Definitions for Quantum polar codes for arbitrary channels

QAlgorithms.Papers.Independent.Intakethird100.P23.amplitudeBlock

For arbitrary natural numbers $b,e,n$, an arbitrary complex matrix $U$ with rows $(r,q)\in\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and Boolean columns, and a Boolean string $u$ of length $N=2^n$, this defines a complex matrix indexed by sequences $s,s'\in\{0,\ldots,b-1\}^N$:

$$A_{s,s'}= \sum_{t\in\{0,\ldots,e-1\}^N} \left(\prod_{k=0}^{N-1}U_{(s_k,t_k),\,c_k(u)}\right) \overline{\left(\prod_{k=0}^{N-1}U_{(s'_k,t_k),\,c_k(u)}\right)}.$$

The Boolean $c_k(u)$ is the parity of the number of true coordinates $u_\ell$ satisfying $(\lfloor k/2^r\rfloor\bmod2)\le(\lfloor\ell/2^{n-1-r}\rfloor\bmod2)$ for every $0\le r<n$. This formula forms the vector’s outer product with its complex conjugate and sums over the second sequence coordinate. There is no normalization hypothesis on $U$. When $n=0$, $N=1$ and $c_0(u)=u_0$; when $e=0$, the sum is empty and the matrix is zero, and when $b=0$, the matrix has an empty index set.

QAlgorithms.Papers.Independent.Intakethird100.P23.amplitudeInformation

For any natural numbers $b,e$ and any complex matrix $U$ with row index $(r,h)\in\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and Boolean column index $a$, define the two $b\times b$ matrices

$$R_a(r,s)=\sum_{h=0}^{e-1}U_{(r,h),a}\overline{U_{(s,h),a}} \qquad(a=0,1).$$

Writing

$$H(M)=-\operatorname{Re}\operatorname{tr}\bigl(\operatorname{cfc}(x\mapsto x\log_2x)(M)\bigr),$$

where $\operatorname{cfc}$ denotes continuous functional calculus, this declaration defines

$$H\!\left(\frac{R_0+R_1}{2}\right) -\frac{H(R_0)+H(R_1)}{2}.$$

The function $x\mapsto x\log_2x$ takes value zero at $x=0$ under the total real-logarithm convention. No normalization or isometry condition is imposed on $U$, so the matrices $R_a$ are not required to have trace one. Both $b=0$ and $e=0$ are included, and either case gives value zero.

QAlgorithms.Papers.Independent.Intakethird100.P23.amplitudeSplit

For any natural numbers $b,e,n$, any complex matrix $U$ with rows indexed by $\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and columns indexed by Boolean values, and any $i\in\{0,\ldots,2^n-1\}$, this defines a Boolean-indexed family of matrices as follows. Put $N=2^n$, identify Boolean values with $0,1$, and write $Q=\{0,1\}^N$, $\mathcal B=\{0,\ldots,b-1\}^N$, and $\mathcal E=\{0,\ldots,e-1\}^N$. Define the encoding $E_n:Q\to Q$ by

$$E_n(u)_j= \left(\sum_{\substack{0\le h<N\\ \forall\,0\le k<n,\; (\lfloor j/2^k\rfloor\bmod2)\le (\lfloor h/2^{n-1-k}\rfloor\bmod2)}}u_h\right)\bmod2,$$

and put $v_u(r,h)=\prod_{k=0}^{N-1}U_{(r_k,h_k),E_n(u)_k}$ for $r\in\mathcal B$, $h\in\mathcal E$. Let $p_i(u)\in Q$ have coordinate $u_j$ when $j<i$, and $0$ otherwise. For each Boolean input $a$, the defined matrix $A_i^a$, indexed by $Q\times\mathcal B$, has entries

$$A_i^a((p,r),(q,s))= \begin{cases} 2^{-(N-1)} \displaystyle\sum_{\substack{u\in Q\\u_i=a,\;p_i(u)=p}} \sum_{h\in\mathcal E}v_u(r,h)\overline{v_u(s,h)},&p=q,\\ 0,&p\ne q. \end{cases}$$

