Definitions for Towards practical and fast Quantum Cryptography

QAlgorithms.Papers.Independent.Intakethird100.P33.FinitePairMoments

For any type $J$ with decidable equality, any complex matrix $\rho$ indexed by occupation configurations $p:J\to\mathbb N$, and any $i,j\in J$, this is the conjunction that each of the following three families, indexed by all such $p$, is summable:

$$p(i)\operatorname{Re}\rho_{p,p},\qquad p(j)\operatorname{Re}\rho_{p,p},\qquad \begin{cases} p(i)\rho_{p,p},&i=j,\\ 0,&i\ne j,\ p(j)=0,\\ \sqrt{(p(i)+1)p(j)}\,\rho_{p,p^{i+,j-}},&i\ne j,\ p(j)>0. \end{cases}$$

Here $p^{i+,j-}$ changes the occupations at $i,j$ to $p(i)+1,p(j)-1$, respectively, and leaves all others unchanged. The first two summability statements concern real families, and the third concerns a complex family. The indices $i,j$ may coincide, and $J$ need not be finite or countable. The predicate does not require $\rho$ to be positive or normalized.

QAlgorithms.Papers.Independent.Intakethird100.P33.OpticalPOVM

For any type $O$ equipped with a finite enumeration, this abbreviates the type of families of complex kernels $E_o:(\mathbb N\times\mathbb N)^2\to\mathbb C$, indexed by $o\in O$, satisfying two requirements: for every $o$, every finite set $S\subseteq\mathbb N\times\mathbb N$, and every function $z:\mathbb N\times\mathbb N\to\mathbb C$, the quantity $\sum_{a\in S}\sum_{b\in S}\overline{z(a)}E_o(a,b)z(b)$ has imaginary part $0$ and nonnegative real part; and for all $a,b\in\mathbb N\times\mathbb N$, $\sum_{o\in O}E_o(a,b)=\delta_{a,b}$. Here $\delta_{a,b}$ is $1$ when $a=b$ and $0$ otherwise. If $O$ is empty, no such family exists, since the completeness requirement would give $0=1$ on the diagonal.

QAlgorithms.Papers.Independent.Intakethird100.P33.Unambiguous

For every real $x$ and every three-outcome family $M=(E_0,E_1,E_2)$ of complex kernels on $\mathbb N\times\mathbb N$, each positive on all finite quadratic forms and satisfying $E_0+E_1+E_2=I$ coefficientwise, this defines the conjunction

$$\operatorname{Re}\!\left(\sum_{p\in\mathbb N^2}\sum_{q\in\mathbb N^2}d_0(p)\overline{d_0(q)}E_1(q,p)\right)=0, \qquad \operatorname{Re}\!\left(\sum_{p\in\mathbb N^2}\sum_{q\in\mathbb N^2}d_1(p)\overline{d_1(q)}E_0(q,p)\right)=0.$$

Here $s=\sqrt{x}$ is the real square root, $c_\alpha(r)=e^{-|\alpha|^2/2}\alpha^r/\sqrt{r!}$, and $d_0(m,n)=c_s(m)c_0(n)$ and $d_1(m,n)=c_0(m)c_s(n)$, with each coefficient function replaced by zero if it is not square summable. The displayed sums are iterated infinite sums in the indicated order. No restriction on $x$ is imposed; for $x\leq0$, both $d_0$ and $d_1$ are the vacuum vector supported at $(0,0)$.

QAlgorithms.Papers.Independent.Intakethird100.P33.aliceBobJoint

For any type $O$, any function $K:\mathbb N^2\times O\to\mathbb R$, arbitrary real $\mu,t$, $o\in O$, and a Boolean $b$, define $c(a,r)=e^{-|a|^2/2}a^r/\sqrt{r!}$ and, with $x=\mu t$, define coefficients

$$d_{x,b}(r,s)= \begin{cases} c(0,r)c(\sqrt{x},s),&b=\mathrm{true},\\ c(\sqrt{x},r)c(0,s),&b=\mathrm{false}. \end{cases}$$

Let $D_{x,b}$ be this coefficient vector if it is square summable, and zero otherwise. The definition is

$$\operatorname{aliceBobJoint}(K,\mu,t,o,b) =\frac12\sum_{p\in\mathbb N^2}|D_{x,b}(p)|^2K(p,o).$$

Real square roots are embedded in $\mathbb C$, with $\sqrt{x}=0$ when $x<0$. There are no positivity or normalization requirements on $K$, no restrictions on $\mu,t$, and no finiteness assumption on $O$; a nonsummable real series is assigned value $0$.

