QuantumInfo.Finite.Pinching

Given a quantum mixed state $\rho$ of dimension $d$, let $M$ be its associated Hermitian operator and $\rho.m$ its underlying $d \times d$ complex matrix. For any eigenvalue $x$ in the spectrum of $\rho.m$, the function `pinching_kraus` returns the projector $P_x$ onto the eigenspace associated with $x$. This is defined via the continuous functional calculus (CFC) as $P_x = f(M)$, where $f$ is the indicator function $f(y) = \delta_{yx}$ (which evaluates to $1$ if $y = x$ and $0$ otherwise). The resulting operator is a $d \times d$ Hermitian matrix.

Let $\rho$ be a quantum state of dimension $d$ and $P_i$ be the pinching Kraus operator (the projector onto the $i$-th eigenspace) associated with an eigenvalue $i$ in the spectrum of $\rho$. Then $P_i$ commutes with the matrix representation of $\rho$, i.e., $P_i \rho = \rho P_i$.

Let $\rho$ be a quantum state represented by a $d \times d$ complex matrix. For any eigenvalue $i$ in the spectrum of $\rho$, let $P_i$ denote the corresponding pinching Kraus operator (the projector onto the $i$-th eigenspace). Then the product of the pinching Kraus operator $P_i$ and the matrix $\rho$ satisfies the relation $P_i \rho = i P_i$, where $i$ is treated as a scalar.

For a quantum state $\rho$ represented by a $d \times d$ complex matrix, the set of distinct real eigenvalues in its spectrum, denoted by $\text{spec}(\rho)$, is a finite set (i.e., it has the `Fintype` property).

Let $\rho$ be a quantum state represented by a $d \times d$ complex matrix. For any two distinct eigenvalues $i, j$ in the spectrum of $\rho$ ($i \neq j$), let $P_i$ and $P_j$ be the corresponding pinching Kraus operators (the projectors onto the respective eigenspaces). Then the product of these operators is zero, i.e., $P_i P_j = 0$.

Let $\rho$ be a quantum state (represented by a $d \times d$ complex matrix) and let $\{P_k\}_k$ be the Kraus operators of the pinching channel with respect to $\rho$. For any $k$ in the spectrum of $\rho$, the Kraus operator $P_k$ is an idempotent matrix, satisfying $P_k^2 = P_k$.

Let $\rho$ be a quantum state with dimension $d$, and let its density matrix be $m(\rho)$. Let $\{P_k\}_{k \in \operatorname{spec}(\rho)}$ be the Kraus operators of the pinching channel with respect to $\rho$, where each $P_i$ is the orthogonal projector onto the eigenspace associated with the eigenvalue $i$ in the spectrum of $\rho$. For any two eigenvalues $i$ and $j$ in the spectrum of $\rho$, the product of their corresponding Kraus operators satisfies: $$P_i P_j = \begin{cases} P_i & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$

For a quantum mixed state $\rho$ of dimension $d$, let $\{P_k\}_k$ be the set of projectors onto the eigenspaces of its density matrix, where $k$ ranges over the distinct real eigenvalues in the spectrum of $\rho$. The sum of these projectors is equal to the identity operator, i.e., $\sum_k P_k = \mathbb{I}$.

Given a quantum state $\rho$, represented as a $d \times d$ density matrix, the pinching channel $\mathcal{E}_\rho$ is a completely positive, trace-preserving (CPTP) map from $d \times d$ matrices to itself. It is defined via a set of Kraus operators $\{P_i\}_i$ where each $P_i$ is the orthogonal projection onto the $i$-th eigenspace of $\rho$. The action of the map on an arbitrary state $\sigma$ is given by $$\mathcal{E}_\rho(\sigma) = \sum_{i} P_i \sigma P_i$$ where the sum runs over the distinct eigenvalues in the spectrum of $\rho$.

Let $\sigma$ and $\rho$ be quantum states (mixed states) of dimension $d$. The Hermitian operator associated with the state resulting from the application of the pinching channel $\mathcal{E}_\sigma$ to the state $\rho$ is itself a Hermitian operator. Specifically, its underlying matrix is obtained by applying the Kraus map $\sum_k P_k (\cdot) P_k$ to the matrix $\rho.M$ (where $P_k$ are the spectral projectors of $\sigma$), and this resulting matrix is guaranteed to be Hermitian because the pinching channel is a hermiticity-preserving map (derived from its property of being completely positive and thus positive).

