QuantumInfo.Entropy.DPI

The sandwiched trace functional $\tilde{Q}_\alpha(\rho\|\sigma)$ is a real-valued function of a parameter $\alpha \in \mathbb{R}$ and two quantum states $\rho$ and $\sigma$. It is defined as: $$\tilde{Q}_\alpha(\rho\|\sigma) = \text{Tr}\left[\left(\sigma^\gamma \rho \sigma^\gamma\right)^\alpha\right]$$ where $\gamma = \frac{1 - \alpha}{2\alpha}$. In this expression, $\rho$ and $\sigma$ represent the density matrices of the states, the matrix powers are defined via functional calculus, and $\text{Tr}$ denotes the trace.

This notation represents the sandwiched trace functional $\tilde{Q}_\alpha(\rho\|\sigma)$ for two quantum states $\rho$ and $\sigma$ and a parameter $\alpha$. It is defined as $$\tilde{Q}_\alpha(\rho\|\sigma) = \text{Tr}\left[\left(\sigma^\gamma \rho \sigma^\gamma\right)^\alpha\right]$$ where $\gamma = \frac{1 - \alpha}{2\alpha}$.

Let $\rho$ and $\sigma$ be quantum states with density matrices $M_\rho$ and $M_\sigma$ such that the kernel condition $\ker M_\sigma \subseteq \ker M_\rho$ is satisfied. For any $\alpha \in (0, 1) \cup (1, \infty)$, the sandwiched Rényi relative entropy $D_\alpha(\rho \| \sigma)$ is related to the sandwiched trace functional $\tilde{Q}_\alpha(\rho \| \sigma)$ by the identity: $$D_\alpha(\rho \| \sigma) = \frac{1}{\alpha - 1} \log \tilde{Q}_\alpha(\rho \| \sigma)$$ where the trace functional is defined as $\tilde{Q}_\alpha(\rho \| \sigma) = \text{Tr}\left[\left(M_\sigma^\gamma M_\rho M_\sigma^\gamma\right)^\alpha\right]$ with $\gamma = \frac{1 - \alpha}{2\alpha}$.

Let $\rho$ and $\sigma$ be quantum states (density matrices) on a finite-dimensional Hilbert space. For any real parameter $\alpha$, the sandwiched trace functional $\tilde{Q}_\alpha(\rho\|\sigma)$ is non-negative, that is, $\tilde{Q}_\alpha(\rho\|\sigma) \ge 0$.

Let $\rho$ and $\sigma$ be quantum states. If the kernel of $\sigma$ is contained in the kernel of $\rho$ ($\ker \sigma \subseteq \ker \rho$), which is equivalent to the condition that the support of $\rho$ is contained in the support of $\sigma$ ($\text{supp}(\rho) \subseteq \text{supp}(\sigma)$), then the sandwiched trace functional $\tilde{Q}_\alpha(\rho\|\sigma)$ is strictly positive: $$\tilde{Q}_\alpha(\rho\|\sigma) > 0.$$

Let $d$ be a finite dimension and $U$ be a $d \times d$ unitary matrix. Let $A$ and $B$ be $d \times d$ Hermitian matrices over $\mathbb{C}$. For a real parameter $\alpha$, let $\gamma = \frac{1 - \alpha}{2\alpha}$. Then the trace functional $\text{Tr}[(B^\gamma A B^\gamma)^\alpha]$ is invariant under the simultaneous unitary conjugation of $A$ and $B$, meaning: $$\text{Tr}\left[ \left( (UBU^\dagger)^\gamma (UAU^\dagger) (UBU^\dagger)^\gamma \right)^\alpha \right] = \text{Tr}\left[ \left( B^\gamma A B^\gamma \right)^\alpha \right].$$ Here, $M^\gamma$ and $M^\alpha$ denote the powers of a Hermitian matrix defined via functional calculus, and $U^\dagger$ denotes the conjugate transpose of $U$.

Let $d$ be a finite dimension. For any real number $\alpha$, any $d \times d$ unitary matrix $U \in U(d)$, and any two mixed quantum states (density matrices) $\rho$ and $\sigma$ of dimension $d$, the sandwiched trace functional $\tilde{Q}_\alpha$ is invariant under simultaneous unitary conjugation. That is, $$\tilde{Q}_\alpha(U \rho U^\dagger \| U \sigma U^\dagger) = \tilde{Q}_\alpha(\rho \| \sigma),$$ where $\tilde{Q}_\alpha(\rho\|\sigma) = \text{Tr}\left[\left(\sigma^\gamma \rho \sigma^\gamma\right)^\alpha\right]$ with $\gamma = \frac{1 - \alpha}{2\alpha}$, and $U^\dagger$ denotes the conjugate transpose of $U$.