QAlgorithms.Papers.Independent.Intakethird100.P33.bobDistribution

For any type $O$, any function $K:\mathbb N^2\times O\to\mathbb R$, arbitrary real $\mu,t$, a Boolean $b$, and $o\in O$, put $x=\mu t$ and

$$c(a,r)=\frac{e^{-|a|^2/2}a^r}{\sqrt{r!}},\qquad d_{x,b}(r,s)= \begin{cases} c(0,r)c(\sqrt{x},s),&b=\mathrm{true},\\ c(\sqrt{x},r)c(0,s),&b=\mathrm{false}. \end{cases}$$

Here square roots of real numbers are embedded in $\mathbb C$. Let $D_{x,b}$ have these coefficients when they are square summable, and be zero otherwise. The definition returns the real number

$$\operatorname{bobDistribution}(K,\mu,t,b,o) =\sum_{p\in\mathbb N^2}|D_{x,b}(p)|^2K(p,o).$$

No positivity, normalization, or summability assumptions are imposed on $K$, and $O$ need not be finite or nonempty. A nonsummable real series has value $0$ under the infinite-sum convention used here. Negative $x$ is allowed and has $\sqrt{x}=0$.

QAlgorithms.Papers.Independent.Intakethird100.P33.boundaryLeft

For every natural number $n$ and every integer index $k$ satisfying $0\le k<n-1$, where subtraction in $\mathbb N$ is truncated at $0$, this returns the index $k$ in $\{0,\ldots,n-1\}$. For $n=0$ or $n=1$, the input index set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P33.boundaryRight

For every natural number $n$ and every integer index $k$ satisfying $0\le k<n-1$, where subtraction in $\mathbb N$ is truncated at $0$, this returns the index $k+1$ in $\{0,\ldots,n-1\}$. For $n=0$ or $n=1$, the input index set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P33.boundaryTotal

For every natural number $n$, occupation function $p:\{0,\ldots,n-1\}\times\{\mathrm{false},\mathrm{true}\}\to\mathbb N$, and index $k$ with $0\le k<n-1$, this defines the natural number

$$\operatorname{boundaryTotal}(p,k) =p(k,\mathrm{true})+p(k+1,\mathrm{false}).$$

Here $n-1$ uses truncated natural-number subtraction; for $n=0$ or $n=1$, there is no admissible $k$.

QAlgorithms.Papers.Independent.Intakethird100.P33.clickJoint

For any type $O$, any function $K:\mathbb N^2\times O\to\mathbb R$, arbitrary real $\mu,t$, $o\in O$, and a Boolean $e$, define $c(a,r)=e^{-|a|^2/2}a^r/\sqrt{r!}$ and let $D_{x,b}$ be the square-summable vector with coefficients $c(\sqrt{x},r)c(0,s)$ for $b=\mathrm{false}$ and $c(0,r)c(\sqrt{x},s)$ for $b=\mathrm{true}$, with the construction returning zero if the specified coefficients are not square summable. Set $t_+=\max(t,0)$ and $u_+=\max(1-t,0)$. For $p=(p_1,p_2)$ and $h=(h_1,h_2)$ in $\mathbb N^2$, define

$$v_b(p,h)=D_{\mu,b}(p_1+h_1,p_2+h_2) \prod_{j=1}^{2} \sqrt{\frac{(p_j+h_j)!}{p_j!\,h_j!}}\, (\sqrt{t_+})^{p_j}(\sqrt{u_+})^{h_j},$$

and let $V_b$ have these coefficients if they are square summable, and be zero otherwise. Define $A_{\mathrm{false}}=\{(r,0):r>0\}$ and $A_{\mathrm{true}}=\{(0,s):s>0\}$. Then the returned real number is

$$\operatorname{clickJoint}(K,\mu,t,o,e)= \frac{\displaystyle\sum_{b\in\{\mathrm{false},\mathrm{true}\}} \sum_{p,h\in\mathbb N^2} \mathbf1_{h\in A_e}|V_b(p,h)|^2K(p,o)} {\displaystyle 2\sum_{h\in A_{\mathrm{false}}} |D_{\mu(1-t),\mathrm{false}}(h)|^2}.$$

