\operatorname{Re}\langle u_n,Eu_n\rangle\bigr) +\pi_1\sum_{n=0}^{\infty}b_n\operatorname{Re}\langle v_n,Ev_n\rangle.$$

The priors have no sign or normalization assumptions, and the kernels are not assumed to admit resolutions. If there are no candidates, the real infimum of the empty set is $0$; in particular, this applies when $X$ is empty. Infinite sums use the total convention of value $0$ for nonsummable families.

QAlgorithms.Papers.Independent.Intakethird100.P54.chernoffAt

For any type $X$, arbitrary kernels $\rho,\sigma:X\times X\to\mathbb C$, and any real $s$, let $H=\ell^2(X;\mathbb C)$ and define $h_t(p)=0$ if $p=0$ and $h_t(p)=p^t$ otherwise, using real powers. A resolution $(w_n,u_n)$ of a kernel $K$ means nonnegative summable weights with sum $1$, vectors in $H$ normalized at positive weights and mutually orthogonal between distinct positive-weight indices, and $K(x,y)=\sum_nw_nu_n(x)\overline{u_n(y)}$ for every $x,y$. Then

$$\operatorname{chernoffAt}(\rho,\sigma,s) = \sup_{\substack{(a_n,u_n)\text{ resolves }\rho\\(b_m,v_m)\text{ resolves }\sigma}} \left\{ \sum_{(n,m)\in\mathbb N^2} h_s(a_n)\,h_{1-s}(b_m)\,|\langle u_n,v_m\rangle|^2 \right\},$$

where the supremum is taken in $\mathbb R$. There is no restriction $0\le s\le1$ in this definition. A zero weight contributes $0$ even when its exponent is zero or negative. The sum is indexed by pairs, and has value $0$ if its family is nonsummable. Resolutions need not exist; the real supremum is $0$ for an empty candidate set, including when $X$ is empty, and also uses the total value $0$ for a set unbounded above.

QAlgorithms.Papers.Independent.Intakethird100.P54.classicalReturned

For arbitrary natural numbers $J,M$, including zero, a splitter $a$ with reflectance $R\in[0,1]$ and real phase $\theta$, and any measure $\nu$ on $\mathbb C^J\times\mathbb C^M$, this defines a complex kernel on pairs of occupations $x=(p,q)$, where $p:\{0,\ldots,J-1\}\to\mathbb N$ and $q:\{0,\ldots,M-1\}\to\mathbb N$. Put $f=\sqrt R\,e^{\mathrm i\theta}$ and

$$h(z,n)=e^{-|z|^2/2}\frac{z^n}{\sqrt{n!}},\qquad g_{\beta,\alpha}(p,q)= \left(\prod_{j=0}^{J-1}h(\beta_j,p_j)\right) \left(\prod_{m=0}^{M-1}h(f\alpha_m,q_m)\right).$$

Let $\psi_{\beta,\alpha}$ be this coefficient function regarded as a vector in the complex space $\ell^2$ of occupation pairs when it belongs to that space, and the zero vector otherwise. The defined kernel is

$$K(x,y)=\int_{\mathbb C^J\times\mathbb C^M} \psi_{\beta,\alpha}(x)\, \overline{\psi_{\beta,\alpha}(y)}\,d\nu(\beta,\alpha).$$

Empty products are $1$. The measure is not required to be a probability measure, and no integrability hypothesis is supplied; each integral uses the total Bochner integral, which is zero when its integrand is not integrable.

QAlgorithms.Papers.Independent.Intakethird100.P54.coherentEnergy

For any natural number $M$, including zero, and any complex amplitudes $\alpha=(\alpha_m)_{m=0}^{M-1}$, this defines the real number

$$\sum_{m=0}^{M-1}|\alpha_m|^2 = \sum_{m=0}^{M-1}\left((\operatorname{Re}\alpha_m)^2+(\operatorname{Im}\alpha_m)^2\right).$$

For $M=0$, this is the empty sum and equals $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.coherentExponent

For any pair $c$ of splitters, each having reflectance $R_b\in[0,1]$ and an arbitrary real phase, the definition assigns

$$\left(\sqrt{R_{\mathrm{true}}}-\sqrt{R_{\mathrm{false}}}\right)^2,$$

where the square roots are nonnegative. No ordering or distinctness assumption is imposed; equal reflectances give zero, and the phases do not affect the value.

QAlgorithms.Papers.Independent.Intakethird100.P54.coherentOpticalProbe

For arbitrary natural numbers $J,M$, complex amplitudes $\beta=(\beta_j)_{j=0}^{J-1}$ and $\alpha=(\alpha_m)_{m=0}^{M-1}$, this defines a vector in $\ell^2(\mathbb N^J\times\mathbb N^M;\mathbb C)$ from the coefficients

$$a(u,v)= \left(\prod_{j=0}^{J-1}e^{-|\beta_j|^2/2}\frac{\beta_j^{u_j}}{\sqrt{u_j!}}\right) \left(\prod_{m=0}^{M-1}e^{-|\alpha_m|^2/2}\frac{\alpha_m^{v_m}}{\sqrt{v_m!}}\right), \qquad (u,v)\in\mathbb N^J\times\mathbb N^M.$$

If this coefficient function is square-summable, the defined vector has these coefficients; otherwise it is the zero vector. Either number of modes may be zero, and each empty product equals $1$. Natural powers use $0^0=1$, and every factorial denominator is positive.

QAlgorithms.Papers.Independent.Intakethird100.P54.countLaw

For any natural number $M$, any splitter $a$ consisting of a real reflectance $R\in[0,1]$ and an arbitrary real phase, and any occupation vectors $N,j\in\mathbb N^M$, this defines the real number

$$\begin{cases} \displaystyle\prod_{m=0}^{M-1}\binom{N_m}{j_m}R^{j_m}(1-R)^{N_m-j_m}, &j_m\le N_m\text{ for every }m,\\[6pt] 0,&\text{otherwise}. \end{cases}$$

The phase does not enter this expression. The exponents use natural-number subtraction, which truncates at zero, although the branch condition ensures $N_m-j_m$ is the ordinary nonnegative difference wherever the product is used. The boundary reflectances $R=0$ and $R=1$ are included, with $0^0=1$. For $M=0$, the branch condition is vacuous and the empty product equals $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.countThreshold

For any pair of splitters $c_0=c(\mathrm{false})$ and $c_1=c(\mathrm{true})$, with respective reflectances $R_0,R_1\in[0,1]$ and arbitrary real phases, and any natural number $K$, this defines the real number

$$\frac{ K\log\!\left(\dfrac{1-R_0}{1-R_1}\right) }{ \log\!\left(\dfrac{R_1(1-R_0)}{R_0(1-R_1)}\right) }.$$

The phases do not enter the expression, and no ordering, distinctness, or strict interior condition is imposed on the reflectances. All divisions and logarithms are total: division by zero equals $0$, and $\log 0=0$. Thus the expression is defined even at boundary reflectances and when its final denominator is zero; in the latter case its value is $0$. For $K=0$, its value is also $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.countingError

For any pair of splitters $c_0=c(\mathrm{false})$ and $c_1=c(\mathrm{true})$, each having a real reflectance $R_b\in[0,1]$ and an arbitrary real phase, and any natural number $K$, this defines

$$\frac12\sum_{j=0}^{K} \min\!\left\{ \binom Kj R_0^j(1-R_0)^{K-j}, \binom Kj R_1^j(1-R_1)^{K-j} \right\}.$$

No ordering or distinctness of the splitters, and no restriction on their phases, is assumed; only their reflectances enter the expression. The endpoints $R_b=0,1$ are included, with $0^0=1$. For $K=0$, the sum has one term and the defined value is $\tfrac12$.

