QuantumInfo.Finite.POVM

Given a Positive Operator-Valued Measure (POVM) $\Lambda$ consisting of $d \times d$ positive semidefinite matrices $M_x$ for $x \in X$, the measurement map $\mathcal{M}_\Lambda$ is a completely positive trace-preserving (CPTP) map from the space of $d$-dimensional quantum states to the space of $(d \times X)$-dimensional quantum-classical states. For a density matrix $\rho$, the map is defined as: $$\mathcal{M}_\Lambda(\rho) = \sum_{x \in X} \sqrt{M_x} \rho \sqrt{M_x} \otimes |x \rangle \langle x|$$ where $\sqrt{M_x}$ is the unique positive semidefinite square root of the operator $M_x$, and $|x \rangle \langle x|$ (represented by `Matrix.single x x 1`) is the projector onto the classical outcome $x$.

Let $\Lambda$ be a Positive Operator-Valued Measure (POVM) over a finite outcome space $X$ and a Hilbert space of dimension $d$, where $\Lambda.mats(x)$ denotes the positive semidefinite operator associated with outcome $x \in X$. For any matrix $m \in M_d(\mathbb{C})$, the application of the induced measurement map $\Lambda.measurementMap$ to $m$ is given by the sum: $$\Lambda.measurementMap(m) = \sum_{x \in X} \left( \sqrt{\Lambda.mats(x)} \cdot m \cdot \sqrt{\Lambda.mats(x)} \right) \otimes (E_{x,x})$$ where $\sqrt{\Lambda.mats(x)}$ is the unique positive semidefinite square root of the operator associated with outcome $x$, and $E_{x,x}$ is the matrix with a $1$ at position $(x, x)$ and $0$ elsewhere. This map represents the transformation of the quantum state into a joint state of the system and a classical register storing the measurement outcome.

Let $\Lambda$ be a Positive Operator-Valued Measure (POVM) indexed by $X$ on a Hilbert space of dimension $d$, with effects $\Lambda_x$. For any Hermitian matrix $m \in \mathcal{H}_d(\mathbb{C})$, the application of the associated measurement map $\mathcal{E}_\Lambda$ to $m$ is given by $$\mathcal{E}_\Lambda(m) = \sum_{x \in X} \left[ \left( \Lambda_x^{1/2} m \Lambda_x^{1/2} \right) \otimes \text{diag}(\delta_{xy})_y \right]$$ where $\Lambda_x^{1/2}$ denotes the unique positive semidefinite square root of the effect $\Lambda_x$, and $\text{diag}(\delta_{xy})_y$ is the diagonal Hermitian matrix whose only non-zero entry is a $1$ at position $(x, x)$.

Given a Positive Operator-Valued Measure (POVM) $\Lambda$ with outcomes in $X$ and representing matrices $\Lambda_x$ for each $x \in X$, and a quantum mixed state $\rho$ (represented by a density matrix $M_\rho$), the probability of obtaining outcome $x$ is given by the Born rule: $$p(x) = \langle \Lambda_x, M_\rho \rangle = \text{Tr}(\Lambda_x^\dagger M_\rho)$$ Since $\Lambda_x$ is positive semidefinite and $\sum_{x \in X} \Lambda_x = I$, this defines a valid probability distribution over the set of outcomes $X$.

Let $\Lambda$ be a positive operator-valued measure (POVM) with outcomes in $X$ and acting on a quantum system $d$. Let $\rho$ be a density matrix representing a mixed state in $d$. The partial trace over the system $d$ of the state resulting from the quantum-classical measurement map, denoted $\text{Tr}_d(\mathcal{E}_{\Lambda}(\rho))$, is equal to the diagonal density matrix representing the classical probability distribution of the measurement outcomes, $\text{ofClassical}(\text{measure}(\Lambda, \rho))$.

Given a Positive Operator-Valued Measure (POVM) $\Lambda$ with outcomes in $X$ and acting on a Hilbert space of dimension $d$, the map `measureDiscard` is defined as the composition of the measurement map with a partial trace over the system state. Specifically, let $\mathcal{M}_\Lambda: \mathcal{L}(\mathbb{C}^d) \to \mathcal{L}(\mathbb{C}^d \otimes \mathbb{C}^{|X|})$ be the CPTP map representing the measurement process. Then `measureDiscard` is the CPTP map $\text{Tr}_d \circ \mathcal{M}_\Lambda: \mathcal{L}(\mathbb{C}^d) \to \mathcal{L}(\mathbb{C}^{|X|})$ which takes an input state $\rho$, performs the measurement, discards the post-measurement quantum state, and retains only the diagonal mixed state representing the probability distribution of classical outcomes in $X$.

Let $\Lambda$ be a Positive Operator-Valued Measure (POVM) with outcomes in $X$ for a quantum system with state space $d$. For any quantum state $\rho$, applying the discard-measurement map $\Lambda.\text{measureDiscard}$ to $\rho$ results in a diagonal density matrix representing the classical probability distribution of the measurement outcomes. Specifically, $\Lambda.\text{measureDiscard}(\rho) = \text{ofClassical}(\Lambda.\text{measure}(\rho))$, where $\Lambda.\text{measure}(\rho)$ is the distribution of outcomes induced by the POVM.

For a Positive Operator-Valued Measure (POVM) $\Lambda$ with outcomes in $X$ and acting on a Hilbert space of dimension $d$, the map $\Lambda.\text{measureForget}$ is a Completely Positive Trace-Preserving (CPTP) map from the space of states of dimension $d$ to itself. It is defined as the composition of the measurement map $\Lambda.\text{measurementMap}$ (which maps a state to a joint state of the system and the classical outcome) and the partial trace $\text{Tr}_X$ over the outcome space. Physically, this represents the process of performing the measurement $\Lambda$ on a quantum state and discarding the resulting outcome, thereby retaining only the post-measurement state.