There is no normalization or isometry hypothesis on $U$. The parameters $b,e,n$ may be zero; when $n=0$, $N=1$, the encoding is the identity, the prefix is always zero, and the factor $2^{-(N-1)}$ is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P23.amplitudeVector

For arbitrary natural numbers $b,e,n$, an arbitrary complex matrix $U$ with rows indexed by $\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and columns indexed by Booleans, and a Boolean string $u$ of length $N=2^n$, this defines a vector in the complex Euclidean space indexed by pairs of sequences $s\in\{0,\ldots,b-1\}^N$ and $t\in\{0,\ldots,e-1\}^N$, with coordinates

$$v(s,t)=\prod_{k=0}^{N-1}U_{(s_k,t_k),\,c_k(u)}.$$

Here $c_k(u)$ is true exactly when the number of indices $\ell$ for which $u_\ell$ is true and

$$\forall\,0\le r<n,\qquad (\lfloor k/2^r\rfloor\bmod2) \le(\lfloor\ell/2^{n-1-r}\rfloor\bmod2)$$

is odd. No isometry or normalization hypothesis is imposed on $U$. The case $n=0$ has $N=1$ and $c_0(u)=u_0$; if $b=0$ or $e=0$, the vector’s index set is empty and the vector is zero.

QAlgorithms.Papers.Independent.Intakethird100.P23.binaryCharacter

For every natural number $N$, including $N=0$, and Boolean strings $x,z$ of length $N$, this defines the complex number

$$(-1)^{\,|\{i:0\le i<N,\ x_i=\mathrm{true}\ \text{and}\ z_i=\mathrm{true}\}|}.$$

It is $1$ when the number of coordinates where both strings are true is even and $-1$ when that number is odd. For $N=0$, the exponent is the empty sum $0$, so the value is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P23.channelRootFidelity

For every finite type $O$ equipped with decidable equality and every assignment $W$ of a complex $O\times O$ matrix to each Boolean, this defines

$$\operatorname{Re}\operatorname{tr}\!\left( \operatorname{cfc}[\sqrt{\phantom t}]\!\left( \operatorname{cfc}[\sqrt{\phantom t}](W(\mathrm{false})) \,W(\mathrm{true})\, \operatorname{cfc}[\sqrt{\phantom t}](W(\mathrm{false})) \right)\right).$$

Each occurrence of $\operatorname{cfc}[\sqrt{\phantom t}]$ denotes the real continuous functional calculus for the total real square-root function, which is zero on negative inputs. No positivity or trace-one assumption is imposed on either matrix. The type $O$ may be empty, in which case the trace and the defined value are zero.

QAlgorithms.Papers.Independent.Intakethird100.P23.consumedSet

For arbitrary natural numbers $b,e,n$, any complex matrix $U$ with rows indexed by $\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and Boolean columns, and any real $\beta$, put $N=2^n$, $Q=\{0,1\}^N$, $\mathcal B=\{0,\ldots,b-1\}^N$, and $\mathcal E=\{0,\ldots,e-1\}^N$. Define $E_n(u)_j$ as the parity of the sum of $u_h$ over $h<N$ satisfying

$$\forall\,0\le k<n,\qquad (\lfloor j/2^k\rfloor\bmod2)\le (\lfloor h/2^{n-1-k}\rfloor\bmod2),$$

and set

$$v_u(r,h)=\prod_{k=0}^{N-1}U_{(r_k,h_k),E_n(u)_k}, \qquad w_x(r,z,h)=2^{-N/2}(-1)^{\sum_{k=0}^{N-1}x_kz_k}v_z(r,h).$$

Let $p_i(u)_j=u_j$ for $j<i$ and zero otherwise, and let $s_i(x)_j=x_j$ for $i<j$ and zero otherwise. For each Boolean $a$, form matrices

$$\begin{aligned} A_i^a((p,r),(q,s)) &=\mathbf1_{p=q}\,2^{-(N-1)} \sum_{\substack{u\in Q\\u_i=a,\;p_i(u)=p}} \sum_{h\in\mathcal E}v_u(r,h)\overline{v_u(s,h)},\\ P_i^a((p,(r,z)),(q,(s,z'))) &=\mathbf1_{p=q}\,2^{-(N-1)} \sum_{\substack{x\in Q\\x_i=a,\;s_i(x)=p}} \sum_{h\in\mathcal E}w_x(r,z,h)\overline{w_x(s,z',h)}. \end{aligned}$$