QAlgorithms.Papers.Independent.Intakethird100.P54.efficiencyScaled

For every splitter $a$, whose reflectance $R$ satisfies $0\le R\le1$ and whose phase $\theta$ is an arbitrary real number, and every real $\eta$ with supplied proofs of $0\le\eta\le1$, the definition constructs a splitter with reflectance $\eta R$ and phase $\theta$, together with proofs that $0\le\eta R\le1$. The cases $\eta=0$, $\eta=1$, $R=0$, and $R=1$ are all included; at $\eta=0$, the resulting reflectance is zero while its phase remains $\theta$.

QAlgorithms.Papers.Independent.Intakethird100.P54.eprAlpha

For any pair $c$ of splitters with reflectances $R_0,R_1\in[0,1]$, indexed respectively by $\mathrm{false}$ and $\mathrm{true}$, and arbitrary real phases, and for every real $N$, the definition assigns

$$\frac{(1-R_0)N+1}{(1-R_1)N+1}.$$

There is no nonnegativity assumption on $N$, nor any ordering or distinctness assumption on the splitters. If the denominator is zero, the quotient is defined to be zero; the phases do not enter the formula.

QAlgorithms.Papers.Independent.Intakethird100.P54.eprBeta

For any pair $c$ of splitters with reflectances $R_0,R_1\in[0,1]$, indexed respectively by $\mathrm{false}$ and $\mathrm{true}$, and arbitrary real phases, the definition assigns

$$\frac{1-R_0}{1-R_1}.$$

No ordering or distinctness is assumed, and the phases do not enter the formula. The endpoint $R_1=1$ is included and gives zero under the convention that division by zero is zero.

QAlgorithms.Papers.Independent.Intakethird100.P54.eprC

For any pair $c$ of splitters with reflectances $R_0,R_1\in[0,1]$, indexed respectively by $\mathrm{false}$ and $\mathrm{true}$, and arbitrary real phases, and for every real $N$, the definition assigns

$$\frac{\left((1-\sqrt{R_0}\sqrt{R_1})N+1\right)^2} {(1-R_0)N+1},$$

using nonnegative square roots. No sign restriction on $N$, ordering, or distinctness is assumed. A zero denominator gives zero, and the phases do not affect the value.

QAlgorithms.Papers.Independent.Intakethird100.P54.eprD

For any pair $c$ of splitters with reflectances $R_0,R_1\in[0,1]$, indexed respectively by $\mathrm{false}$ and $\mathrm{true}$, and arbitrary real phases, and for every real $N$, the definition assigns

$$(1-R_1)N.$$

There is no nonnegativity assumption on $N$, nor any ordering or distinctness requirement. The value is zero when $N=0$ or $R_1=1$, and neither phase enters the formula.

QAlgorithms.Papers.Independent.Intakethird100.P54.eprDerivativeSign

For any pair $c$ of splitters with reflectances $R_0,R_1\in[0,1]$, indexed respectively by $\mathrm{false}$ and $\mathrm{true}$, and arbitrary real phases, and for every real $N$, the definition assigns the real number

$$(1-R_1)N\log\!\left(\frac{1-R_0}{1-R_1}\right) - \frac{\left((1-\sqrt{R_0}\sqrt{R_1})N+1\right)^2} {(1-R_0)N+1} \log\!\left(\frac{(1-R_0)N+1}{(1-R_1)N+1}\right).$$

No sign restriction on $N$, ordering, or distinctness is assumed. Division by zero is zero, $\log0=0$, and $\log x=\log|x|$ for negative $x$. The phases do not enter this expression, and the definition itself asserts no derivative identity or sign property.

QAlgorithms.Papers.Independent.Intakethird100.P54.eprOptimizer

For any pair $c$ of splitters with reflectances $R_0,R_1\in[0,1]$, indexed respectively by $\mathrm{false}$ and $\mathrm{true}$, and arbitrary real phases, and for every real $N$, define the auxiliary real numbers

$$A=\frac{(1-R_0)N+1}{(1-R_1)N+1},\qquad B=\frac{1-R_0}{1-R_1},\qquad C=\frac{\left((1-\sqrt{R_0}\sqrt{R_1})N+1\right)^2}{(1-R_0)N+1}, \qquad D=(1-R_1)N.$$

The definition assigns

$$\begin{cases} 0,&0\le D\log B-C\log A,\\[2mm] \displaystyle\min\!\left\{1,\, \frac{\log\!\left(\frac{C\log A}{D\log B}\right)} {\log(B/A)}\right\},&D\log B-C\log A<0. \end{cases}$$

Equality in the branch condition selects zero. No sign restriction on $N$, ordering, or distinctness is assumed. All divisions and logarithms are total, with division by zero equal to zero, $\log0=0$, and $\log x=\log|x|$ for negative $x$. The second branch caps the expression above by $1$ without imposing a lower bound of zero, and no minimizing property is asserted.

QAlgorithms.Papers.Independent.Intakethird100.P54.exponentGain

For any pair $c$ of splitters with reflectances $R_0,R_1\in[0,1]$, indexed respectively by $\mathrm{false}$ and $\mathrm{true}$, and arbitrary real phases, set

$$s_*= \frac{\log\!\left( \frac{-R_1\log(R_0/R_1)} {(1-R_1)\log((1-R_0)/(1-R_1))} \right)} {\log\!\left(\frac{(1-R_0)R_1}{(1-R_1)R_0}\right)}.$$

The definition assigns

$$\frac{-\log\!\left( R_0^{s_*}R_1^{1-s_*}+ (1-R_0)^{s_*}(1-R_1)^{1-s_*} \right)} {(\sqrt{R_1}-\sqrt{R_0})^2}.$$

No ordering, distinctness, or phase restriction is assumed. Division by zero is zero, $\log0=0$, and $\log x=\log|x|$ for negative $x$; powers are real powers with $0^0=1$ and $0^t=0$ for $t\ne0$. In particular, equal reflectances make the denominator zero and the defined quotient zero.

