QuantumInfo.Finite.Entropy.DPI

Given a set of $d \times d$ complex matrices, for $p \in \mathbb{R}$ and a quantum state $\sigma$ (represented as a density matrix), the weighted norm $\|\cdot\|_{p, \sigma}$ is defined for a matrix $X$ as \[ \|X\|_{p, \sigma} = \left\| \sigma^{1/(2p)} X \sigma^{1/(2p)} \right\|_p \] where $\|\cdot\|_p$ denotes the Schatten $p$-norm and $\sigma^{1/(2p)}$ is defined via the continuous functional calculus for Hermitian matrices.

For a matrix $A \in \text{Mat}_{d \times d}(\mathbb{C})$, the spectral norm (or operator norm) $\|A\|_\infty$ is defined as the square root of the maximum eigenvalue of the matrix $A^\dagger A$, where $A^\dagger$ denotes the conjugate transpose of $A$. If the index set of the matrix is empty, the norm is defined to be $0$.

For a quantum state $\sigma$ represented as a density matrix in $\text{MState } d$ and a complex matrix $X \in \text{Mat}_{d \times d}(\mathbb{C})$, the weighted norm for $p = \infty$ is defined as the spectral norm $\|X\|_\infty$ of the matrix $X$. In this case, the norm is independent of the state $\sigma$.

Given a quantum state $\sigma$ represented as a density matrix in $\text{Mat}_d(\mathbb{C})$, the map $\Gamma_\sigma$ is defined for any matrix $X \in \text{Mat}_d(\mathbb{C})$ as $$\Gamma_\sigma(X) = \sigma^{1/2} X \sigma^{1/2}$$ where $\sigma^{1/2}$ is the unique positive semidefinite square root of $\sigma$ obtained via the continuous functional calculus.

Given a quantum state $\sigma$ (represented as a density matrix in $\mathbb{C}^{d \times d}$) and a complex matrix $X \in \mathbb{C}^{d \times d}$, the map $\Gamma_\sigma^{-1}$ is defined as $$\Gamma_\sigma^{-1}(X) = \sigma^{-1/2} X \sigma^{-1/2}$$ where $\sigma^{-1/2}$ is computed via the continuous functional calculus for Hermitian matrices using the power function $x \mapsto x^{-1/2}$.

Given a completely positive trace-preserving (CPTP) map $\Phi: \text{Mat}_d(\mathbb{C}) \to \text{Mat}_{d_2}(\mathbb{C})$ and a density matrix (state) $\sigma \in \text{MState}(d)$, the operator $T_{\Phi, \sigma}: \text{Mat}_d(\mathbb{C}) \to \text{Mat}_{d_2}(\mathbb{C})$ is defined by the composition $$T_{\Phi, \sigma} = \Gamma_{\Phi(\sigma)}^{-1} \circ \Phi \circ \Gamma_\sigma$$ Specifically, for an input matrix $X \in \text{Mat}_d(\mathbb{C})$, the value is $T_{\Phi, \sigma}(X) = \Gamma_{\Phi(\sigma)}^{-1}(\Phi(\Gamma_\sigma(X)))$, where $\Gamma_\sigma$ is a map associated with the state $\sigma$ and $\Gamma_{\Phi(\sigma)}^{-1}$ is the inverse map associated with the evolved state $\Phi(\sigma)$.

Let $\mathcal{M}_d$ denote the space of $d \times d$ complex matrices and $\text{MState}(d)$ denote the space of density matrices (quantum states). Given a real number $p$, a reference state $\sigma \in \text{MState}(d)$, a CPTP (completely positive trace-preserving) map $\Phi: \mathcal{M}_d \to \mathcal{M}_{d_2}$, and a linear map $\Psi: \mathcal{M}_d \to \mathcal{M}_{d_2}$, the induced norm (with respect to weighted $p$-norms) is defined as the supremum: $$\|\Psi\|_{p, \sigma \to \Phi(\sigma)} = \sup_{X \in \mathcal{M}_d, \|X\|_{p, \sigma} \neq 0} \frac{\|\Psi(X)\|_{p, \Phi(\sigma)}}{\|X\|_{p, \sigma}}$$ where $\| \cdot \|_{p, \sigma}$ denotes the weighted $p$-norm relative to the state $\sigma$.