For a real number $\alpha$, a $d \times d$ Hermitian matrix $H$, and quantum states (density matrices) $\rho$ and $\sigma$, the variational function $f_\alpha$ is defined as $$f_\alpha(H, \rho, \sigma) = \alpha \text{Tr}(\rho H) - (\alpha - 1) \text{Tr} \left[ \left( \sigma^{-\gamma} H \sigma^{-\gamma} \right)^{\frac{\alpha}{\alpha - 1}} \right]$$ where $\gamma = \frac{1 - \alpha}{2\alpha}$ and the matrix powers are defined via functional calculus. For $\alpha > 1$, this function is linear in $\rho$ and convex in $\sigma$ for a fixed $H \geq 0$, and its supremum over all $H \geq 0$ characterizes the sandwiched Rényi trace functional $\tilde{Q}_\alpha(\rho\|\sigma) = \text{Tr}\left[ \left( \sigma^\gamma \rho \sigma^\gamma \right)^\alpha \right]$.

Given a real number $\alpha$ and two quantum states (density matrices) $\rho$ and $\sigma$ acting on a $d$-dimensional Hilbert space, the Hermitian matrix $\hat{H}(\alpha, \rho, \sigma)$ is defined as $$\hat{H} = \sigma^\gamma (\sigma^\gamma \rho \sigma^\gamma)^{\alpha-1} \sigma^\gamma$$ where $\gamma = \frac{1-\alpha}{2\alpha}$ and the matrix powers are defined via the functional calculus for Hermitian matrices. This matrix serves as the optimizer in the variational formula for the sandwiched trace functional $\tilde{Q}_\alpha(\rho\|\sigma) = \text{Tr}[(\sigma^\gamma \rho \sigma^\gamma)^\alpha]$.

For any quantum states $\rho$ and $\sigma$ on a $d$-dimensional Hilbert space and a real number $\alpha$, the Hermitian matrix $\hat{H}(\alpha, \rho, \sigma) = \sigma^\gamma (\sigma^\gamma \rho \sigma^\gamma)^{\alpha-1} \sigma^\gamma$, where $\gamma = \frac{1-\alpha}{2\alpha}$, is positive semidefinite, written as $0 \le \hat{H}(\alpha, \rho, \sigma)$.

Let $\alpha > 1$ be a real number and let $\rho$ and $\sigma$ be quantum states (density matrices) on a $d$-dimensional Hilbert space. Let $\gamma = \frac{1-\alpha}{2\alpha}$ and define the optimizer matrix $\hat{H}$ as $$\hat{H} = \sigma^\gamma (\sigma^\gamma \rho \sigma^\gamma)^{\alpha-1} \sigma^\gamma.$$ The congruence transformation of $\hat{H}$ by $\sigma^{-\gamma}$ satisfies the identity: $$\sigma^{-\gamma} \hat{H} \sigma^{-\gamma} = (\sigma^\gamma \rho \sigma^\gamma)^{\alpha-1}.$$ Here, $\sigma^{-\gamma}$ is defined via the functional calculus on the support of $\sigma$, and the conjugation operation $A \mapsto B A B^\dagger$ is applied to the Hermitian matrix $\hat{H}$.

Let $\alpha > 1$ be a real number and let $\rho$ and $\sigma$ be quantum states (density matrices) acting on a $d$-dimensional Hilbert space. Define $\gamma = \frac{1-\alpha}{2\alpha}$ and the Hermitian matrix $\hat{H} = \sigma^\gamma (\sigma^\gamma \rho \sigma^\gamma)^{\alpha-1} \sigma^\gamma$. Then the Hilbert–Schmidt inner product of $\rho$ and $\hat{H}$ is equal to the sandwiched trace functional: $$\langle \rho, \hat{H} \rangle_{\mathbb{R}} = \text{Tr}[(\sigma^\gamma \rho \sigma^\gamma)^\alpha]$$ where $\langle A, B \rangle_{\mathbb{R}} = \text{Tr}(AB)$.

Let $\alpha > 1$ be a real number and let $\rho$ and $\sigma$ be quantum states (density matrices) on a $d$-dimensional Hilbert space. Let $\gamma = \frac{1-\alpha}{2\alpha}$ and define the Hermitian matrix $\hat{H}$ as $$\hat{H} = \sigma^\gamma (\sigma^\gamma \rho \sigma^\gamma)^{\alpha-1} \sigma^\gamma.$$ For the variational function $f_\alpha$ defined by $$f_\alpha(H, \rho, \sigma) = \alpha \text{Tr}(\rho H) - (\alpha - 1) \text{Tr} \left[ \left( \sigma^{-\gamma} H \sigma^{-\gamma} \right)^{\frac{\alpha}{\alpha - 1}} \right],$$ it holds that $f_\alpha$ evaluated at $\hat{H}$ equals the sandwiched trace functional $\tilde{Q}_\alpha(\rho\|\sigma)$: $$f_\alpha(\hat{H}, \rho, \sigma) = \tilde{Q}_\alpha(\rho\|\sigma) = \text{Tr}\left[\left(\sigma^\gamma \rho \sigma^\gamma\right)^\alpha\right].$$

Let $A$ be a $d \times d$ positive semidefinite Hermitian matrix over $\mathbb{C}$. For any non-zero real number $\gamma$, the product of the matrix powers $A^\gamma$ and $A^{-\gamma}$ (defined via functional calculus) is equal to the orthogonal projection onto the support of $A$, denoted $\Pi_A$: $$A^\gamma A^{-\gamma} = \Pi_A$$ where the support of $A$ is the subspace spanned by eigenvectors corresponding to non-zero eigenvalues.