QAlgorithms.Papers.Independent.Intakethird100.P54.fock

For any natural number $M\ge0$ and any occupation tuple $N:\{0,\ldots,M-1\}\to\mathbb N$, this defines a vector in $\ell^2(\mathcal N_0\times\mathcal N_M;\mathbb C)$, where $\mathcal N_j=\{0,\ldots,j-1\}\to\mathbb N$, by

$$f_N(a,n)= \begin{cases} 1,&(a,n)=(0,N),\\ 0,&\text{otherwise}. \end{cases}$$

Here $\mathcal N_0$ contains exactly one function, denoted $0$, so this is the unit coordinate vector at the pair consisting of that empty function and $N$. When $M=0$, the entire coordinate index set is a singleton and the vector’s sole coordinate is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.fockCoefficient

For any pair of splitters $c_0=c(\mathrm{false})$ and $c_1=c(\mathrm{true})$, with respective real reflectances $R_0,R_1\in[0,1]$ and arbitrary real phases, and any real number $s$, this defines

$$R_0^{\,s}R_1^{\,1-s}+(1-R_0)^{\,s}(1-R_1)^{\,1-s},$$

where the powers are real powers. There is no restriction that $s$ lie in $[0,1]$, nor any ordering or distinctness assumption on the reflectances; the phases do not enter the formula. For a positive base $x$, $x^t=\exp(t\log x)$; at a zero base, the convention is $0^0=1$ and $0^t=0$ for every nonzero real $t$, including negative $t$.

QAlgorithms.Papers.Independent.Intakethird100.P54.fockOptimizer

For any pair of splitters $c_0=c(\mathrm{false})$ and $c_1=c(\mathrm{true})$, with respective reflectances $R_0,R_1\in[0,1]$ and arbitrary real phases, this defines the real number

$$\frac{ \log\!\left( \frac{-R_1\log(R_0/R_1)} {(1-R_1)\log((1-R_0)/(1-R_1))} \right) }{ \log\!\left( \frac{(1-R_0)R_1}{(1-R_1)R_0} \right) }.$$

Only the reflectances enter this expression. There is no ordering, distinctness, strict interior, or logarithm-argument positivity assumption. Division by zero equals $0$; the real logarithm satisfies $\log 0=0$ and, for negative arguments, $\log x=\log|x|$. Consequently the formula assigns a real value even at boundary reflectances or vanishing denominators, and a zero final denominator makes the defined value $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.fockVector

For any type $X$ and function $a:X\to\mathbb C$, this defines a vector in $\ell^2(X;\mathbb C)$: if $a$ is square-summable, the vector has coordinates $a(x)$; otherwise, it is the zero vector. No normalization is required. If $X$ is empty, the result is the unique zero vector.

QAlgorithms.Papers.Independent.Intakethird100.P54.kernelProduct

For any type $X$ and any two complex kernels $\rho,\sigma:X\times X\to\mathbb C$, this defines their kernel product by

$$(\rho\mathbin{\star}\sigma)(x,y)=\sum_{z\in X}\rho(x,z)\sigma(z,y) \qquad(x,y\in X).$$

There is no finiteness, countability, boundedness, positivity, or summability assumption. The sum is the total complex summation operation, equal to zero when its summand is not summable. Empty index types are included.

QAlgorithms.Papers.Independent.Intakethird100.P54.lossAmplitude

For every natural number $M$, every splitter $a$ with reflectance $R\in[0,1]$ and real phase $\theta$, and every pair of occupation functions $n,k:\{0,\ldots,M-1\}\to\mathbb N$, this defines the complex number

$$\begin{cases} \displaystyle\prod_{m=0}^{M-1} \left(\sqrt{\binom{n_m}{k_m}}\, e^{i n_m\theta}\, (\sqrt R)^{\,n_m-k_m}\, (\sqrt{1-R})^{\,k_m}\right), & \text{if }k_m\le n_m\text{ for every }m,\\[4pt] 0,&\text{otherwise}. \end{cases}$$

The subtraction in each exponent is natural-number subtraction, and the condition in the first branch ensures it equals the ordinary nonnegative difference. Zero exponents give the value $1$, including for a zero base, so the formula includes $R=0$ and $R=1$. For $M=0$, the condition is vacuous and the empty product equals $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.lossBlock

For any type $I$, natural number $M\ge0$, splitter pair $c$, vector $\psi\in\ell^2(I\times\mathcal N_M;\mathbb C)$, and $k\in\mathcal N_M=\{0,\ldots,M-1\}\to\mathbb N$, write $R_b\in[0,1]$ and $\theta_b\in\mathbb R$ for the reflectance and phase of $c(b)$, for $b=0,1$. Let $L_{b,k}$ be the vector whose coordinate at $(i,q)$ is

$$\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m},$$

with the entire vector defined as zero if this coordinate function is not square-summable. The definition returns the complex linear subspace

$$\operatorname{span}_{\mathbb C}\{L_{0,k},L_{1,k}\} =\{\alpha L_{0,k}+\beta L_{1,k}:\alpha,\beta\in\mathbb C\}.$$

The two vectors need not be nonzero or independent, so the subspace may have dimension zero, one, or two. No normalization of $\psi$ is required. The case $M=0$ uses the unique empty occupation tuple and product $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.lossBlockDifference

Let $I$ be any type, $M\ge0$ a natural number, $\mathcal N_M=\{0,\ldots,M-1\}\to\mathbb N$, $c$ a splitter pair, $\psi,v\in\ell^2(I\times\mathcal N_M;\mathbb C)$, $k\in\mathcal N_M$, and $w_0,w_1\in\mathbb R$. Write $R_b\in[0,1]$ and $\theta_b\in\mathbb R$ for the reflectance and phase of $c(b)$. Define $L_{b,k}$, for $b=0,1$, from the coordinates

$$\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m},$$

using the zero vector if this function is not square-summable. This definition returns the vector

$$w_0\langle L_{0,k},v\rangle L_{0,k} -w_1\langle L_{1,k},v\rangle L_{1,k},$$

where the inner product is conjugate-linear in its first argument. The real weights are unrestricted, neither vector must be normalized, and $v$ need not lie in the span of the two sector vectors. For $M=0$, occupation tuples are empty and the product is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.lossOverlap

For any type $I$, natural number $M\ge0$, vector $\psi\in\ell^2(I\times\mathcal N_M;\mathbb C)$, splitter pair $c$, and $k\in\mathcal N_M$, where $\mathcal N_M=\{0,\ldots,M-1\}\to\mathbb N$, write $R_b\in[0,1]$ and $\theta_b\in\mathbb R$ for the reflectance and phase of splitter $c(b)$, for $b=0,1$. For each $b$, let $L_{b,k}$ have coordinates

$$\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m}$$

when this function is square-summable, and let $L_{b,k}=0$ otherwise. This defines the complex number $\langle L_{0,k},L_{1,k}\rangle$, using the Hilbert inner product conjugate-linear in its first argument. The vector $\psi$ need not be normalized. The case $M=0$ is included, with one empty occupation tuple and empty product $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.lossOverlapTerm