Given a mixed state $\sigma \in \text{MState}(d)$, a completely positive trace-preserving (CPTP) map $\Phi: \text{MState}(d) \to \text{MState}(d_2)$, and a linear map between matrices $\Psi: \text{Mat}_d(\mathbb{C}) \to \text{Mat}_{d_2}(\mathbb{C})$, the induced infinity-norm $\left\| \Psi \right\|_{\sigma \to \Phi(\sigma), \infty}$ is defined as the supremum of the ratio of weighted infinity-norms: $$\sup_{X \neq 0} \frac{\| \Psi(X) \|_{\infty, \Phi(\sigma)}}{\| X \|_{\infty, \sigma}}$$ where $\| \cdot \|_{\infty, \rho}$ denotes the weighted infinity-norm relative to a state $\rho$.

Given a quantum state $\sigma$ (represented as a density matrix in $\text{Mat}_d(\mathbb{C})$) and a completely positive trace-preserving (CPTP) map $\Phi: \text{Mat}_d(\mathbb{C}) \to \text{Mat}_{d_2}(\mathbb{C})$, the map $T_{\sigma, \Phi}$ is the linear map from $\text{Mat}_d(\mathbb{C})$ to $\text{Mat}_{d_2}(\mathbb{C})$ defined by the operation $T_{\text{op}}$. Specifically, it is the linear map version of the operator $X \mapsto \Phi(\Gamma_\sigma(X))$ or its related variants used in the study of Rényi relative entropy and the Data Processing Inequality.

Let $d$ be a finite-dimensional Hilbert space index and $\sigma$ be a quantum state (density matrix) in $\text{MState}_d$. Let $\sigma^{1/2}$ denote the square root of the matrix $\sigma$, calculated via the continuous functional calculus. The linear map $\Gamma_\sigma$ is defined as the matrix conjugation map from $\text{Mat}_d(\mathbb{C})$ to $\text{Mat}_d(\mathbb{C})$ given by $X \mapsto \sigma^{1/2} X \sigma^{1/2}$.

For a quantum mixed state $\sigma$ of dimension $d$ and any $d \times d$ complex matrix $X$, the evaluation of the linear map $\Gamma_{\text{map}}(\sigma)$ at $X$ is equal to the matrix $\Gamma(\sigma, X)$.

Let $\sigma$ be a quantum state (represented as an element of `MState d`). The associated linear map $\Gamma_{\sigma}$, defined by $\Gamma_{\sigma}(X) = \sigma^{1/2} X \sigma^{1/2}$, is a completely positive matrix map.

Let $\sigma$ be a quantum state represented by a matrix $M_\sigma$ in the space of $d \times d$ complex matrices. The linear map $\Gamma_{\sigma}^{-1}: \text{Mat}_d(\mathbb{C}) \to \text{Mat}_d(\mathbb{C})$ is defined as the matrix conjugation map $X \mapsto M_\sigma^{-1/2} X (M_\sigma^{-1/2})^H$. Here, $M_\sigma^{-1/2}$ is obtained by applying the continuous functional calculus to the matrix $M_\sigma$ with the power function $f(x) = x^{-1/2}$.

Let $\sigma$ be a quantum mixed state of dimension $d$ and $X$ be a $d \times d$ complex matrix. The evaluation of the inverse $\Gamma$-map of $\sigma$, denoted as $\Gamma_{\sigma}^{-1}$, at the matrix $X$ is equal to the matrix-defined operation $\text{Gamma\_inv}(\sigma, X)$.

Given a mixed state $\sigma$ (represented as a density matrix $M$ in the space of $d \times d$ complex matrices), $\sigma^{-1/2}$ is defined as the matrix obtained by applying the continuous functional calculus to $M$ with the function $f(x) = x^{-1/2}$.

Let $\sigma$ be a quantum state (represented as a density matrix of type `MState d`). The linear map $\Gamma_\sigma^{-1}$ (denoted as `Gamma_inv_map σ`) is equal to the matrix conjugation map $X \mapsto \sigma^{-1/2} X (\sigma^{-1/2})^H$, where $\sigma^{-1/2}$ is the inverse square root of the density matrix $\sigma$ (denoted as `sigma_inv_sqrt σ`).

Let $\sigma$ be a quantum state (density matrix) of dimension $d$. The linear map $\Gamma_{\sigma}^{-1}: \text{Mat}_d(\mathbb{C}) \to \text{Mat}_d(\mathbb{C})$, defined as the inverse of the map associated with $\sigma$, is completely positive.

