PhyslibAlpha.QuantumMechanics.StinespringDilation

Given a finite family of matrices $\{K_i\}_{i \in r}$ (known as Kraus operators) and a matrix $\rho$, the function computes the sum $\sum_{i \in r} K_i \rho K_i^\dagger$, where $K_i^\dagger$ denotes the conjugate transpose of $K_i$. This corresponds to the Kraus representation of a completely positive map acting on an operator $\rho$.

Let $R$ be a star-ring equipped with a partial order and $q, r$ be finite index sets. Let $\{K_i\}_{i \in r}$ be a finite collection of $q \times q$ matrices over $R$. If $\rho$ is a $q \times q$ positive semi-definite matrix, then the result of the Kraus operator sum $\sum_{i \in r} K_i \rho K_i^\dagger$ is also a positive semi-definite matrix, where $K_i^\dagger$ denotes the conjugate transpose of $K_i$.

A collection of matrices $\{K_i\}_{i \in r}$ in $M_q(R)$ defines a quantum channel if they satisfy the completeness relation $\sum_{i \in r} K_i^\dagger K_i = I$, where $I$ is the $q \times q$ identity matrix and $K_i^\dagger$ denotes the conjugate transpose of $K_i$.

A collection of matrices $\{K_i\}_{i \in r}$ (Kraus operators) in $\text{Mat}_q(R)$, where $R$ is a field such as $\mathbb{R}$ or $\mathbb{C}$, defines a **quantum operation** if they satisfy the condition $$\sum_{i \in r} K_i^\dagger K_i \le I$$ where $K_i^\dagger$ denotes the conjugate transpose (adjoint) of $K_i$, and $I$ is the identity matrix. The inequality $\le$ denotes the Loewner order, signifying that the difference $I - \sum_i K_i^\dagger K_i$ is a positive semi-definite matrix. This condition characterizes a trace-non-increasing completely positive map.

Let $R$ be a ring equipped with a partial order and a star-ring structure, and $d$ be a finite index set. A density matrix $\rho$ is defined as a $d \times d$ matrix over $R$ that is positive semidefinite ($\rho \succeq 0$) and has a trace equal to 1 ($\text{Tr}(\rho) = 1$).

Let $R$ be a field (specifically an `RCLike` field such as $\mathbb{R}$ or $\mathbb{C}$) and $d$ be a natural number representing the dimension. Given two $d \times d$ density matrices $\rho_0$ and $\rho_1$, and a scalar $t \in R$ such that $0 \le t$ and $0 \le 1 - t$ (i.e., $0 \le t \le 1$), their convex combination is a new density matrix defined as: $$\rho = t \rho_0 + (1 - t) \rho_1$$ This construction confirms that the set of density matrices is closed under real convex combinations, as the resulting matrix $\rho$ is positive semidefinite and satisfies $\text{Tr}(\rho) = 1$.

Given a ring $R$ and a matrix $\rho$ whose rows are indexed by the Cartesian product $m \times n$ and columns by $m' \times n$, the function $\text{tr}_2$ (the partial trace over the second system) maps $\rho$ to a matrix with rows indexed by $m$ and columns by $m'$. Its $(i, j)$-th entry is defined by the sum over the second index: $$(\text{tr}_2 \rho)_{i,j} = \sum_{k \in n} \rho_{(i, k), (j, k)}$$ where $n$ is a finite index set.

Let $R$ be a ring and $m, r$ be finite sets. Given a collection of matrices $\{K_i\}_{i \in r}$ where each $K_i \in \text{Mat}_m(R)$, the Stinespring operator $V$ is a matrix in $\text{Mat}_{m \times r, m}(R)$ defined such that for any $x, y \in m$ and $i \in r$, the entry $V_{(x, i), y}$ is equal to the $(x, y)$-th entry of the matrix $K_i$. In block matrix notation, this is equivalent to $V = \sum_{i \in r} e_i \otimes K_i$, where $e_i$ are the standard basis vectors of $R^r$.

Let $R$ be a ring, and let $m$ and $r$ be finite index sets. Given a collection of matrices $\{K_k\}_{k \in r}$ where each $K_k \in \text{Mat}_m(R)$, let $V$ be the Stinespring operator. For any row index $(i, k) \in m \times r$ and column index $j \in m$, the entry of the matrix $V$ is given by $$V_{(i, k), j} = (K_k)_{i, j}$$ where $(K_k)_{i, j}$ denotes the $(i, j)$-th entry of the $k$-th matrix $K_k$.