For any type $I$, natural number $M\ge0$, splitter pair $c$, vector $\psi\in\ell^2(I\times\mathcal N_M;\mathbb C)$, and occupation tuples $k,n\in\mathcal N_M=\{0,\ldots,M-1\}\to\mathbb N$, put

$$p_\psi(n)=\sum_{i\in I}|\psi(i,n)|^2.$$

For $b=0,1$, write $R_b\in[0,1]$ and $\theta_b\in\mathbb R$ for splitter $c(b)$’s reflectance and phase, and define

$$A_b(n,k)= \begin{cases} \displaystyle\prod_{m=0}^{M-1} \sqrt{\binom{n_m}{k_m}}\, e^{\,\mathrm i n_m\theta_b} (\sqrt{R_b})^{n_m-k_m}(\sqrt{1-R_b})^{k_m}, & k_m\le n_m\text{ for every }m,\\ 0,&\text{otherwise}. \end{cases}$$

The definition returns $p_\psi(n)\,\overline{A_0(n,k)}A_1(n,k)$, as a complex number. In particular, it returns zero if any $k_m>n_m$. No normalization or orthogonality condition on $\psi$ is imposed. For $M=0$, both tuples are the unique empty tuple, the condition is vacuous, and both amplitude products equal $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.lossProbability

Let $I$ be any type, let $M\ge0$ be a natural number, and write $\mathcal N_M=\{0,\ldots,M-1\}\to\mathbb N$. Let $\psi\in\ell^2(I\times\mathcal N_M;\mathbb C)$, let $c$ assign a splitter to each Boolean $b$, and fix $b$ and $k\in\mathcal N_M$. Write $R_b\in[0,1]$ and $\theta_b\in\mathbb R$ for that splitter’s reflectance and phase. Define $L_{b,k}$ to be the square-summable vector with coordinates

$$L_{b,k}(i,q)=\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m},$$

if this coordinate function is square-summable, and the zero vector otherwise. The definition returns $\|L_{b,k}\|^2$. No normalization of $\psi$, ordering of the two reflectances, or relation between their phases is assumed. When $M=0$, $\mathcal N_M$ contains just the empty tuple and the product is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.lossSector

For any type $I$, natural number $M$, splitter with reflectance $R\in[0,1]$ and phase $\theta\in\mathbb R$, square-summable complex family $\psi\in\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}})$, and occupation vector $k\in\mathbb N^{\{0,\ldots,M-1\}}$, form the complex family

$$s_k(i,q)=\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta} (\sqrt R)^{q_m}(\sqrt{1-R})^{k_m}.$$

The defined vector is this family regarded as an element of $\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}})$ if it is square-summable, and is the zero vector otherwise. No unit-norm condition is imposed on $\psi$ or on the resulting vector. Endpoint reflectances and zero exponents are included, with $0^0=1$; for $M=0$, the product is empty and the resulting vector is $\psi$.

QAlgorithms.Papers.Independent.Intakethird100.P54.lowerError

For every real number $F$, this defines

$$\operatorname{lowerError}(F)=\frac{1-\sqrt{1-F}}{2}.$$

There is no restriction $0\le F\le1$. The real square root is defined to be zero on negative inputs, so the value is $1/2$ whenever $F\ge1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.ndsChernoffTerm

For any type $I$, natural number $M\ge0$, splitter pair $c$, vector $\psi\in\ell^2(I\times\mathcal N_M;\mathbb C)$, real $s$, and $k\in\mathcal N_M=\{0,\ldots,M-1\}\to\mathbb N$, write $R_b\in[0,1]$ and $\theta_b\in\mathbb R$ for splitter $c(b)$’s parameters. Define $L_{b,k}$, for $b=0,1$, from the coordinates

$$\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m},$$

using zero if this function is not square-summable, and put $p_b=\|L_{b,k}\|^2$. This defines

$$\begin{cases} 0,&p_0=0\text{ or }p_1=0,\\ p_0^{\,s-1}p_1^{-s}\,|\langle L_{0,k},L_{1,k}\rangle|^2, &\text{otherwise}, \end{cases}$$

where powers are real powers. In the second case both bases are positive. The parameter $s$ is not restricted to $[0,1]$, and no normalization or occupation-sector orthogonality of $\psi$ is assumed. The case $M=0$ uses the unique empty tuple and product $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.numberExponent

For any pair $c$ of splitters indexed by $b\in\{\mathrm{false},\mathrm{true}\}$, each with reflectance $R_b\in[0,1]$ and an arbitrary real phase, write $R_0=R_{\mathrm{false}}$ and $R_1=R_{\mathrm{true}}$, and set

$$s_*= \frac{\log\!\left( \frac{-R_1\log(R_0/R_1)} {(1-R_1)\log((1-R_0)/(1-R_1))} \right)} {\log\!\left(\frac{(1-R_0)R_1}{(1-R_1)R_0}\right)}.$$

The definition assigns

$$-\log\!\left(R_0^{s_*}R_1^{1-s_*} +(1-R_0)^{s_*}(1-R_1)^{1-s_*}\right).$$

The phases do not enter this expression. No ordering or distinctness of the reflectances is required, and endpoint reflectances are included. All expressions use total real operations: division by zero is zero, $\log 0=0$, and for negative $x$, $\log x=\log|x|$; the powers are real powers, with $0^0=1$ and $0^t=0$ for $t\ne0$. The definition makes no assertion that $s_*$ minimizes any function or belongs to $[0,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P54.opticalBayesRisk

For any type $X$, let $H=\ell^2(X;\mathbb C)$ and let $\rho,\sigma:X\times X\to\mathbb C$ be arbitrary kernels. A resolution of a kernel $K$ consists of weights $w_n\ge0$ and vectors $u_n\in H$, indexed by $n\in\mathbb N$, such that the weights are summable with sum $1$, every vector with positive weight has norm $1$, distinct vectors with positive weights are orthogonal, and $K(x,y)=\sum_{n=0}^{\infty}w_nu_n(x)\overline{u_n(y)}$ for every $x,y$. Given resolutions $A=(a_n,u_n)$ of $\rho$ and $B=(b_n,v_n)$ of $\sigma$, any continuous complex-linear operator $E:H\to H$, and any real numbers $\pi_0,\pi_1$, the definition is

$$\operatorname{opticalBayesRisk}(A,B,E,\pi_0,\pi_1) =\pi_0\sum_{n=0}^{\infty}a_n\bigl(1-\operatorname{Re}\langle u_n,Eu_n\rangle\bigr) +\pi_1\sum_{n=0}^{\infty}b_n\operatorname{Re}\langle v_n,Ev_n\rangle.$$

There is no assumption here that $E$ is an effect or that the real numbers $\pi_0,\pi_1$ are nonnegative or sum to $1$. Zero-weight vectors need not be normalized or orthogonal. The infinite sums use the total convention that a nonsummable family has sum $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.opticalChernoff

For any type $X$ and arbitrary kernels $\rho,\sigma:X\times X\to\mathbb C$, put $H=\ell^2(X;\mathbb C)$ and $h_t(p)=0$ for $p=0$, with $h_t(p)=p^t$ otherwise. For each $s\in[0,1]$, form the real supremum

$$C_s= \sup_{\substack{(a_n,u_n)\text{ resolves }\rho\\(b_m,v_m)\text{ resolves }\sigma}} \left\{ \sum_{(n,m)\in\mathbb N^2} h_s(a_n)\,h_{1-s}(b_m)\,|\langle u_n,v_m\rangle|^2 \right\}.$$