With $f(\rho,\sigma)=\operatorname{Re}\operatorname{tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}$, using continuous functional calculus, this defines

$$\left\{i<N: 2^{-N^\beta}\le f(A_i^0,A_i^1) \ \text{and}\ 2^{-N^\beta}\le f(P_i^0,P_i^1) \right\}.$$

Thus equality with the threshold is included in each condition. There is no normalization or isometry hypothesis on $U$, and $\beta$ is any real number. Zero values of $b,e,n$ are allowed; when $n=0$, $N=1$, the encoding is the identity, the prefix and suffix strings are zero, and the threshold is $1/2$.

QAlgorithms.Papers.Independent.Intakethird100.P23.environmentBlock

For arbitrary natural numbers $b,e,n$, an arbitrary complex matrix $U$ with rows $(r,q)\in\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and Boolean columns, and a Boolean string $u$ of length $N=2^n$, this defines a complex matrix indexed by sequences $t,t'\in\{0,\ldots,e-1\}^N$:

$$E_{t,t'}= \sum_{s\in\{0,\ldots,b-1\}^N} \left(\prod_{k=0}^{N-1}U_{(s_k,t_k),\,c_k(u)}\right) \overline{\left(\prod_{k=0}^{N-1}U_{(s_k,t'_k),\,c_k(u)}\right)}.$$

The Boolean $c_k(u)$ is the parity of the number of true coordinates $u_\ell$ satisfying $(\lfloor k/2^r\rfloor\bmod2)\le(\lfloor\ell/2^{n-1-r}\rfloor\bmod2)$ for every $0\le r<n$. This formula forms the vector’s outer product with its complex conjugate and sums over the first sequence coordinate. There is no normalization hypothesis on $U$. When $n=0$, $N=1$ and $c_0(u)=u_0$; when $b=0$, the sum is empty and the matrix is zero, and when $e=0$, the matrix has an empty index set.

QAlgorithms.Papers.Independent.Intakethird100.P23.environmentSplit

For any natural numbers $b,e,n$, any complex matrix $U$ with row index $(r,h)\in\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and Boolean column index, and any $i<2^n$, this defines a Boolean-indexed family of matrices. Put $N=2^n$, $Q=\{0,1\}^N$, $\mathcal B=\{0,\ldots,b-1\}^N$, and $\mathcal E=\{0,\ldots,e-1\}^N$. Interpret Boolean values as $0,1$, and define

$$E_n(u)_j= \left(\sum_{\substack{0\le h<N\\ \forall\,0\le k<n,\; (\lfloor j/2^k\rfloor\bmod2)\le (\lfloor h/2^{n-1-k}\rfloor\bmod2)}}u_h\right)\bmod2, \qquad v_u(r,h)=\prod_{k=0}^{N-1}U_{(r_k,h_k),E_n(u)_k}.$$

Let $p_i(u)_j=u_j$ when $j<i$, and $0$ otherwise. For each Boolean $a$, the defined matrix $F_i^a$, indexed by $Q\times\mathcal E$, has entries

$$F_i^a((p,h),(q,h'))= \begin{cases} 2^{-(N-1)} \displaystyle\sum_{\substack{u\in Q\\u_i=a,\;p_i(u)=p}} \sum_{r\in\mathcal B}v_u(r,h)\overline{v_u(r,h')},&p=q,\\ 0,&p\ne q. \end{cases}$$

The matrix $U$ is arbitrary, with no normalization or isometry assumption. Zero values of $b,e,n$ are included; for $n=0$, $N=1$, the encoding is the identity, the prefix is zero, and the splitting factor is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P23.goodAmplitude