Let $R$ be a star ring and $m, r$ be finite sets. Given a collection of matrices $\{K_i\}_{i \in r}$ in $\text{Mat}_m(R)$ and a matrix $\rho \in \text{Mat}_m(R)$, the Stinespring dilation of $\rho$ with respect to $\{K_i\}$ is the matrix in $\text{Mat}_{m \times r}(R)$ defined as $V \rho V^\dagger$, where $V$ is the Stinespring operator associated with the collection $\{K_i\}$, and $V^\dagger$ denotes the conjugate transpose of $V$.

Let $R$ be a star ring and $m, r$ be finite sets. For a collection of matrices $\{K_i\}_{i \in r}$ in $\text{Mat}_m(R)$, the Stinespring form is the operator that maps a matrix $\rho \in \text{Mat}_m(R)$ to the partial trace over the second system of its Stinespring dilation. Mathematically, it is the map: $$\rho \mapsto \text{tr}_2(V \rho V^\dagger) = \sum_{i \in r} K_i \rho K_i^\dagger$$ where $V$ is the Stinespring operator associated with the collection $\{K_i\}$, and $\text{tr}_2$ is the partial trace over the index set $r$.

Let $R$ be a star-ring and $m, r$ be finite sets. Given a collection of $m \times m$ matrices $\{K_i\}_{i \in r}$ over $R$, let $V$ be the Stinespring operator (the matrix in $\text{Mat}_{(m \times r), m}(R)$ formed by stacking the matrices $K_i$ vertically). Then the sum of the products of the adjoints of $K_i$ with $K_i$ is equal to the product of the adjoint of $V$ with $V$: $$\sum_{i \in r} K_i^\dagger K_i = V^\dagger V$$ where $M^\dagger$ denotes the conjugate transpose (adjoint) of a matrix $M$.

Let $R$ be either the real or complex field, and let $m$ and $r$ be finite sets. Given a collection of matrices $\{K_i\}_{i \in r}$ in $\text{Mat}_m(R)$ such that $\sum_{i \in r} K_i^\dagger K_i \le I$, where $I$ is the identity matrix, let $V \in \text{Mat}_{m \times r, m}(R)$ be the Stinespring operator defined by the entries $V_{(x, i), y} = (K_i)_{x, y}$ for $x, y \in m$ and $i \in r$. Then $V^\dagger V \le I$.

Let $R$ be a star-ring and $m, r$ be finite sets. Given a collection of matrices $\{K_i\}_{i \in r}$ in $\text{Mat}_m(R)$ that satisfy the condition $\sum_{i \in r} K_i^\dagger K_i = I$ (where $K_i^\dagger$ denotes the conjugate transpose and $I$ is the identity matrix), the Stinespring operator $V \in \text{Mat}_{m \times r, m}(R)$ is an isometry, satisfying $V^\dagger V = I$.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be finite sets, and let $\{K_i\}_{i \in r}$ be a collection of matrices in $\text{Mat}_m(R)$ satisfying the identity $\sum_{i \in r} K_i^\dagger K_i = I$. Let $V$ be the Stinespring operator in $\text{Mat}_{m \times r, m}(R)$ defined such that its entries are $V_{(x, i), y} = (K_i)_{x, y}$ for $i \in r$ and $x, y \in m$. Then the columns of $V$ form an orthonormal set in $R^{m \times r}$.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $n$ and $m$ be finite types, and let $s$ be an $n \times m$ matrix with entries in $R$. If the columns of $s$ form an orthonormal set in the Hilbert space $R^n$, then the cardinality of $n$ is equal to the cardinality of the orthonormal basis obtained by extending this set to the whole space.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be finite sets, and let $\{K_i\}_{i \in r}$ be a collection of matrices in $\text{Mat}_m(R)$ satisfying the identity $\sum_{i \in r} K_i^\dagger K_i = I_m$, where $K_i^\dagger$ denotes the conjugate transpose. The columns of the Stinespring operator $V \in \text{Mat}_{m \times r, m}(R)$ form an orthonormal set in $R^{m \times r}$. The cardinality of the orthonormal basis for $R^{m \times r}$ obtained by extending this orthonormal set is equal to the cardinality of the product set $m \times r$.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be natural numbers. Consider a collection of $m \times m$ matrices $\{K_i\}_{i \in \text{Fin } r}$ that satisfies the condition $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$, where $K_i^\dagger$ denotes the conjugate transpose. Let $V$ be the Stinespring operator (an $mr \times m$ matrix) whose entries are defined by $V_{(x, i), y} = (K_i)_{x, y}$. Let $\text{Range}(V)$ be the subspace of $R^{m \times r}$ spanned by the columns of $V$, and let $\text{Range}(V)^\perp$ be its orthogonal complement. If $u'$ is an orthonormal basis for $\text{Range}(V)^\perp$, then the cardinality of $u'$ is equal to the cardinality of the set $\text{Fin } m \times \text{Fin } (r - 1)$, which is $m(r-1)$.