All real amplitudes in these formulas are embedded in $\mathbb C$, and negative arguments of real square roots give $0$. There are no probability or summability assumptions on $K$ and no restrictions on $\mu,t$. Each nonsummable real series is assigned value $0$; if the denominator is $0$, the quotient is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P33.coherenceTerm

For any type $J$ with decidable equality, complex matrix $\rho$ indexed by functions $J\to\mathbb N$, indices $i,j\in J$, and occupation configuration $p:J\to\mathbb N$, this defines the complex number

$$C_{\rho,i,j}(p)= \begin{cases} p(i)\rho_{p,p},&i=j,\\ 0,&i\ne j\ \text{and }p(j)=0,\\ \sqrt{(p(i)+1)p(j)}\,\rho_{p,p^{i+,j-}}, &i\ne j\ \text{and }p(j)>0, \end{cases}$$

where $p^{i+,j-}$ agrees with $p$ away from $i,j$, has value $p(i)+1$ at $i$, and value $p(j)-1$ at $j$. The square-root factor is regarded as complex. The first branch takes precedence when $i=j$; the subtraction in the last branch occurs only when $p(j)>0$. No positivity, normalization, or summability condition on $\rho$, and no finiteness or countability condition on $J$, is required.

QAlgorithms.Papers.Independent.Intakethird100.P33.countingOutcome

For every pair $(m,n)\in\mathbb N\times\mathbb N$, this defines an outcome in $\{0,1,2\}$: it is $0$ when $m>0$ and $n=0$, it is $1$ when $m=0$ and $n>0$, and it is $2$ otherwise. Thus both $(0,0)$ and every pair with both entries positive give outcome $2$.

QAlgorithms.Papers.Independent.Intakethird100.P33.crossCounting

For every natural number $n$, word $w:\{0,\ldots,n-1\}\to\{0,1,2\}$, and real number $x$, this defines a complex matrix indexed by occupation configurations $p,q:\{0,\ldots,n-1\}\times\{\mathrm{false},\mathrm{true}\}\to\mathbb N$. Define $s_w(i,b)$ to be $1$ for $(w_i,b)=(0,\mathrm{false})$ or $(1,\mathrm{true})$, $0$ for $(w_i,b)=(0,\mathrm{true})$ or $(1,\mathrm{false})$, and $1$ for either value of $b$ when $w_i=2$. Put $a_{i,b}=\sqrt{x}$ if $s_w(i,b)=1$, and $a_{i,b}=0$ otherwise, regarding these amplitudes as complex numbers, and define $c_a(r)=e^{-|a|^2/2}a^r/\sqrt{r!}$. The matrix entry is

$$\begin{cases} \displaystyle \left(\prod_{i,b}c_{a_{i,b}}(p(i,b))\right) \overline{\left(\prod_{i,b}c_{a_{i,b}}(q(i,b))\right)}, & \displaystyle p(k,\mathrm{true})+p(k+1,\mathrm{false}) =q(k,\mathrm{true})+q(k+1,\mathrm{false}) \quad\text{for every }0\le k<n-1,\\[4pt] 0,&\text{otherwise}. \end{cases}$$

Here $n-1$ is natural-number subtraction, truncated at zero, so for $n=0$ or $n=1$ the boundary condition is vacuous. There is no restriction on $x$; the real square root used here is zero when $x<0$. For $n=0$, both products are empty and the sole matrix entry is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P33.dataState