For arbitrary natural numbers $b,e,n$, any complex matrix $U$ indexed by $(\{0,\ldots,b-1\}\times\{0,\ldots,e-1\})\times\{0,1\}$, and any real $\beta$, this defines the following set of indices. Put $N=2^n$, $Q=\{0,1\}^N$, $\mathcal B=\{0,\ldots,b-1\}^N$, and $\mathcal E=\{0,\ldots,e-1\}^N$. Define $E_n(u)_j$ as the parity of the sum of $u_h$ over those $h<N$ satisfying

$$\forall\,0\le k<n,\qquad (\lfloor j/2^k\rfloor\bmod2)\le (\lfloor h/2^{n-1-k}\rfloor\bmod2),$$

and set $v_u(r,h)=\prod_{k=0}^{N-1}U_{(r_k,h_k),E_n(u)_k}$. For $i<N$, let $p_i(u)_j=u_j$ if $j<i$, and $0$ otherwise, and define matrices on $Q\times\mathcal B$ by

$$A_i^a((p,r),(q,s))= \begin{cases} 2^{-(N-1)} \displaystyle\sum_{\substack{u\in Q\\u_i=a,\;p_i(u)=p}} \sum_{h\in\mathcal E}v_u(r,h)\overline{v_u(s,h)},&p=q,\\ 0,&p\ne q. \end{cases}$$

The resulting finite set is

$$\left\{i<N: \operatorname{Re}\operatorname{tr} \sqrt{\sqrt{A_i^0}\,A_i^1\,\sqrt{A_i^0}} <2^{-N^\beta}\right\},$$

where the matrix square roots are given by continuous functional calculus. No isometry or normalization is assumed for $U$, and $\beta$ is unrestricted. The parameters $b,e,n$ may be zero; when $n=0$, $N=1$, the encoding is the identity and the threshold is $1/2$.

QAlgorithms.Papers.Independent.Intakethird100.P23.goodEnvironment

For arbitrary natural numbers $b,e,n$, any complex matrix $U$ indexed by $(\{0,\ldots,b-1\}\times\{0,\ldots,e-1\})\times\{0,1\}$, and any real $\beta$, put $N=2^n$, $Q=\{0,1\}^N$, $\mathcal B=\{0,\ldots,b-1\}^N$, and $\mathcal E=\{0,\ldots,e-1\}^N$. Define $E_n(u)_j$ to be the parity of the sum of $u_h$ over indices $h<N$ satisfying

$$\forall\,0\le k<n,\qquad (\lfloor j/2^k\rfloor\bmod2)\le (\lfloor h/2^{n-1-k}\rfloor\bmod2),$$

and write $v_u(r,h)=\prod_{k=0}^{N-1}U_{(r_k,h_k),E_n(u)_k}$. Let $p_i(u)_j=u_j$ for $j<i$, and $0$ otherwise. Define matrices indexed by $Q\times\mathcal E$ by

$$F_i^a((p,h),(q,h'))= \begin{cases} 2^{-(N-1)} \displaystyle\sum_{\substack{u\in Q\\u_i=a,\;p_i(u)=p}} \sum_{r\in\mathcal B}v_u(r,h)\overline{v_u(r,h')},&p=q,\\ 0,&p\ne q. \end{cases}$$

The defined finite set is

$$\left\{i<N: \operatorname{Re}\operatorname{tr} \sqrt{\sqrt{F_i^0}\,F_i^1\,\sqrt{F_i^0}} <2^{-N^\beta}\right\},$$

where matrix square roots use continuous functional calculus. The matrix $U$ need not be an isometry or normalized, and $\beta$ is any real number. The values $b,e,n=0$ are included; when $n=0$, $N=1$, the encoding is the identity, the prefix is zero, and the threshold is $1/2$.