Let $\sigma$ and $\rho$ be quantum states (mixed states) of dimension $d$. The Hermitian operator associated with the state resulting from the pinching channel $\mathcal{E}_\sigma$ applied to $\rho$ is given by the sum of conjugations: $$(\mathcal{E}_\sigma(\rho)).M = \sum_{k \in \text{spec}(\sigma)} P_k \rho.M P_k$$ where $P_k$ (denoted in the formal statement as `(pinching_kraus σ k).mat`) are the spectral projectors onto the eigenspaces of the state $\sigma$, and $\rho.M$ is the Hermitian operator of the state $\rho$.

Let $\rho$ and $\sigma$ be quantum states in $\text{MState } d$. Let $\mathcal{E}_\sigma$ denote the pinching channel with respect to $\sigma$, and let $P_i$ be the projector onto the $i$-th eigenspace of $\sigma$, where $i \in \text{spec}(\sigma)$ is an eigenvalue in the spectrum of the matrix $\sigma.m$. Then the matrix of the pinched state $\mathcal{E}_\sigma(\rho)$ commutes with the projector $P_i$, i.e., $[\mathcal{E}_\sigma(\rho).m, P_i] = 0$.

Let $\sigma$ and $\rho$ be quantum states (density matrices) in $\text{MState } d$. Let $\mathcal{E}_\sigma$ denote the pinching channel with respect to $\sigma$, defined as $\mathcal{E}_\sigma(\rho) = \sum_i P_i \rho P_i$, where $P_i$ are the spectral projectors of $\sigma$. Then the resulting matrix $\mathcal{E}_\sigma(\rho)$ commutes with the matrix $\sigma$.

Let $\rho$ be a quantum mixed state of dimension $d$. Let $\mathcal{E}_\rho$ denote the pinching channel with respect to the eigenspaces of $\rho$. Then $\mathcal{E}_\rho(\rho) = \rho$. Specifically, the state $\rho$ is a fixed point of its own pinching map.

Let $\rho$ and $\sigma$ be two quantum states represented by density matrices in $\mathbb{C}^{d \times d}$, and let $M_\rho$ and $M_\sigma$ be their associated Hermitian operators. Let $\mathcal{E}_\sigma$ denote the pinching channel with respect to $\sigma$ (the projection of a state onto the diagonal blocks corresponding to the eigenspaces of $\sigma$). It holds that: $$M_\rho \le |\text{spec}(\sigma)| \cdot \mathcal{E}_\sigma(M_\rho)$$ where $\le$ denotes the Loewner order (positive semi-definite order) and $|\text{spec}(\sigma)|$ is the number of distinct eigenvalues in the spectrum of the matrix $\sigma$.

Let $\rho$ and $\sigma$ be density states in a $d$-dimensional Hilbert space. Let $\mathcal{E}_\sigma$ denote the pinching channel with respect to $\sigma$, defined by $\mathcal{E}_\sigma(\rho) = \sum_i P_i \rho P_i$ where $P_i$ are the projectors onto the eigenspaces of $\sigma$. If $\sigma$ is positive definite (denoted $\sigma > 0$), then the kernel of the linear operator associated with $\sigma$ is a subspace of the kernel of the linear operator associated with the pinched state $\mathcal{E}_\sigma(\rho)$. That is, $\ker(\sigma) \subseteq \ker(\mathcal{E}_\sigma(\rho))$.

Let $\rho$ and $\sigma$ be quantum states of dimension $d$, and let $\mathcal{E}_\sigma$ denote the pinching channel with respect to the state $\sigma$. The pinching channel is idempotent, meaning that applying it twice to $\rho$ is equivalent to applying it once: $\mathcal{E}_\sigma(\mathcal{E}_\sigma(\rho)) = \mathcal{E}_\sigma(\rho)$.

Let $\rho$ and $\sigma$ be quantum states in the space of $d \times d$ complex matrices, denoted by $\text{MState } d$. Let $\mathcal{E}_\sigma$ be the pinching channel with respect to the state $\sigma$, and let $\mathcal{E}_\sigma(\rho)$ be the image of $\rho$ under this channel. For any real-valued function $f: \mathbb{R} \to \mathbb{R}$, let $f(\mathcal{E}_\sigma(\rho))$ be the operator defined by applying $f$ to the eigenvalues of $\mathcal{E}_\sigma(\rho)$ via the continuous functional calculus. Then the Hilbert-Schmidt inner product satisfies: $$\langle \rho, f(\mathcal{E}_\sigma(\rho)) \rangle = \langle \mathcal{E}_\sigma(\rho), f(\mathcal{E}_\sigma(\rho)) \rangle$$ where $\langle A, B \rangle = \text{Tr}(A^\dagger B)$ (with $A, B$ being Hermitian).