Let $\sigma$ be a quantum state (density matrix) in the state space $\text{MState}(d)$ and $\Phi: \text{MState}(d) \to \text{MState}(d_2)$ be a completely positive trace-preserving (CPTP) map with underlying linear map $\Phi_{\text{map}}$. Let $\Gamma_{\sigma}$ denote the map $X \mapsto \sigma^{1/2} X \sigma^{1/2}$ and $\Gamma_{\sigma}^{-1}$ denote the map $X \mapsto \sigma^{-1/2} X \sigma^{-1/2}$. The map $T_{\sigma, \Phi}$ is equal to the composition of the linear maps $\Gamma_{\Phi(\sigma)}^{-1} \circ \Phi_{\text{map}} \circ \Gamma_{\sigma}$.

Let $\sigma \in \text{MState}(d)$ be a matrix state and $\Phi: \text{MState}(d) \to \text{MState}(d_2)$ be a completely positive trace-preserving (CPTP) map. Let $T_{\sigma, \Phi}$ be the linear matrix map defined by $T_{\sigma, \Phi} = \Gamma_{\Phi(\sigma)}^{-1} \circ \Phi \circ \Gamma_{\sigma}$, where $\Gamma$ and $\Gamma^{-1}$ are specific maps associated with the states. Then the map $T_{\sigma, \Phi}$ is completely positive.

Let $\sigma$ be a quantum state (represented as an element of `MState d`) and $\Phi$ be a Completely Positive Trace Preserving (CPTP) map from dimension $d$ to $d_2$. Then the associated matrix map $T_{\sigma, \Phi}$ (defined as `T_map σ Φ`) is a positive map. Specifically, for any positive semi-definite matrix $x$, $T_{\sigma, \Phi}(x)$ is also positive semi-definite.

Let $\sigma$ be a mixed quantum state of dimension $d$ and $X$ be a $d \times d$ complex matrix. The weighted norm of $X$ with respect to $\sigma$ for $p = 1$, denoted as $\text{weighted\_norm}(1, \sigma, X)$, is equal to the trace norm (Schatten $1$-norm) of the matrix $\Gamma_\sigma(X)$, where $\Gamma_\sigma(X)$ is the operator defined by the action of $\sigma$ on $X$. Mathematically, this is expressed as: $$\text{weighted\_norm}(1, \sigma, X) = \|\Gamma_\sigma(X)\|_1$$

Let $d$ and $d_2$ be finite-dimensional Hilbert space dimensions. Given two quantum states $\sigma \in \text{MState}(d)$ and $\sigma' \in \text{MState}(d_2)$, and a linear map between matrix spaces $\Psi: \text{Mat}_d(\mathbb{C}) \to \text{Mat}_{d_2}(\mathbb{C})$, the $(p, q)$-induced norm of $\Psi$ relative to $\sigma$ and $\sigma'$ is defined as: $$\|\Psi\|_{p \to q, \sigma \to \sigma'} = \sup_{X \in \text{Mat}_d(\mathbb{C}), \|X\|_{p, \sigma} \neq 0} \frac{\|\Psi(X)\|_{q, \sigma'}}{\|X\|_{p, \sigma}}$$ where $\| \cdot \|_{p, \sigma}$ denotes the weighted norm with respect to the state $\sigma$.

Let $d$ be a finite type with decidable equality. For any Hermitian matrix $A \in \mathbb{C}^{d \times d}$ and any functions $f, g: \mathbb{R} \to \mathbb{R}$, the product of the matrices obtained by applying the continuous functional calculus (CFC) to $A$ with functions $f$ and $g$ is equal to the matrix obtained by applying the continuous functional calculus to $A$ with the pointwise product function $f \cdot g$. That is, $$f(A) \cdot g(A) = (f \cdot g)(A)$$ where $f(A)$ denotes the result of the functional calculus applied to $A$.

For any quantum mixed state $\sigma$ of finite dimension $d$, the operator $\Gamma_\sigma$ applied to the identity matrix $I$ is equal to the underlying matrix representation of the state $\sigma$, denoted as $\sigma.M.\text{mat}$.

Let $\sigma$ be a quantum mixed state of dimension $d$ with a positive definite density matrix $m(\sigma)$. Let $M_\sigma$ be the Hermitian operator associated with $\sigma$, and let $\Gamma_\sigma^{-1}$ be the inverse of the map $\Gamma_\sigma$. Then applying $\Gamma_\sigma^{-1}$ to the matrix representation of $M_\sigma$ yields the identity matrix, i.e., $\Gamma_\sigma^{-1}(M_\sigma) = \mathbb{1}$.