For every real number $x$ and Boolean $\mathrm{bit}$, this defines a vector in $\ell^2(\mathbb N\times\mathbb N;\mathbb C)$. Put $s=\sqrt{x}$, using the real square root, which is $0$ when $x\leq0$, and define $c_\alpha(r)=e^{-|\alpha|^2/2}\alpha^r/\sqrt{r!}$ for $\alpha\in\mathbb C$ and $r\in\mathbb N$, with $\alpha^0=1$ even when $\alpha=0$. If $\mathrm{bit}$ is false, the vector has coefficients $c_s(m)c_0(n)$; if $\mathrm{bit}$ is true, it has coefficients $c_0(m)c_s(n)$. In either case, the coefficient function is used as an $\ell^2$ vector if it is square summable, and the zero vector is used otherwise. There is no assumption that $x$ is positive; for $x\leq0$, both Boolean inputs give the vector with coefficient $1$ at $(0,0)$ and $0$ elsewhere.

QAlgorithms.Papers.Independent.Intakethird100.P33.discriminationSuccess

For every real $x$ and every three-outcome family $M=(E_0,E_1,E_2)$ of complex kernels on $\mathbb N\times\mathbb N$, each positive on finite quadratic forms and summing coefficientwise to the identity, this defines the real number

$$\frac12\left[ \operatorname{Re}\!\left(\sum_{p\in\mathbb N^2}\sum_{q\in\mathbb N^2}d_0(p)\overline{d_0(q)}E_0(q,p)\right) + \operatorname{Re}\!\left(\sum_{p\in\mathbb N^2}\sum_{q\in\mathbb N^2}d_1(p)\overline{d_1(q)}E_1(q,p)\right) \right].$$

Here $s=\sqrt{x}$ is the real square root, $c_\alpha(r)=e^{-|\alpha|^2/2}\alpha^r/\sqrt{r!}$, and $d_0(m,n)=c_s(m)c_0(n)$ and $d_1(m,n)=c_0(m)c_s(n)$, with either coefficient function replaced by zero if it is not square summable. No unambiguity hypothesis is required. For $x\leq0$, both vectors are the vacuum vector supported at $(0,0)$.

QAlgorithms.Papers.Independent.Intakethird100.P33.eveClickProbability

For arbitrary real numbers $\mu,t$, put $x=\mu(1-t)$ and define

$$c(a,r)=\frac{e^{-|a|^2/2}a^r}{\sqrt{r!}},\qquad d_x(r,s)=c(\sqrt{x},r)c(0,s),$$

with real square roots embedded in $\mathbb C$. Let $D_x$ be the vector with coefficients $d_x$ if those coefficients are square summable, and the zero vector otherwise. This is the literal construction of the state used here. The definition returns

$$\operatorname{eveClickProbability}(\mu,t) =\sum_{\substack{(r,s)\in\mathbb N^2\\r>0,\ s=0}}|D_x(r,s)|^2.$$

There are no sign or interval assumptions on $\mu,t$; the real square root is $0$ for negative arguments, so $x\le0$ gives the vacuum coefficients and the displayed value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P33.eveReadout

For every pair $(r,s)\in\mathbb N^2$, this defines an optional Boolean value by

$$\operatorname{eveReadout}(r,s)= \begin{cases} \operatorname{some}(\mathrm{false}),&r>0\text{ and }s=0,\\ \operatorname{some}(\mathrm{true}),&r=0\text{ and }s>0,\\ \operatorname{none},&\text{otherwise}. \end{cases}$$

Thus both $(0,0)$ and every pair with both coordinates positive return no value.

QAlgorithms.Papers.Independent.Intakethird100.P33.interceptResend

For every natural number $n$, word $w:\{0,\ldots,n-1\}\to\{0,1,2\}$, and real $x$, this defines a complex matrix indexed by occupation functions $p,q:\{0,\ldots,n-1\}\times\{\mathrm{false},\mathrm{true}\}\to\mathbb N$. Declare $(i,\mathrm{false})$ occupied precisely when $w_i\in\{0,2\}$, and $(i,\mathrm{true})$ occupied precisely when $w_i\in\{1,2\}$. For each mode $j$, let $z_j=x$ when occupied and $z_j=0$ otherwise, and put

$$r_j(m)= \begin{cases} e^{-z_j},&m=0,\\ 1-e^{-z_j},&m=1,\\ 0,&m\ge2. \end{cases}$$

The matrix entry is

$$\operatorname{interceptResend}(w,x)_{p,q} =\begin{cases} \displaystyle\prod_j r_j(p(j)),&p=q,\\ 0,&p\ne q, \end{cases}$$

with the real product embedded in $\mathbb C$. No sign assumption is imposed on $x$, so the factors need not be nonnegative. When $n=0$, the unique entry is the empty product $1$.