Here a resolution $(w_n,u_n)$ of $K$ has nonnegative summable weights summing to $1$, vectors in $H$ normalized whenever their weights are positive, orthogonality between distinct positive-weight vectors, and $K(x,y)=\sum_nw_nu_n(x)\overline{u_n(y)}$ for all $x,y$. The definition is $\operatorname{opticalChernoff}(\rho,\sigma)=\inf\{C_s:0\le s\le1\}$, a real infimum; both endpoints are included, and zero weights retain value $0$ at exponent zero. No existence of resolutions is assumed. Empty or unbounded-above candidate sets for the inner real supremum have value $0$, and a nonsummable infinite family has sum $0$; in particular, empty $X$ gives value $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.opticalCopies

For arbitrary natural numbers $J,M,L$, including zero, and any vector $\psi\in\ell^2(\mathbb N^J\times\mathbb N^M;\mathbb C)$, this defines a vector in $\ell^2((\mathbb N^J)^L\times\mathbb N^{LM};\mathbb C)$ from the coefficient function

$$a(i,n)=\prod_{\ell=0}^{L-1}\psi\!\left(i_\ell,\bigl(n_{\ell M+m}\bigr)_{m=0}^{M-1}\right),$$

where $i=(i_\ell)_{\ell=0}^{L-1}$ is a list of $L$ occupation vectors in $\mathbb N^J$, and $n\in\mathbb N^{LM}$. If $a$ is square-summable, the defined vector has exactly these coefficients; otherwise it is the zero vector. There is no normalization assumption on $\psi$. When $L=0$, the product is empty and equals $1$, and the output index set is a singleton; $J=0$ and $M=0$ are also allowed, with their occupation vectors interpreted as functions on the empty set.

QAlgorithms.Papers.Independent.Intakethird100.P54.opticalError

For any type $X$ and arbitrary kernels $\rho,\sigma:X\times X\to\mathbb C$, let $H=\ell^2(X;\mathbb C)$. The value $\operatorname{opticalError}(\rho,\sigma)$ is the real infimum of

$$\frac12\sum_{n=0}^{\infty}a_n\bigl(1-\operatorname{Re}\langle u_n,Eu_n\rangle\bigr) +\frac12\sum_{n=0}^{\infty}b_n\operatorname{Re}\langle v_n,Ev_n\rangle$$

over the following data: nonnegative summable weights $a_n,b_n$ each summing to $1$; vectors $u_n,v_n\in H$ normalized at positive weights and orthogonal within each family between distinct positive-weight indices; representations $\rho(x,y)=\sum_na_nu_n(x)\overline{u_n(y)}$ and $\sigma(x,y)=\sum_nb_nv_n(x)\overline{v_n(y)}$ for every $x,y$; and a continuous complex-linear operator $E:H\to H$ such that $\langle u,Ev\rangle=\overline{\langle v,Eu\rangle}$ for all $u,v$ and $0\le\operatorname{Re}\langle v,Ev\rangle\le\|v\|^2$ for all $v$. Thus the two weights multiplying the risks are exactly $1/2$. Existence of these data is not assumed; the real infimum is $0$ if the candidate set is empty, including when $X$ is empty. Infinite sums have value $0$ for nonsummable families.

QAlgorithms.Papers.Independent.Intakethird100.P54.opticalKernel

For any type $X$, an optical kernel on $X$ is an arbitrary function $X\times X\to\mathbb C$. No finiteness, symmetry, positivity, or normalization condition is imposed; $X$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P54.opticalPureKernel

For every type $X$ and every square-summable complex function $\psi\in\ell^2(X;\mathbb C)$, this defines the complex-valued kernel on $X\times X$ by

$$K_\psi(x,y)=\psi(x)\overline{\psi(y)} \qquad(x,y\in X).$$

No unit-norm assumption is imposed: the zero vector gives the zero kernel. Empty $X$ is also allowed, in which case the kernel has no entries.

QAlgorithms.Papers.Independent.Intakethird100.P54.outputChernoff

For any type $I$, $M\in\mathbb N$, splitters $c_b$ indexed by $b\in\{0,1\}$ with reflectances $R_b\in[0,1]$ and phases $\theta_b\in\mathbb R$, and any $\psi\in\ell^2(X;\mathbb C)$ where $Q=\mathbb N^{\{0,\ldots,M-1\}}$ and $X=I\times Q$, define

$$f_{b,k}(i,q)=\psi(i,q+k) \prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m}, \qquad K_b(x,y)=\sum_{k\in Q}f_{b,k}(x)\overline{f_{b,k}(y)}.$$

For each $s\in[0,1]$, let $C_s$ be the real supremum of $\sum_{(n,m)\in\mathbb N^2}h_s(a_n)h_{1-s}(b_m)|\langle u_n,v_m\rangle|^2$ over resolutions $(a_n,u_n)$ of $K_0$ and $(b_n,v_n)$ of $K_1$, where $h_t(0)=0$ and $h_t(p)=p^t$ for $p\ne0$. A resolution $(w_n,z_n)$ consists of nonnegative summable weights summing to $1$, vectors in $\ell^2(X;\mathbb C)$ normalized at positive weights and orthogonal between distinct positive-weight indices, and $K(x,y)=\sum_nw_nz_n(x)\overline{z_n(y)}$ for every $x,y$. The definition is $\operatorname{outputChernoff}(c,\psi)=\inf\{C_s:0\le s\le1\}$ in $\mathbb R$, including both endpoints and retaining $h_0(0)=0$. No normalization of $\psi$, ordering or distinctness of splitters, or restriction on phases is assumed. The case $M=0$ uses a singleton $Q$ and empty products equal to $1$. Infinite sums are $0$ for nonsummable families, and empty or unbounded-above candidate sets for the inner real supremum have value $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.outputChernoffAt

For any type $I$, $M\in\mathbb N$, splitters $c_b$ indexed by $b\in\{0,1\}$ with reflectances $R_b\in[0,1]$ and phases $\theta_b\in\mathbb R$, any $\psi\in\ell^2(X;\mathbb C)$ where $Q=\mathbb N^{\{0,\ldots,M-1\}}$ and $X=I\times Q$, and any real $s$, define

$$f_{b,k}(i,q)=\psi(i,q+k) \prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m}, \qquad K_b(x,y)=\sum_{k\in Q}f_{b,k}(x)\overline{f_{b,k}(y)}.$$