Let $\Phi$ be a completely positive trace-preserving (CPTP) map from the space of density matrices $\mathcal{M}_d$ to $\mathcal{M}_{d_2}$. For any matrix state $\sigma$ of dimension $d$, the underlying matrix of the image $\Phi(\sigma)$ is equal to the underlying linear map of $\Phi$ applied to the matrix $\sigma.M.mat$. That is, $(\Phi \sigma).M.mat = \Phi.map(\sigma.M.mat)$.

Let $\sigma$ be a quantum state (matrix state) of dimension $d$, and let $\Phi$ be a completely positive trace-preserving (CPTP) map from $d \times d$ matrices to $d_2 \times d_2$ matrices. Suppose that the image of the state under the map, $\Phi(\sigma)$, is positive definite. Then the associated linear map $T_{\sigma, \Phi}$ is unital, i.e., $T_{\sigma, \Phi}(\mathbb{I}) = \mathbb{I}$, where $\mathbb{I}$ denotes the identity matrix.

Let $\sigma$ be a mixed state (density matrix) in a $d$-dimensional Hilbert space, and let $\Phi$ be a Completely Positive Trace-Preserving (CPTP) map from $d$-dimensional matrices to $d_2$-dimensional matrices. Then the associated map $T_{\sigma, \Phi}$ (often referred to as the Petz recovery map or a related transition map in the context of the Data Processing Inequality) is a completely positive matrix map.

Let $\sigma$ be a quantum mixed state of dimension $d$ such that its underlying matrix $\sigma.m$ is positive definite. For any $d \times d$ complex matrix $X$, the composition of the map $\Gamma_{\sigma}^{-1}$ followed by $\Gamma_{\sigma}$ is the identity, such that $\Gamma_{\sigma}(\Gamma_{\sigma}^{-1}(X)) = X$.

Let $A$ be a Hermitian matrix of dimension $d$ and $M$ be a real number. If $A \le M I$, where $I$ is the identity matrix and $\le$ denotes the Loewner order (the positive semi-definite order), then every eigenvalue $\lambda_i$ of $A$ satisfies $\lambda_i \le M$.

Let $A$ be a Hermitian matrix of dimension $d$ over the complex numbers $\mathbb{C}$. Given a real number $M$, if all eigenvalues $\lambda_i$ of $A$ satisfy $\lambda_i \leq M$ for all indices $i$, then $A$ is less than or equal to $M$ times the identity matrix $I$ in the Loewner order, i.e., $A \leq M \cdot I$.

Let $m, n, k$ be finite types. For any complex matrices $X \in \mathbb{C}^{k \times n}$ and $Y \in \mathbb{C}^{k \times m}$, the block matrix $$\begin{pmatrix} Y^\dagger Y & Y^\dagger X \\ X^\dagger Y & X^\dagger X \end{pmatrix}$$ is positive semidefinite, where $M^\dagger$ denotes the conjugate transpose of a matrix $M$.

Let $m$ and $n$ be finite types. For any $m \times n$ complex matrix $X$, the block matrix $$\begin{pmatrix} I & X \\ X^\dagger & X^\dagger X \end{pmatrix}$$ is positive semidefinite, where $I$ is the $m \times m$ identity matrix and $X^\dagger$ denotes the conjugate transpose of $X$.

Let $\rho$ and $\sigma$ be two quantum states (density matrices) in a finite-dimensional Hilbert space of dimension $d$. For any parameter $\alpha \ge 1$ and any completely positive trace-preserving (CPTP) map $\Phi$ from states of dimension $d$ to states of dimension $d_2$, the sandwiched Rényi relative entropy $D_{\alpha}$ satisfies the inequality: $$D_{\alpha}(\Phi(\rho) \| \Phi(\sigma)) \le D_{\alpha}(\rho \| \sigma)$$ where $D_{\alpha}(\rho \| \sigma)$ denotes the sandwiched Rényi relative entropy between $\rho$ and $\sigma$.

For any real number $\alpha \ge 1$, any matrix states $\rho$ and $\sigma$ in a Hilbert space of dimension $d$, and any completely positive trace-preserving (CPTP) map $\Phi$ from dimension $d$ to dimension $d_2$, the sandwiched Rényi relative entropy satisfies the inequality: $$\tilde{D}_\alpha(\Phi(\rho) \| \Phi(\sigma)) \le \tilde{D}_\alpha(\rho \| \sigma)$$ where $\tilde{D}_\alpha$ denotes the sandwiched Rényi relative entropy of order $\alpha$.