QAlgorithms.Papers.Independent.Intakethird100.P23.goodPhase

For any natural numbers $b,e,n$, any complex matrix $U$ with rows indexed by $\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and Boolean columns, and any real $\beta$, put $N=2^n$, $Q=\{0,1\}^N$, $\mathcal B=\{0,\ldots,b-1\}^N$, and $\mathcal E=\{0,\ldots,e-1\}^N$. With Boolean values identified with $0,1$, let $E_n(z)_j$ be the parity of the sum of $z_h$ over $h<N$ satisfying

$$\forall\,0\le k<n,\qquad (\lfloor j/2^k\rfloor\bmod2)\le (\lfloor h/2^{n-1-k}\rfloor\bmod2),$$

and define

$$w_x(r,z,h)=2^{-N/2}(-1)^{\sum_{k=0}^{N-1}x_kz_k} \prod_{k=0}^{N-1}U_{(r_k,h_k),E_n(z)_k}.$$

For $i<N$, set $s_i(x)_j=x_j$ when $i<j$, and $0$ otherwise. For Boolean $a$, define matrices indexed by $Q\times(\mathcal B\times Q)$ by

$$P_i^a((p,(r,z)),(q,(s,z')))= \begin{cases} 2^{-(N-1)} \displaystyle\sum_{\substack{x\in Q\\x_i=a,\;s_i(x)=p}} \sum_{h\in\mathcal E}w_x(r,z,h)\overline{w_x(s,z',h)},&p=q,\\ 0,&p\ne q. \end{cases}$$

The declaration defines the finite set

$$\left\{i<N: \operatorname{Re}\operatorname{tr} \sqrt{\sqrt{P_i^0}\,P_i^1\,\sqrt{P_i^0}} <2^{-N^\beta}\right\},$$

using continuous functional calculus for matrix square roots. There is no normalization or isometry requirement on $U$, and no restriction on $\beta$. Zero values of $b,e,n$ are allowed; $n=0$ gives $N=1$, identity encoding, a zero suffix, and threshold $1/2$.

QAlgorithms.Papers.Independent.Intakethird100.P23.goodSet

For any natural number $N$, any finite type $O$ equipped with decidable equality, any family $W_i^a$ of complex $O\times O$ matrices indexed by $i\in\{0,\ldots,N-1\}$ and Boolean $a$, and any real number $t$, this defines the finite set

$$\left\{i<N: \operatorname{Re}\operatorname{tr} \mathcal R\!\left(\mathcal R(W_i^0)\,W_i^1\,\mathcal R(W_i^0)\right)<t \right\},$$

where $\mathcal R(M)$ means applying the real function $x\mapsto\sqrt{x}$ to $M$ using the total continuous functional calculus. The comparison is strictly $<t$, and no positivity, trace, or normalization condition is imposed on the matrices. The threshold $t$ is unrestricted, $O$ may be empty, and when $N=0$ the resulting set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P23.indexRate

For every natural number $n$ and every finite subset $S\subseteq\{0,\ldots,2^n-1\}$, this defines the real number

$$\frac{|S|}{2^n},$$

with both numerator and denominator interpreted as real numbers. The empty set is allowed and gives $0$. The denominator is always positive, including when $n=0$; in that case the ambient index set has one element, so the value is $0$ or $1$.

QAlgorithms.Papers.Independent.Intakethird100.P23.informationSet

For any natural numbers $b,e,n$, any complex matrix $U$ indexed by $(\{0,\ldots,b-1\}\times\{0,\ldots,e-1\})\times\{0,1\}$, and any real $\beta$, this defines a finite set as follows. Put $N=2^n$, $Q=\{0,1\}^N$, $\mathcal B=\{0,\ldots,b-1\}^N$, and $\mathcal E=\{0,\ldots,e-1\}^N$. Let $E_n(u)_j$ be the parity of the sum of $u_h$ over $h<N$ for which

$$\forall\,0\le k<n,\qquad (\lfloor j/2^k\rfloor\bmod2)\le (\lfloor h/2^{n-1-k}\rfloor\bmod2).$$

Define

$$v_u(r,h)=\prod_{k=0}^{N-1}U_{(r_k,h_k),E_n(u)_k}, \qquad w_x(r,z,h)=2^{-N/2}(-1)^{\sum_{k=0}^{N-1}x_kz_k}v_z(r,h).$$