QAlgorithms.Papers.Independent.Intakethird100.P33.monitorFlux

For any type $J$ with decidable equality, complex matrix $\rho$ indexed by configurations $p:J\to\mathbb N$, indices $i,j\in J$, and port $r\in\{0,1\}$, this defines

$$\frac{A_i+A_j+\varepsilon_r\,2\operatorname{Re}C}{4}, \qquad A_k=\sum_p p(k)\operatorname{Re}\rho_{p,p}, \qquad \varepsilon_0=1,\quad\varepsilon_1=-1,$$

where

$$C=\sum_p \begin{cases} p(i)\rho_{p,p},&i=j,\\ 0,&i\ne j,\ p(j)=0,\\ \sqrt{(p(i)+1)p(j)}\,\rho_{p,p^{i+,j-}},&i\ne j,\ p(j)>0, \end{cases}$$

and $p^{i+,j-}$ changes only the occupations at $i,j$ to $p(i)+1,p(j)-1$. All sums range over every function $J\to\mathbb N$. No summability, positivity, normalization, or distinctness of $i,j$ is assumed. Each sum is a totalized infinite sum: if its family is not summable, that sum has value zero.

QAlgorithms.Papers.Independent.Intakethird100.P33.monitoringOutput

For arbitrary complex numbers $a,b$, put $\alpha=i(a+b)/2$ and $\beta=(-a+b)/2$. This defines the vector in $\ell^2(\mathbb N\times\mathbb N;\mathbb C)$ whose coefficient at $(m,n)$ is

$$e^{-(|\alpha|^2+|\beta|^2)/2}\frac{\alpha^m\beta^n}{\sqrt{m!}\sqrt{n!}},$$

provided this coefficient function is square summable, and defines the zero vector otherwise. Powers with exponent $0$ equal $1$, including when their base is $0$. There are no restrictions on $a$ or $b$.

QAlgorithms.Papers.Independent.Intakethird100.P33.numberTerm

For any type $J$, any complex matrix $\rho$ indexed by functions $J\to\mathbb N$, any $i\in J$, and any occupation configuration $p:J\to\mathbb N$, this defines the real number

$$p(i)\,\operatorname{Re}\rho_{p,p}.$$

No finiteness or countability assumption on $J$, or positivity, normalization, or summability assumption on $\rho$, is imposed; in particular, the value need not be nonnegative.

QAlgorithms.Papers.Independent.Intakethird100.P33.opticalClick

For every vector $v\in\ell^2(\mathbb N\times\mathbb N;\mathbb C)$ and every port $j\in\{0,1\}$, this defines the real number

$$\sum_{(m,n)\in\mathbb N^2} \begin{cases} |v(m,n)|^2,&j=0\text{ and }m>0,\text{ or }j=1\text{ and }n>0,\\ 0,&\text{otherwise}. \end{cases}$$

No unit-norm assumption is imposed on $v$; the selected occupation must be strictly positive, so occupation $0$ contributes nothing.

QAlgorithms.Papers.Independent.Intakethird100.P33.opticalVisibility

For any type $J$ with decidable equality, complex matrix $\rho$ indexed by configurations $p:J\to\mathbb N$, and indices $i,j\in J$, define

$$A=\sum_p p(i)\operatorname{Re}\rho_{p,p} +\sum_p p(j)\operatorname{Re}\rho_{p,p}, \qquad C=\sum_p \begin{cases} p(i)\rho_{p,p},&i=j,\\ 0,&i\ne j,\ p(j)=0,\\ \sqrt{(p(i)+1)p(j)}\,\rho_{p,p^{i+,j-}},&i\ne j,\ p(j)>0, \end{cases}$$

where $p^{i+,j-}$ changes the occupations at $i,j$ to $p(i)+1,p(j)-1$. This definition is the ratio

$$\frac{F_0-F_1}{F_0+F_1}, \qquad F_0=\frac{A+2\operatorname{Re}C}{4}, \qquad F_1=\frac{A-2\operatorname{Re}C}{4}.$$

No positivity, normalization, summability, or distinctness assumption is imposed. A nonsummable family has totalized sum zero, and division by zero has value zero; consequently the defined visibility is zero whenever $F_0+F_1=0$.