With $h_t(p)=0$ if $p=0$ and $h_t(p)=p^t$ otherwise, $\operatorname{outputChernoffAt}(c,\psi,s)$ is the real supremum of

$$\sum_{(n,m)\in\mathbb N^2}h_s(a_n)h_{1-s}(b_m)|\langle u_n,v_m\rangle|^2$$

over resolutions $(a_n,u_n)$ of $K_0$ and $(b_n,v_n)$ of $K_1$. A resolution $(w_n,z_n)$ has nonnegative summable weights summing to $1$, vectors in $\ell^2(X;\mathbb C)$ normalized at positive weights and orthogonal between distinct positive-weight indices, and the representation $K(x,y)=\sum_nw_nz_n(x)\overline{z_n(y)}$ for every $x,y$. There is no restriction on $s$, no normalization assumption on $\psi$, and no ordering, distinctness, or phase restriction on the splitters. Zero weights give $0$ even at zero or negative exponents. The case $M=0$ is allowed, with empty products equal to $1$. Nonsummable families have sum $0$, and an empty or unbounded-above candidate set has real supremum $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.outputError

For any type $I$, $M\in\mathbb N$, splitters $c_b$ indexed by $b\in\{0,1\}$ with reflectances $R_b\in[0,1]$ and phases $\theta_b\in\mathbb R$, and any $\psi\in H=\ell^2(X;\mathbb C)$ where $Q=\mathbb N^{\{0,\ldots,M-1\}}$ and $X=I\times Q$, set

$$f_{b,k}(i,q)=\psi(i,q+k) \prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m}, \qquad K_b(x,y)=\sum_{k\in Q}f_{b,k}(x)\overline{f_{b,k}(y)}.$$

The value $\operatorname{outputError}(c,\psi)$ is the real infimum of

$$\frac12\sum_{n=0}^{\infty}a_n\bigl(1-\operatorname{Re}\langle u_n,Eu_n\rangle\bigr) +\frac12\sum_{n=0}^{\infty}b_n\operatorname{Re}\langle v_n,Ev_n\rangle$$

over nonnegative summable weights $a_n,b_n$ each summing to $1$, vectors $u_n,v_n\in H$ normalized at positive weights and orthogonal within each family between distinct positive-weight indices, representations $K_0(x,y)=\sum_na_nu_n(x)\overline{u_n(y)}$ and $K_1(x,y)=\sum_nb_nv_n(x)\overline{v_n(y)}$, and continuous complex-linear operators $E:H\to H$ satisfying $\langle u,Ev\rangle=\overline{\langle v,Eu\rangle}$ for all $u,v$ and $0\le\operatorname{Re}\langle v,Ev\rangle\le\|v\|^2$ for all $v$. No normalization of $\psi$, ordering or distinctness of the splitters, or restriction on phases is required. The case $M=0$ uses a singleton $Q$ and empty products equal to $1$. If no admissible resolutions and operator exist, the real infimum is $0$; nonsummable infinite families have sum $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.outputFidelity

For any type $I$, any natural number $M$, two splitters $c_0=c(\mathrm{false})$ and $c_1=c(\mathrm{true})$ with reflectances $R_b\in[0,1]$ and phases $\theta_b\in\mathbb R$, and any $\psi\in\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}};\mathbb C)$, put $Q=\mathbb N^{\{0,\ldots,M-1\}}$, $X=I\times Q$, $r_b=\sqrt{R_b}$, and $t_b=\sqrt{1-R_b}$. Define, for $k,q\in Q$,

$$f_{b,k}(i,q)=\psi(i,q+k) \prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b}\, r_b^{q_m}t_b^{k_m}, \qquad K_b(x,y)=\sum_{k\in Q}f_{b,k}(x)\overline{f_{b,k}(y)}.$$

Then $\operatorname{outputFidelity}(c,\psi)$ is the real supremum of $|\langle z,w\rangle|^2$ over all $z,w\in\ell^2(X\times\mathbb N;\mathbb C)$ with $\|z\|=\|w\|=1$ and, for every $x,y\in X$,

$$\sum_{e=0}^{\infty}z(x,e)\overline{z(y,e)}=K_0(x,y), \qquad \sum_{e=0}^{\infty}w(x,e)\overline{w(y,e)}=K_1(x,y).$$

No normalization of $\psi$, ordering or distinctness of the splitters, or restriction on their phases is assumed. The case $M=0$ is included, with $Q$ a singleton and empty products equal to $1$. The real supremum is $0$ if no such purifications exist; infinite sums have value $0$ for nonsummable families.

QAlgorithms.Papers.Independent.Intakethird100.P54.photons

For every natural number $M$, including $M=0$, and every occupation function $n:\{0,\ldots,M-1\}\to\mathbb N$, this defines the natural number $\sum_{m=0}^{M-1}n_m$. For $M=0$, the sum is empty and equals zero.

QAlgorithms.Papers.Independent.Intakethird100.P54.productOpticalProbe

For any natural number $M\ge0$, any function $J:\{0,\ldots,M-1\}\to\mathbb N$, and any family

$$\psi_m\in\ell^2(\mathcal N_{J(m)}\times\mathcal N_1;\mathbb C), \qquad \mathcal N_j=\{0,\ldots,j-1\}\to\mathbb N,$$

this defines a vector in $\ell^2((\prod_{m=0}^{M-1}\mathcal N_{J(m)})\times\mathcal N_M;\mathbb C)$ from the coordinate function

$$f(a,n)=\prod_{m=0}^{M-1}\psi_m(a_m,\underline{n_m}),$$

where $a_m\in\mathcal N_{J(m)}$, $n\in\mathcal N_M$, and $\underline{n_m}$ is the function on the singleton $\{0\}$ with value $n_m$. The result has these coordinates if $f$ is square-summable, and is the zero vector otherwise. No normalization of the input vectors is required, and any $J(m)$ may be zero. For $M=0$, both coordinate factors are singletons and the empty product gives the vector with sole coordinate $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.purificationMarginal

For any type $X$ and square-summable complex family $z\in\ell^2(X\times\mathbb N)$, this defines the complex-valued kernel on $X$ by

$$K_z(x,y)=\sum_{e=0}^{\infty}z(x,e)\,\overline{z(y,e)} \qquad(x,y\in X).$$

No unit-norm condition is imposed on $z$. In particular, $z=0$ gives the zero kernel, and if $X$ is empty the resulting kernel has no entries.

QAlgorithms.Papers.Independent.Intakethird100.P54.purificationOverlap

For any type $I$, $M\in\mathbb N$, two splitters $c_0=c(\mathrm{false})$, $c_1=c(\mathrm{true})$ with reflectances $R_0,R_1\in[0,1]$ and arbitrary real phases $\theta_0,\theta_1$, and any $\psi\in\ell^2(I\times Q;\mathbb C)$ with $Q=\mathbb N^{\{0,\ldots,M-1\}}$, put

$$P_n=\sum_{q\in Q} \begin{cases} \displaystyle\sum_{i\in I}|\psi(i,q)|^2,&\displaystyle\sum_{m=0}^{M-1}q_m=n,\\ 0,&\text{otherwise}, \end{cases} \qquad \kappa=\sqrt{R_0}\sqrt{R_1}+\sqrt{1-R_0}\sqrt{1-R_1}.$$

The definition is the real number

$$\operatorname{purificationOverlap}(c,\psi) =\left|\sum_{n=0}^{\infty}P_n e^{\,\mathrm i n(\theta_1-\theta_0)}\kappa^n\right|^2.$$

No normalization or energy assumption on $\psi$, or ordering or distinctness assumption on the splitters, is required. The sum includes $n=0$, whose exponential and power factors are $1$, including when $\kappa=0$. The case $M=0$ is included: the sole occupation has photon count $0$, so the expression becomes $\bigl(\sum_{i\in I}|\psi(i,0)|^2\bigr)^2$, where $0$ denotes the empty occupation. Infinite sums have value $0$ for nonsummable families.