Given a natural number $n$ and indices $k, i$ in the set $\{0, \dots, n-1\}$ such that $i \neq k$, the function returns an index in the set $\{0, \dots, n-2\}$ defined by shifting indices greater than $k$ down by one: $$f(i) = \begin{cases} i - 1 & \text{if } i > k \\ i & \text{if } i < k \end{cases}$$

For any natural number $n$ and elements $k, x, y \in \{0, \dots, n-1\}$, let $f: \{i \in \text{Fin } n \mid i \neq k\} \to \{0, \dots, n-2\}$ be the function defined by $$f(i) = \begin{cases} i & \text{if } i < k \\ i - 1 & \text{if } i > k \end{cases}$$ If $x \neq k$, $y \neq k$, and $f(x) = f(y)$, then $x = y$. In other words, the function that maps an index $i$ (excluding $k$) to its predecessor above $k$ is injective.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be natural numbers. Consider a collection of $m \times m$ matrices $\{K_i\}_{i \in \text{Fin } r}$ satisfying the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$, where $K_i^\dagger$ denotes the conjugate transpose. Let $V$ be the Stinespring operator, defined as the $mr \times m$ matrix where $V_{(a, i), b} = (K_i)_{a, b}$. Let $\text{Range}(V)^\perp$ be the orthogonal complement of the subspace spanned by the columns of $V$ in $R^{m \times r}$. Given a fixed index $z \in \text{Fin } r$, and an index $x = (j, i) \in \text{Fin } m \times \text{Fin } r$ such that $i \neq z$, `onbPart` returns the vector in $R^{m \times r}$ corresponding to the basis element $u_{(j, f_z(i))}$ of a chosen orthonormal basis $\{u_w\}$ of $\text{Range}(V)^\perp$. Here, $f_z(i)$ is the index in $\text{Fin } (r-1)$ obtained by shifting the index $i$ to skip $z$.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be natural numbers, and let $\{K_i\}_{i \in \text{Fin } r}$ be a collection of $m \times m$ matrices satisfying the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$. For a fixed index $z \in \text{Fin } r$, let $x, y \in \text{Fin } m \times \text{Fin } r$ be indices such that their second components are not equal to $z$ (i.e., $x_2 \neq z$ and $y_2 \neq z$). If $x \neq y$, then the inner product of the vectors defined by `onbPart` for $x$ and $y$ is zero: $$\langle \text{onbPart}(y), \text{onbPart}(x) \rangle = 0$$

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be natural numbers. Consider a collection of $m \times m$ matrices $\{K_i\}_{i \in \text{Fin } r}$ satisfying the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$, where $K_i^\dagger$ denotes the conjugate transpose. Let $V$ be the Stinespring operator, defined as the $mr \times m$ matrix where $V_{(a, i), b} = (K_i)_{a, b}$. For a fixed index $z \in \text{Fin } r$ and an index $x = (j, i) \in \text{Fin } m \times \text{Fin } r$ such that $i \neq z$, the vector `onbPart` (which represents a basis element of the orthogonal complement $(\text{Range } V)^\perp$) has a norm of 1, i.e., $\|\text{onbPart}\|_2 = 1$.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be natural numbers. Consider a collection of $m \times m$ matrices $\{K_i\}_{i \in \text{Fin } r}$ satisfying the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$, where $K_i^\dagger$ denotes the conjugate transpose. For a fixed index $z \in \text{Fin } r$, the Stinespring dilation $U$ (also known as the unitary dilation) is an $(mr) \times (mr)$ matrix constructed as follows: For any row index $x = (x_1, x_2) \in \text{Fin } m \times \text{Fin } r$ and column index $y = (y_1, y_2) \in \text{Fin } m \times \text{Fin } r$, the entries of $U$ are: $$U_{x, (y_1, y_2)} = \begin{cases} (K_{x_2})_{x_1, y_1} & \text{if } y_2 = z \\ (\text{onbPart } y)_x & \text{if } y_2 \neq z \end{cases}$$ Here, the columns where $y_2 = z$ form the Stinespring operator $V$, and the columns where $y_2 \neq z$ are formed by the components of an orthonormal basis $(\text{onbPart } y)$ for the orthogonal complement of the range of $V$. This construction results in a unitary (or orthogonal) matrix $U$.