Let $p_i(u)$ retain $u_j$ for $j<i$ and set all other coordinates to zero, and let $s_i(x)$ retain $x_j$ for $i<j$ and set all other coordinates to zero. For Boolean $a$, define

$$\begin{aligned} A_i^a((p,r),(q,s)) &=\mathbf1_{p=q}\,2^{-(N-1)} \sum_{\substack{u\in Q\\u_i=a,\;p_i(u)=p}} \sum_{h\in\mathcal E}v_u(r,h)\overline{v_u(s,h)},\\ P_i^a((p,(r,z)),(q,(s,z'))) &=\mathbf1_{p=q}\,2^{-(N-1)} \sum_{\substack{x\in Q\\x_i=a,\;s_i(x)=p}} \sum_{h\in\mathcal E}w_x(r,z,h)\overline{w_x(s,z',h)}. \end{aligned}$$

Writing $f(\rho,\sigma)=\operatorname{Re}\operatorname{tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}$, with square roots given by continuous functional calculus, the defined set is

$$\left\{i<N: f(A_i^0,A_i^1)<2^{-N^\beta} \ \text{and}\ f(P_i^0,P_i^1)<2^{-N^\beta} \right\}.$$

No isometry or normalization is required of $U$; $\beta$ is unrestricted, and $b,e,n$ may be zero. For $n=0$, $N=1$, the encoding is the identity, both retained-coordinate strings are zero, and the threshold is $1/2$.

QAlgorithms.Papers.Independent.Intakethird100.P23.phaseBlock

For arbitrary natural numbers $b,e,n$, an arbitrary complex matrix $U$ with rows in $\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and Boolean columns, and a Boolean string $x$ of length $N=2^n$, this defines a complex matrix indexed by pairs $(s,u)$, where $s\in\{0,\ldots,b-1\}^N$ and $u$ is a Boolean string of length $N$, as

$$P_{(s,u),(s',u')}= 2^{-N}(-1)^{h(x,u)+h(x,u')} \sum_{t\in\{0,\ldots,e-1\}^N} \left(\prod_{k=0}^{N-1}U_{(s_k,t_k),\,c_k(u)}\right) \overline{\left(\prod_{k=0}^{N-1}U_{(s'_k,t_k),\,c_k(u')}\right)}.$$

Here $h(x,u)$ counts coordinates where both strings are true, and $c_k(u)$ is the parity of the number of true coordinates $u_\ell$ satisfying $(\lfloor k/2^r\rfloor\bmod2)\le(\lfloor\ell/2^{n-1-r}\rfloor\bmod2)$ for every $0\le r<n$. Thus the definition takes the phase vector’s outer product and sums over its final sequence coordinate. No normalization hypothesis is imposed on $U$. For $n=0$, $N=1$ and $c_0(u)=u_0$; for $e=0$ the sum is empty and the matrix is zero, and for $b=0$ its index set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P23.phaseInformation

For arbitrary natural numbers $b,e$ and any complex matrix $U$ indexed by $(\{0,\ldots,b-1\}\times\{0,\ldots,e-1\})\times\{0,1\}$, define, for Boolean $a$, a matrix $P_a$ indexed by $\{0,\ldots,b-1\}\times\{0,1\}$ through

$$P_a((r,z),(s,z'))= \frac12(-1)^{az+az'} \sum_{h=0}^{e-1}U_{(r,h),z}\overline{U_{(s,h),z'}},$$

where Boolean values in the exponent are interpreted as integers $0,1$. Let

$$H(M)=-\operatorname{Re}\operatorname{tr}\bigl(\operatorname{cfc}(x\mapsto x\log_2x)(M)\bigr),$$

using continuous functional calculus and value zero for $x\log_2x$ at $x=0$. The defined real number is

$$H\!\left(\frac{P_0+P_1}{2}\right) -\frac{H(P_0)+H(P_1)}{2}.$$

There is no normalization or isometry assumption on $U$, and the matrices $P_a$ need not have trace one. The parameters $b,e$ may be zero; either zero case gives value zero.