QAlgorithms.Papers.Independent.Intakethird100.P33.printedCount

For arbitrary real numbers $x,p_d,f$, pattern $a\in\{0,1,2,3\}$, and port $r\in\{0,1\}$, this defines

$$\frac{\bigl(S+(1-S)p_d\bigr)(1-f)}{4}, \qquad S= \begin{cases} x,&a=2,\ r=0,\\ 0,&a=2,\ r=1,\ \text{or }a=1,\\ x/4,&a=0\ \text{or }a=3. \end{cases}$$

None of $x,p_d,f$ is assumed nonnegative or bounded by one.

QAlgorithms.Papers.Independent.Intakethird100.P33.printedSignal

For every real number $x$, pattern $a\in\{0,1,2,3\}$, and port $r\in\{0,1\}$, this defines the real number

$$S(x,a,r)= \begin{cases} x,&a=2,\ r=0,\\ 0,&a=2,\ r=1,\\ 0,&a=1,\\ x/4,&a=0\ \text{or }a=3. \end{cases}$$

There is no sign or size restriction on $x$.

QAlgorithms.Papers.Independent.Intakethird100.P33.ratePolynomial

For arbitrary real numbers $t,t_B,\eta,f,\mu$, this defines the real polynomial

$$\operatorname{ratePolynomial}(t,t_B,\eta,f,\mu) =\mu\,t\,t_B\,\eta\,(1-f)\bigl(1-\mu(1-t)\bigr).$$

No bounds or sign conditions are imposed on any argument.

QAlgorithms.Papers.Independent.Intakethird100.P33.resendWeight

For every real $x$, Boolean $b$, and natural number $m$, set $z=x$ if $b=\mathrm{true}$ and $z=0$ otherwise. This defines

$$\operatorname{resendWeight}(x,b,m)= \begin{cases} e^{-z},&m=0,\\ 1-e^{-z},&m=1,\\ 0,&m\ge2. \end{cases}$$

For $b=\mathrm{false}$ this is $1$ at $m=0$ and $0$ elsewhere. No nonnegativity assumption is imposed on $x$; when $b=\mathrm{true}$ and $x<0$, the value at $m=1$ is negative.

QAlgorithms.Papers.Independent.Intakethird100.P33.sourceOccupation

For every natural number $n$, every word $w:\{0,\ldots,n-1\}\to\{0,1,2\}$, and every pair $(r,b)$ consisting of an index $r<n$ and a Boolean $b$, this defines a natural number as follows: if $w(r)=0$, it is $1$ for $b=\mathrm{false}$ and $0$ for $b=\mathrm{true}$; if $w(r)=1$, it is $0$ for $b=\mathrm{false}$ and $1$ for $b=\mathrm{true}$; if $w(r)=2$, it is $1$ for either Boolean. When $n=0$, there is no possible argument $(r,b)$.

QAlgorithms.Papers.Independent.Intakethird100.P33.splitBobDensity

For every real $t$ and every $v\in\ell^2(\mathbb N\times\mathbb N;\mathbb C)$, put

$$C_{t,v}(p,e)=v(p_1+e_1,p_2+e_2) \prod_{r=1}^{2} \sqrt{\frac{(p_r+e_r)!}{p_r!\,e_r!}}\, \bigl(\sqrt{\max(t,0)}\bigr)^{p_r} \bigl(\sqrt{\max(1-t,0)}\bigr)^{e_r},$$

and let $S_{t,v}=C_{t,v}$ if this function is square summable on $\mathbb N^2\times\mathbb N^2$, and $S_{t,v}=0$ otherwise. This defines the complex kernel on $\mathbb N^2$ by $\rho(p,q)=\sum_{e\in\mathbb N^2}S_{t,v}(p,e)\overline{S_{t,v}(q,e)}$ for all $p,q\in\mathbb N^2$. No normalization of $v$ or restriction on $t$ is imposed; if the square-summability test fails, the entire kernel is zero.