Let $R$ be a ring and $m, r$ be index sets. Given a fixed index $z \in r$, a matrix $S$ of size $(m \times r) \times m$, and a matrix $M$ of size $(m \times r) \times (m \times r)$, the general dilation is the $(m \times r) \times (m \times r)$ matrix whose entries are defined as follows: for any row index $x \in m \times r$ and column index $y = (y_1, y_2) \in m \times r$, the entry is $S_{x, y_1}$ if $y_2 = z$, and $M_{x, y}$ if $y_2 \neq z$. Effectively, this construction replaces the $z$-th block-column of $M$ with the matrix $S$.

Let $R$ be a ring and $m, r$ be index sets where $r$ is finite. Given a collection of matrices $\{K_i\}_{i \in r}$ where each $K_i \in \text{Mat}_m(R)$, a fixed index $z \in r$, and a matrix $M \in \text{Mat}_{m \times r}(R)$, the dilation is the $(m \times r) \times (m \times r)$ matrix $D$ obtained by replacing the $z$-th block-column of $M$ with the Stinespring operator $V$ associated with $\{K_i\}$. Specifically, for any row index $x = (x_1, x_2) \in m \times r$ and column index $y = (y_1, y_2) \in m \times r$, the entries of the matrix are defined as: $$D_{(x_1, x_2), (y_1, y_2)} = \begin{cases} (K_{x_2})_{x_1, y_1} & \text{if } y_2 = z \\ M_{(x_1, x_2), (y_1, y_2)} & \text{if } y_2 \neq z \end{cases}$$ This construction represents a general, not necessarily unitary, dilation of the operators $\{K_i\}$.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m, r$ be natural numbers and $\{K_i\}_{i \in \text{Fin } r}$ be a collection of $m \times m$ matrices over $R$ satisfying the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$. Let $V$ be the Stinespring operator, defined as the $(m \times r) \times m$ matrix such that $V_{(a, i), b} = (K_i)_{a, b}$. For a fixed index $z \in \text{Fin } r$, define a collection of vectors $\{v_y\}_{y \in \text{Fin } m \times \text{Fin } r}$ in $R^{m \times r}$ as follows: - If $y = (j, z)$ for some $j \in \text{Fin } m$, then $v_y$ is the $j$-th column of the Stinespring operator $V$. - If $y = (j, i)$ for $i \neq z$, then $v_y$ is the vector provided by `onbPart`, which belongs to a chosen orthonormal basis of the orthogonal complement $(\text{Range } V)^\perp$. Then the collection of vectors $\{v_y\}_{y \in \text{Fin } m \times \text{Fin } r}$ is orthonormal.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be natural numbers. Suppose $\{K_i\}_{i \in \text{Fin } r}$ is a collection of $m \times m$ matrices over $R$ that satisfy the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$, where $I_m$ is the $m \times m$ identity matrix and $K_i^\dagger$ is the conjugate transpose of $K_i$. Let $V$ be the Stinespring operator, defined as the $mr \times m$ matrix with entries $V_{(a, i), b} = (K_i)_{a, b}$ for $a, b \in \text{Fin } m$ and $i \in \text{Fin } r$. For a fixed index $z \in \text{Fin } r$, we define a set of vectors $\{v_y\}_{y \in \text{Fin } m \times \text{Fin } r}$ in $R^{mr}$ such that: - If $y.2 = z$, $v_y$ is the $y.1$-th column of the Stinespring operator $V$. - If $y.2 \neq z$, $v_y$ is the basis vector of the orthogonal complement $(\text{Range } V)^\perp$ defined by `onbPart`. Then the collection of vectors $\{v_y\}$ is orthonormal with respect to the standard $L^2$ inner product.

Let $R$ be a field that is either the real numbers $\mathbb{R}$ or the complex numbers $\mathbb{C}$ (specifically, an `RCLike` field). For a finite set $t$ and a function $\beta : t \to R$, if the $L^2$-norm of $\beta$ is equal to 1, then the sum of the products of the conjugates of its components with the components themselves is 1. That is, if $\|\beta\|_2 = 1$, then $$\sum_{x \in t} \overline{\beta(x)} \beta(x) = 1$$ where $\overline{\beta(x)}$ denotes the conjugate of $\beta(x)$ in $R$.