QAlgorithms.Papers.Independent.Intakethird100.P23.phaseSplit

For arbitrary natural numbers $b,e,n$, a complex matrix $U$ indexed by $(\{0,\ldots,b-1\}\times\{0,\ldots,e-1\})\times\{0,1\}$, and $i\in\{0,\ldots,2^n-1\}$, this defines a Boolean-indexed matrix family. Put $N=2^n$, $Q=\{0,1\}^N$, $\mathcal B=\{0,\ldots,b-1\}^N$, and $\mathcal E=\{0,\ldots,e-1\}^N$, with Boolean values interpreted as integers $0,1$. For $u\in Q$, define

$$E_n(u)_j= \left(\sum_{\substack{0\le h<N\\ \forall\,0\le k<n,\; (\lfloor j/2^k\rfloor\bmod2)\le (\lfloor h/2^{n-1-k}\rfloor\bmod2)}}u_h\right)\bmod2.$$

For $x,z\in Q$, $r\in\mathcal B$, and $h\in\mathcal E$, set

$$w_x(r,z,h)=2^{-N/2}(-1)^{\sum_{k=0}^{N-1}x_kz_k} \prod_{k=0}^{N-1}U_{(r_k,h_k),E_n(z)_k}.$$

Let $s_i(x)_j=x_j$ for $i<j$, and $s_i(x)_j=0$ otherwise. For each Boolean $a$, the defined matrix $P_i^a$, indexed by $Q\times(\mathcal B\times Q)$, is

$$P_i^a((p,(r,z)),(q,(s,z')))= \begin{cases} 2^{-(N-1)} \displaystyle\sum_{\substack{x\in Q\\x_i=a,\;s_i(x)=p}} \sum_{h\in\mathcal E}w_x(r,z,h)\overline{w_x(s,z',h)},&p=q,\\ 0,&p\ne q. \end{cases}$$

No isometry or normalization condition is imposed on $U$, and $b,e,n$ may be zero. In particular, $n=0$ gives $N=1$, identity encoding, an always-zero suffix, and splitting factor $1$.

QAlgorithms.Papers.Independent.Intakethird100.P23.phaseVector

For arbitrary natural numbers $b,e,n$, an arbitrary complex matrix $U$ with rows in $\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and Boolean columns, and a Boolean string $x$ of length $N=2^n$, this defines a complex Euclidean vector indexed by triples $(s,u,t)$, where $s\in\{0,\ldots,b-1\}^N$, $u$ is a Boolean string of length $N$, and $t\in\{0,\ldots,e-1\}^N$, by

$$v_x((s,u),t)= \frac{(-1)^{\,|\{k:x_k=\mathrm{true},\ u_k=\mathrm{true}\}|}}{\sqrt{2^N}} \prod_{k=0}^{N-1}U_{(s_k,t_k),\,c_k(u)}.$$

Here $c_k(u)$ is true exactly when the number of indices $\ell$ with $u_\ell=\mathrm{true}$ and $(\lfloor k/2^r\rfloor\bmod2)\le(\lfloor\ell/2^{n-1-r}\rfloor\bmod2)$ for every $0\le r<n$ is odd. No normalization or isometry hypothesis is imposed on $U$. For $n=0$, $N=1$ and $c_0(u)=u_0$; if $b=0$ or $e=0$, the vector’s index set is empty and the vector is zero.

QAlgorithms.Papers.Independent.Intakethird100.P23.polarEncode

For every natural number $n$ and Boolean string $u$ indexed by $\{0,\ldots,2^n-1\}$, this defines another Boolean string on the same index set. Its coordinate $j$ is true exactly when the following natural-number sum is odd:

$$\sum_{\substack{0\le i<2^n\\ \forall\,0\le k<n:\; (\lfloor j/2^k\rfloor\bmod2) \le(\lfloor i/2^{n-1-k}\rfloor\bmod2)}} \mathbf1_{\{u_i=\mathrm{true}\}}.$$

Here $\mathbf1$ is $1$ when its condition holds and $0$ otherwise. In particular, $n=0$ gives a string of length one, and the output equals that input string.

QAlgorithms.Papers.Independent.Intakethird100.P23.polarMatrixEntry

For every natural number $n$ and indices $0\le i,j<2^n$, this defines a Boolean value that is true exactly when

$$\forall k\in\{0,\ldots,n-1\},\qquad \left\lfloor\frac{j}{2^k}\right\rfloor\bmod 2 \;\le\; \left\lfloor\frac{i}{2^{n-1-k}}\right\rfloor\bmod 2.$$

Thus each binary digit of $j$ is compared with the oppositely positioned binary digit of $i$. When $n=0$, the only indices are $i=j=0$, the condition has no instances, and the value is true.