QAlgorithms.Papers.Independent.Intakethird100.P33.splitCoefficients

For every real $t$, every $v\in\ell^2(\mathbb N\times\mathbb N;\mathbb C)$, and every pair of occupation pairs $((p_1,p_2),(e_1,e_2))$, this defines the complex number

$$v(p_1+e_1,p_2+e_2) \prod_{r=1}^{2} \left[ \sqrt{\frac{(p_r+e_r)!}{p_r!\,e_r!}}\, \bigl(\sqrt{\max(t,0)}\bigr)^{p_r} \bigl(\sqrt{\max(1-t,0)}\bigr)^{e_r} \right].$$

The real factors are embedded in $\mathbb C$. There is no assumption that $0\leq t\leq1$ or that $v$ has norm $1$; the maxima arise from converting $t$ and $1-t$ to nonnegative reals. Zero powers, including $0^0$, equal $1$.

QAlgorithms.Papers.Independent.Intakethird100.P33.splitState

For every real $t$ and every $v\in\ell^2(\mathbb N\times\mathbb N;\mathbb C)$, define, for $p,e\in\mathbb N^2$,

$$C_{t,v}(p,e)=v(p_1+e_1,p_2+e_2) \prod_{r=1}^{2} \sqrt{\frac{(p_r+e_r)!}{p_r!\,e_r!}}\, \bigl(\sqrt{\max(t,0)}\bigr)^{p_r} \bigl(\sqrt{\max(1-t,0)}\bigr)^{e_r}.$$

This declaration returns the vector in $\ell^2(\mathbb N^2\times\mathbb N^2;\mathbb C)$ with coefficient function $C_{t,v}$ if $\sum_{p,e}|C_{t,v}(p,e)|^2<\infty$, and returns the zero vector otherwise. No restriction on $t$ or normalization of $v$ is assumed, and $0^0$ is interpreted as $1$.

QAlgorithms.Papers.Independent.Intakethird100.P33.substitutedWord

For every natural number $n$ and every word $w:\{0,\ldots,n-1\}\to\{0,1,2\}$, this defines a vector in $\ell^2(\{0,\ldots,n-1\}\times\mathrm{Bool}\to\mathbb N;\mathbb C)$ with coefficient $1$ at the single occupation function $s_w$ and coefficient $0$ at every other occupation function. The function $s_w$ assigns the pair of occupations $(1,0)$ to the Boolean positions $(\mathrm{false},\mathrm{true})$ when $w(r)=0$, assigns $(0,1)$ when $w(r)=1$, and assigns $(1,1)$ when $w(r)=2$. This includes $n=0$, when there is exactly one occupation function, the empty function, and the vector has coefficient $1$ there.

QAlgorithms.Papers.Independent.Intakethird100.P33.trainDensity

For every natural number $n$, word $w:\{0,\ldots,n-1\}\to\{0,1,2\}$, and real $x$, this defines a complex matrix indexed by occupation functions $p,q:\{0,\ldots,n-1\}\times\{\mathrm{false},\mathrm{true}\}\to\mathbb N$. Put

$$a_{i,\mathrm{false}}= \begin{cases}\sqrt{x},&w_i=0\text{ or }2,\\0,&w_i=1,\end{cases} \qquad a_{i,\mathrm{true}}= \begin{cases}\sqrt{x},&w_i=1\text{ or }2,\\0,&w_i=0,\end{cases} \qquad c(a,r)=\frac{e^{-|a|^2/2}a^r}{\sqrt{r!}},$$

where real square roots are embedded in $\mathbb C$. Its entries are

$$\operatorname{trainDensity}(w,x)_{p,q} =\left(\prod_{i,b}c(a_{i,b},p(i,b))\right) \overline{\left(\prod_{i,b}c(a_{i,b},q(i,b))\right)}.$$

There is no assumption that $x\ge0$; for $x<0$, $\sqrt{x}=0$. When $n=0$, both products are empty and equal $1$, so the unique matrix entry is $1$.