QAlgorithms.Papers.Independent.Intakethird100.P54.purificationOverlapTerm

For any type $I$, $M\in\mathbb N$, two splitters $c_0=c(\mathrm{false})$, $c_1=c(\mathrm{true})$ with reflectances $R_0,R_1\in[0,1]$ and arbitrary real phases $\theta_0,\theta_1$, any $\psi\in\ell^2(I\times Q;\mathbb C)$ with $Q=\mathbb N^{\{0,\ldots,M-1\}}$, and any $n\in\mathbb N$, define

$$P_n=\sum_{q\in Q} \begin{cases} \displaystyle\sum_{i\in I}|\psi(i,q)|^2,&\displaystyle\sum_{m=0}^{M-1}q_m=n,\\ 0,&\text{otherwise}, \end{cases} \qquad \kappa=\sqrt{R_0}\sqrt{R_1}+\sqrt{1-R_0}\sqrt{1-R_1}.$$

Then $\operatorname{purificationOverlapTerm}(c,\psi,n)$ is the complex number $P_ne^{\,\mathrm i n(\theta_1-\theta_0)}\kappa^n$. No normalization or energy assumption on $\psi$, or ordering or distinctness assumption on the splitters, is imposed. The index $n=0$ is included, with $\kappa^0=1$ even when $\kappa=0$. If $M=0$, $Q$ contains just the empty occupation and its photon count is $0$. All infinite sums use the total convention of value $0$ for nonsummable families.

QAlgorithms.Papers.Independent.Intakethird100.P54.returned

For any type $I$, natural number $M$, splitter with reflectance $R\in[0,1]$ and phase $\theta\in\mathbb R$, and square-summable complex family $\psi\in\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}})$, define

$$s_k(i,q)=\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta} (\sqrt R)^{q_m}(\sqrt{1-R})^{k_m}.$$

The returned kernel is the complex-valued function of two indices $x,y\in I\times\mathbb N^{\{0,\ldots,M-1\}}$ given by

$$K(x,y)=\sum_{k\in\mathbb N^{\{0,\ldots,M-1\}}} s_k(x)\,\overline{s_k(y)}.$$

These are the coefficient families themselves, without applying the square-summability test used to construct a loss-sector vector. No normalization of $\psi$ is assumed. The reflectances $0$ and $1$ are included, with $0^0=1$. When $M=0$, the sum has one occupation-vector index and $K(x,y)=\psi(x)\overline{\psi(y)}$.

QAlgorithms.Papers.Independent.Intakethird100.P54.sectorCoefficients

For any type $I$, natural number $M$, splitter $a$ with reflectance $R\in[0,1]$ and phase $\theta\in\mathbb R$, square-summable complex family $\psi\in\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}})$, occupation vector $k\in\mathbb N^{\{0,\ldots,M-1\}}$, and $x=(i,q)$ in the same index set as $\psi$, this defines the complex number

$$\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta} (\sqrt R)^{q_m}(\sqrt{1-R})^{k_m}.$$

Addition of occupation vectors is componentwise. The conditional in the underlying amplitude takes its product branch because $k_m\le q_m+k_m$ for every $m$. No normalization of $\psi$ is required. The definition includes $R=0$, $R=1$, and zero exponents, with $0^0=1$. When $M=0$, the occupation vectors are the unique empty vectors and the empty product is $1$, so the coefficient is $\psi(i,q)$.

QAlgorithms.Papers.Independent.Intakethird100.P54.sectorRisk

For any type $I$, natural number $M\ge0$, splitter pair $c$, vector $\psi\in H=\ell^2(I\times\mathcal N_M;\mathbb C)$, bounded complex-linear operator $E:H\to H$, and real numbers $\pi_0,\pi_1$, where $\mathcal N_M=\{0,\ldots,M-1\}\to\mathbb N$, write $R_b\in[0,1]$ and $\theta_b\in\mathbb R$ for splitter $c(b)$’s reflectance and phase. Let $L_{b,k}$ have coordinates

$$\psi(i,q+k)\prod_{m=0}^{M-1} \sqrt{\binom{q_m+k_m}{k_m}}\, e^{\,\mathrm i(q_m+k_m)\theta_b} (\sqrt{R_b})^{q_m}(\sqrt{1-R_b})^{k_m},$$

or be zero if this coordinate function is not square-summable. The defined real number is

$$\pi_0\sum_{k\in\mathcal N_M} \left(\|L_{0,k}\|^2-\operatorname{Re}\langle L_{0,k},EL_{0,k}\rangle\right) +\pi_1\sum_{k\in\mathcal N_M} \operatorname{Re}\langle L_{1,k},EL_{1,k}\rangle.$$

There is no assumption that $E$ is an effect, that $\psi$ is normalized, or that the weights are nonnegative or sum to one. Each infinite sum uses the total summation convention, which assigns zero to a nonsummable family. For $M=0$, the index set has one empty tuple and the coordinate product is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.signalLaw

For any type $I$, natural number $M$, square-summable complex family $\psi\in\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}})$, and occupation vector $n\in\mathbb N^{\{0,\ldots,M-1\}}$, this defines the real number

$$p_\psi(n)=\sum_{i\in I}|\psi(i,n)|^2.$$

The type $I$ need not be countable or nonempty, and $\psi$ need not have norm $1$; in particular, these values are not required by the definition to have total mass $1$. If $I$ is empty, the value is $0$. If $M=0$, there is one occupation vector and its value is $\|\psi\|^2$.

QAlgorithms.Papers.Independent.Intakethird100.P54.splitterOverlap

For any channel pair $c$, consisting of splitters with reflectances $R_0,R_1\in[0,1]$ and arbitrary real phases, this defines the real number $\sqrt{R_0}\sqrt{R_1}+\sqrt{1-R_0}\sqrt{1-R_1}$. The phases do not enter this expression, and no validity or zero-phase hypothesis is required.