QAlgorithms.Papers.Independent.Intakethird100.P23.splitChannel

For every natural number $N$, every finite type $O$ with its supplied finite enumeration, every assignment $W$ of a complex $O\times O$ matrix to each Boolean string of length $N$, every Boolean $\mathrm{reverse}$, every index $0\le i<N$, and every Boolean $a$, this defines a matrix indexed by $(v,o)$ with $v$ a Boolean string of length $N$ and $o\in O$:

$$S_{(v,o),(v',o')}= \begin{cases} 2^{-(N-1)} \displaystyle\sum_{\substack{u:\{0,\ldots,N-1\}\to\mathrm{Bool}\\ u_i=a,\ R_i(u)=v}}(W(u))_{o,o'},&v=v',\\ 0,&v\ne v'. \end{cases}$$

If $\mathrm{reverse}$ is false, $R_i(u)_j=u_j$ for $j<i$ and is false elsewhere; if $\mathrm{reverse}$ is true, $R_i(u)_j=u_j$ for $i<j$ and is false elsewhere. The exponent $N-1$ uses natural-number subtraction. Supplying $i$ requires $N\ge1$, so there are no inputs to this definition with $N=0$. The type $O$ may be empty, yielding an empty matrix. No positivity or trace condition is required of $W(u)$.

QAlgorithms.Papers.Independent.Intakethird100.P23.symmetricCoherentInformation

For any natural numbers $b,e$, including zero, and any complex matrix $U$ with rows indexed by $\{0,\ldots,b-1\}\times\{0,\ldots,e-1\}$ and columns indexed by the two Boolean values, define matrices $B_a$ and $E_a$, for $a\in\{0,1\}$, by

$$(B_a)_{r,s}=\sum_{z=0}^{e-1}U_{(r,z),a}\overline{U_{(s,z),a}}, \qquad (E_a)_{z,w}=\sum_{r=0}^{b-1}U_{(r,z),a}\overline{U_{(r,w),a}}.$$

Here the one-coordinate function indices occurring in the code have been identified with their single coordinates. For a square complex matrix $K$, put $S(K)=-\operatorname{Re}\operatorname{tr}(f(K))$, where $f(x)=x\log_2x$ is applied through the real continuous functional calculus, using the total real logarithm with $\log_2 0=0$. The declaration defines the real number

$$S\!\left(\frac{B_0+B_1}{2}\right) - S\!\left(\frac{E_0+E_1}{2}\right).$$

There is no assumption that $U^\dagger U=I$, or that these matrices have trace one. Empty index sets are allowed; if $b=0$ or $e=0$, the displayed expression is zero.

QAlgorithms.Papers.Independent.Intakethird100.P23.symmetricInformation

For every finite type $O$ equipped with decidable equality and every assignment $W$ of a complex $O\times O$ matrix to each Boolean, this defines the real number

$$H\!\left(\frac{W(\mathrm{false})+W(\mathrm{true})}{2}\right) -\frac{H(W(\mathrm{false}))+H(W(\mathrm{true}))}{2}, \qquad H(M)=-\operatorname{Re}\operatorname{tr} \left(\operatorname{cfc}[t\mapsto t\log_2 t](M)\right).$$

Here $\operatorname{cfc}$ is the real continuous functional calculus on matrices, and the real logarithm is the total real logarithm used in the supplied definition, with $\log_2 0=0$. The matrices are arbitrary: positive semidefiniteness and trace one are not hypotheses. The type $O$ may be empty, in which case the traces and the resulting value are zero.

QAlgorithms.Papers.Independent.Intakethird100.P23.threshold

For every natural number $n$ and every real number $\beta$, this defines the real number

$$2^{-(2^n)^\beta},$$

where the inner $2^n$ is first computed as a natural number and then regarded as a real number, and the powers with real exponents are real powers. There is no restriction on $\beta$. When $n=0$, the value is $\tfrac12$ for every $\beta$.