Let $R$ be a field that is either the real or complex numbers ($\mathbb{R}$ or $\mathbb{C}$). Let $m$ and $r$ be finite index sets, and let $\alpha$ be a square matrix with entries in $R$ indexed by $(m \times r) \times (m \times r)$. If the rows of $\alpha$ (viewed as vectors with the standard $L^2$ inner product) form an orthonormal set, then $\alpha \alpha^* = I$, where $\alpha^*$ denotes the conjugate transpose of $\alpha$ and $I$ is the identity matrix.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be natural numbers and $\{K_i\}_{i \in \text{Fin } r}$ be a collection of $m \times m$ matrices over $R$ satisfying the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$, where $K_i^\dagger$ denotes the conjugate transpose. For any fixed index $z \in \text{Fin } r$, let $U$ be the Stinespring dilation (also known as the unitary dilation) associated with $\{K_i\}$ at $z$. Then the transpose of $U$, denoted $U^T$, is a unitary matrix.

Let $R$ be a field that is either the real numbers $\mathbb{R}$ or the complex numbers $\mathbb{C}$. Let $m$ and $r$ be natural numbers. Suppose $\{K_i\}_{i \in \text{Fin } r}$ is a collection of $m \times m$ matrices over $R$ that satisfies the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$, where $K_i^\dagger$ denotes the conjugate transpose and $I_m$ is the $m \times m$ identity matrix. For any fixed index $z \in \text{Fin } r$, the Stinespring dilation matrix $U$ (constructed from $\{K_i\}$ and $z$) is a unitary matrix. That is, $U U^\dagger = U^\dagger U = I_{mr}$.

Let $R$ be a field of real or complex numbers. Let $e$ be a square matrix indexed by a finite set $w$, and let $\rho$ be a square matrix indexed by a set $m$. If the trace of $e$ is equal to 1, then the partial trace over the second system of the Kronecker product $\rho \otimes e$ is equal to $\rho$. That is, if $\text{tr}(e) = 1$, then: $$\text{tr}_2(\rho \otimes e) = \rho$$

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$ and $m, r$ be natural numbers. Given a collection of $m \times m$ matrices $\{K_i\}_{i \in \text{Fin } r}$ satisfying the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I$, and a fixed index $z \in \text{Fin } r$, the Stinespring unitary form of an $m \times m$ matrix $\rho$ is defined as: $$\text{tr}_2 \left( U (\rho \otimes E_{z,z}) U^\dagger \right)$$ where $U$ is the unitary Stinespring dilation of the Kraus operators $\{K_i\}$ at the index $z$, $E_{z,z}$ is the $r \times r$ matrix with the entry $1$ at $(z, z)$ and $0$ elsewhere, and $\text{tr}_2$ is the partial trace over the second system.

Let $R$ be a field that is either $\mathbb{R}$ or $\mathbb{C}$. Let $m$ and $r$ be natural numbers. Given a collection of $m \times m$ matrices $\{K_i\}_{i \in \text{Fin } r}$ satisfying the completeness relation $\sum_{i \in \text{Fin } r} K_i^\dagger K_i = I_m$, and a fixed reference index $z \in \text{Fin } r$, let $U$ be the unitary Stinespring dilation matrix (as defined by `Ud`). For any $m \times m$ matrix $\rho$ and $r \times r$ matrix $e$, the general Stinespring unitary form is the $m \times m$ matrix defined by: $$\text{tr}_2(U (\rho \otimes e) U^\dagger)$$ where $\otimes$ denotes the Kronecker product, $U^\dagger$ is the conjugate transpose of $U$, and $\text{tr}_2$ is the partial trace over the second system.

Let $R$ be a field (either $\mathbb{R}$ or $\mathbb{C}$), and let $m$ and $r$ be finite index sets. Given a collection of $m \times m$ matrices $\{K_i\}_{i \in r} \subseteq \text{Mat}_m(R)$, define the Stinespring operator $U = \sum_{i \in r} e_i \otimes K_i^\dagger$, where $K_i^\dagger$ is the conjugate transpose of $K_i$ and $e_i$ are the standard basis vectors. For any $m \times m$ matrix $\rho$, the following equality holds: $$U^\dagger (\rho \otimes I_r) U = \sum_{i \in r} K_i \rho K_i^\dagger$$ where $I_r$ is the $r \times r$ identity matrix and $U^\dagger$ denotes the conjugate transpose of $U$.