Let $\rho$ and $\sigma$ be two quantum states represented as Hermitian matrices in $\text{MState } d$. Let $\mathcal{E}_\sigma$ denote the pinching channel with respect to $\sigma$, defined as $\mathcal{E}_\sigma(\rho) = \sum_k P_k \rho P_k$, where the $P_k$ are the projectors onto the eigenspaces of $\sigma$. For any real-valued function $f: \mathbb{R} \to \mathbb{R}$, let $f(\sigma)$ denote the operator obtained by applying the continuous functional calculus to $\sigma$ with respect to $f$. The Hilbert-Schmidt inner product satisfies the identity: $$\langle \mathcal{E}_\sigma(\rho), f(\sigma) \rangle = \langle \rho, f(\sigma) \rangle$$ where $\langle A, B \rangle = \text{Tr}(A^\dagger B)$.

Let $\rho$ and $\sigma$ be two quantum states in a $d$-dimensional Hilbert space. The pinching channel $\mathcal{E}_\sigma$ with respect to $\sigma$ is defined such that the state $\mathcal{E}_\sigma(\rho)$ has an associated Hermitian operator given by the sum: $$(\mathcal{E}_\sigma(\rho)).M = \sum_{\lambda \in \text{spec}(\sigma)} P_\lambda \rho.M P_\lambda$$ where $\{P_\lambda\}_{\lambda \in \text{spec}(\sigma)}$ are the projectors (Kraus operators) onto the eigenspaces corresponding to the distinct eigenvalues $\lambda$ of $\sigma$, and $A.conj(P)$ denotes the conjugation of the operator $A$ by $P$ (i.e., $P A P^\dagger$).

For any two quantum states $\rho$ and $\sigma$ represented by $d \times d$ complex matrices, the kernel of the matrix representing the pinching of $\rho$ with respect to $\sigma$, denoted by $\mathcal{E}_\sigma(\rho)$, is a subspace of the kernel of the matrix representing $\rho$. That is, $\ker(\mathcal{E}_\sigma(\rho)) \subseteq \ker(\rho)$. Here, the pinching channel $\mathcal{E}_\sigma$ is defined relative to the eigenspaces of $\sigma$.

Let $\sigma$ be a quantum state (density matrix) acting on a finite-dimensional Hilbert space $\mathbb{C}^d$. Let $P_k$ (denoted as `pinching_kraus σ k`) be the orthogonal projector onto the eigenspace of $\sigma$ corresponding to the eigenvalue $k$. For any vector $v \in \mathbb{C}^d$, if $v$ is in the kernel of $\sigma$ (i.e., $\sigma v = 0$), then for any non-zero eigenvalue $k \neq 0$ of $\sigma$, the vector $v$ also lies in the kernel of the projector $P_k$, such that $P_k v = 0$.

Let $\rho$ and $\sigma$ be quantum states (mixed states) in a $d$-dimensional Hilbert space. Let $\mathcal{E}_\sigma$ denote the pinching channel with respect to the eigenspaces of $\sigma$, and let $\text{ker}(\rho)$ and $\text{ker}(\sigma)$ denote the kernels of the Hermitian operators associated with $\rho$ and $\sigma$, respectively. If the kernel of $\sigma$ is contained in the kernel of $\rho$ ($\text{ker}(\sigma) \subseteq \text{ker}(\rho)$), then the kernel of $\sigma$ is also contained in the kernel of the pinched state $\mathcal{E}_\sigma(\rho)$ (i.e., $\text{ker}(\sigma) \subseteq \text{ker}(\mathcal{E}_\sigma(\rho))$).

Let $\rho$ and $\sigma$ be quantum states in $\text{MState } d$. Let $\mathcal{E}_\sigma$ denote the pinching channel with respect to the eigenspaces of $\sigma$. The quantum relative entropy $D(\rho \Vert \sigma)$ satisfies the Pythagorean-like identity: $$D(\rho \Vert \sigma) = D(\rho \Vert \mathcal{E}_\sigma(\rho)) + D(\mathcal{E}_\sigma(\rho) \Vert \sigma)$$ where $D(\cdot \Vert \cdot)$ denotes the quantum relative entropy.