QAlgorithms.Papers.Independent.Intakethird100.P54.squaredOpticalFidelity

For any type $X$ and arbitrary complex-valued kernels $\rho,\sigma:X\times X\to\mathbb C$, this defines the real number

$$\sup\left\{ |\langle z,w\rangle|^2: \begin{array}{l} z,w\in\ell^2(X\times\mathbb N),\quad \|z\|=\|w\|=1,\\ \displaystyle\rho(x,y)=\sum_{e=0}^{\infty}z(x,e)\overline{z(y,e)} \quad\text{for all }x,y\in X,\\ \displaystyle\sigma(x,y)=\sum_{e=0}^{\infty}w(x,e)\overline{w(y,e)} \quad\text{for all }x,y\in X \end{array} \right\}.$$

Here $\langle\cdot,\cdot\rangle$ is the complex Hilbert-space inner product. Existence of the two unit vectors is not assumed. If either kernel has no such representation, the set is empty and the total real supremum used by the definition has value $0$; this includes the case $X=\varnothing$. Whenever the set is nonempty, its elements lie in $[0,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P54.supportWeightPower

For arbitrary real numbers $p,t$, the definition is

$$\operatorname{supportWeightPower}(p,t)= \begin{cases} 0,&p=0,\\ p^t,&p\ne0, \end{cases}$$

where the second branch uses the total real-power function. In particular, the value at $p=0$ is $0$ for every exponent, including $t=0$ and negative exponents. No nonnegativity assumption on $p$ or restriction on $t$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P54.targetCoherent

For arbitrary real numbers $R,N$, this defines

$$\operatorname{targetCoherent}(R,N) =\frac{1-\sqrt{\,1-e^{-RN}\,}}{2}.$$

No sign or range restriction is imposed on either argument. The real square root equals zero on negative arguments, so the value is $1/2$ when $RN<0$, and it is $0$ when $RN=0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.targetLower

For arbitrary real numbers $R,N$, this defines

$$\operatorname{targetLower}(R,N) =\frac{1-\sqrt{\,1-(1-R)^N\,}}{2},$$

where the power is the total real-power operation. There is no assumption that $R\in[0,1]$ or that $N\ge0$. The real square root is also total and equals zero on negative arguments; thus the defined value is $1/2$ whenever $(1-R)^N>1$.

QAlgorithms.Papers.Independent.Intakethird100.P54.tmsvCoefficients

For any natural number $M$, any real number $N$, and any pair of occupation vectors $u,v\in\mathbb N^M$, this defines the complex coefficient

$$a_{M,N}(u,v)= \begin{cases} \displaystyle\prod_{m=0}^{M-1} \left(\sqrt{\frac1{N+1}}\right) \left(\sqrt{\frac{N}{N+1}}\right)^{v_m},&u=v,\\[6pt] 0,&u\ne v, \end{cases}$$

with the real factors regarded as complex numbers. There is no restriction that $N$ be nonnegative: division by zero is assigned the value $0$, the real square root of a negative number is $0$, and natural powers satisfy $0^0=1$. For $M=0$, both occupation vectors are the unique empty function, so they are equal and the empty product gives coefficient $1$ for every real $N$.

QAlgorithms.Papers.Independent.Intakethird100.P54.tmsvProbe

For any natural number $M$ and any real number $N$, this defines a vector in $\ell^2(\mathbb N^M\times\mathbb N^M;\mathbb C)$ using the coefficient function

$$a(u,v)= \begin{cases} \displaystyle\prod_{m=0}^{M-1} \sqrt{\frac1{N+1}} \left(\sqrt{\frac{N}{N+1}}\right)^{v_m},&u=v,\\[6pt] 0,&u\ne v. \end{cases}$$

If $a$ is square-summable, the vector has exactly these coefficients; otherwise it is the zero vector. The real factors are embedded into $\mathbb C$. No nonnegativity assumption is imposed on $N$; division by zero equals $0$, real square roots of negative numbers equal $0$, and $0^0=1$. When $M=0$, the coefficient function is $1$ on its singleton index set, and hence defines the vector with that single coefficient.

QAlgorithms.Papers.Independent.Intakethird100.P54.totalCountLaw

For any splitter $a$ with real reflectance $R\in[0,1]$ and arbitrary real phase, and any natural numbers $K,j$, including zero, this defines

$$\begin{cases} \displaystyle\binom Kj R^j(1-R)^{K-j},&j\le K,\\[4pt] 0,&j>K. \end{cases}$$

The phase does not enter the formula. Natural-number subtraction truncates at zero, but in the first branch $K-j$ is the ordinary nonnegative difference. The reflectance endpoints are permitted and natural powers use $0^0=1$; in particular, for $K=0$ the value is $1$ at $j=0$ and $0$ at every $j>0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.totalLaw

For any type $I$, natural number $M$, square-summable complex family $\psi\in\ell^2(I\times\mathbb N^{\{0,\ldots,M-1\}})$, and natural number $n$, this defines

$$P_\psi(n)= \sum_{q\in\mathbb N^{\{0,\ldots,M-1\}}} \begin{cases} \displaystyle\sum_{i\in I}|\psi(i,q)|^2, &\displaystyle\sum_{m=0}^{M-1}q_m=n,\\ 0,&\text{otherwise}. \end{cases}$$

Thus it sums the squared coordinate magnitudes over occupation vectors whose total occupation is exactly $n$. There is no unit-norm assumption on $\psi$, and $n=0$ is included. When $M=0$, the unique occupation vector has total occupation $0$, so $P_\psi(0)=\|\psi\|^2$ and $P_\psi(n)=0$ for every $n>0$. If $I$ is empty, all these values are $0$.

QAlgorithms.Papers.Independent.Intakethird100.P54.vacuumEfficiency

For every real $\eta$, given proofs of $0\le\eta$ and $\eta\le1$, the definition constructs a splitter whose reflectance is $\eta$ and whose phase is $0$, retaining those proofs as the required reflectance bounds. Both endpoints $\eta=0$ and $\eta=1$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P54.weightedTraceNorm

For any type $X$, any complex kernels $\rho,\sigma:X\times X\to\mathbb C$, and arbitrary real weights $w_0,w_1$, let $H=\ell^2(X;\mathbb C)$. A resolution of a kernel $K$ consists of weights $p_n\ge0$, indexed by $n\in\mathbb N$, with $\sum_n p_n=1$ and summable weight sequence, and vectors $a_n\in H$ such that $\|a_n\|=1$ whenever $p_n>0$, distinct vectors with positive weights are orthogonal, and $K(x,y)=\sum_n p_n a_n(x)\overline{a_n(y)}$ for every $x,y$. Vectors with zero weights need not be normalized or satisfy the orthogonality condition. The definition is

$$\sup_{(p,a),(q,b),T} \left[ w_0\sum_n p_n\operatorname{Re}\langle a_n,Ta_n\rangle -w_1\sum_n q_n\operatorname{Re}\langle b_n,Tb_n\rangle \right],$$

where $(p,a)$ resolves $\rho$, $(q,b)$ resolves $\sigma$, and $T:H\to H$ is bounded and complex linear, satisfies $\langle u,Tv\rangle=\overline{\langle v,Tu\rangle}$ for every $u,v$, and satisfies $\left|\operatorname{Re}\langle u,Tu\rangle\right|\le\|u\|^2$ for every $u$. The weights $w_0,w_1$ are not required to be nonnegative or sum to $1$, and the kernels are not assumed to possess resolutions. If either has no resolution, the supremum is over the empty set and the defined real value is $0$.