Let $A$ and $B$ be $d \times d$ Hermitian matrices over $\mathbb{C}$. If the kernel of $A$ is contained in the kernel of $B$ ($\ker A \subseteq \ker B$), then $P_A B = B$, where $P_A$ is the orthogonal projector onto the support of $A$ (the subspace $(\ker A)^\perp$).

Let $\rho$ and $\sigma$ be quantum states (mixed states) on a $d$-dimensional Hilbert space, and let $H$ be a $d \times d$ Hermitian matrix. If the kernel of $\sigma$ is contained in the kernel of $\rho$ ($\ker \sigma \subseteq \ker \rho$), then for any non-zero real number $\gamma$, the Hilbert-Schmidt inner product satisfies: $$\langle \rho, H \rangle_{\mathbb{R}} = \langle \sigma^\gamma \rho \sigma^\gamma, \sigma^{-\gamma} H \sigma^{-\gamma} \rangle_{\mathbb{R}}$$ where $\sigma^\gamma$ is the real power of the density matrix $\sigma$ defined via functional calculus, and the inner product is defined as $\langle A, B \rangle_{\mathbb{R}} = \text{Tr}(AB)$.

Let $\alpha > 1$ be a real number and let $\rho$ and $\sigma$ be quantum states (density matrices) acting on a $d$-dimensional Hilbert space. Suppose the kernel of $\sigma$ is contained in the kernel of $\rho$ ($\ker \sigma \subseteq \ker \rho$), which is equivalent to the support of $\rho$ being contained in the support of $\sigma$. For any positive semidefinite Hermitian matrix $H \geq 0$, the variational function $f_\alpha$ satisfies the inequality $$f_\alpha(H, \rho, \sigma) \leq f_\alpha(\hat{H}, \rho, \sigma)$$ where the optimizer $\hat{H}$ is defined as $\hat{H} = \sigma^\gamma (\sigma^\gamma \rho \sigma^\gamma)^{\alpha-1} \sigma^\gamma$ with $\gamma = \frac{1-\alpha}{2\alpha}$. This shows that $\hat{H}$ is a maximizer for the variational formula of the sandwiched trace functional $Q_{\alpha}(\rho\|\sigma)$.

For any real number $\alpha > 1$ and quantum states (density matrices) $\rho$ and $\sigma$ such that the kernel of $\sigma$ is contained in the kernel of $\rho$ ($\ker \sigma \subseteq \ker \rho$), the sandwiched trace functional $\tilde{Q}_\alpha(\rho\|\sigma) = \text{Tr}\left[\left(\sigma^\gamma \rho \sigma^\gamma\right)^\alpha\right]$ (where $\gamma = \frac{1 - \alpha}{2\alpha}$) is equal to the supremum of the variational function $f_\alpha$ over all positive semidefinite Hermitian matrices $H \geq 0$: $$\tilde{Q}_\alpha(\rho\|\sigma) = \sup_{H \geq 0} \left( \alpha \text{Tr}(\rho H) - (\alpha - 1) \text{Tr} \left[ \left( \sigma^{-\gamma} H \sigma^{-\gamma} \right)^{\frac{\alpha}{\alpha - 1}} \right] \right).$$ The supremum is achieved by the optimizer $\hat{H} = \sigma^\gamma (\sigma^\gamma \rho \sigma^\gamma)^{\alpha-1} \sigma^\gamma$.

Let $\alpha > 1$ be a real number and $H$ be a $d \times d$ positive semidefinite Hermitian matrix ($H \geq 0$). For a fixed quantum state $\rho$, the variational function $f_\alpha(H, \rho, \sigma)$ is convex with respect to the state $\sigma$. Specifically, for any finite collection of quantum states $\{\sigma_i\}$ and weights $w_i \geq 0$ such that $\sum_i w_i = 1$, let $\sigma_{\text{mix}} = \sum_i w_i \sigma_i$. Then the function defined by $$f_\alpha(H, \rho, \sigma) = \alpha \text{Tr}(\rho H) - (\alpha - 1) \text{Tr} \left[ \left( \sigma^{-\gamma} H \sigma^{-\gamma} \right)^{\frac{\alpha}{\alpha - 1}} \right]$$ where $\gamma = \frac{1 - \alpha}{2\alpha}$, satisfies the inequality: $$f_\alpha(H, \rho, \sigma_{\text{mix}}) \leq \sum_i w_i f_\alpha(H, \rho, \sigma_i).$$ This convexity arises because for $p = \frac{\alpha}{\alpha-1} > 1$, the map $A \mapsto \text{Tr}(A^p)$ is convex on positive semidefinite matrices, and the map $\sigma \mapsto \sigma^{-\gamma} H \sigma^{-\gamma}$ is concave in $\sigma$ (by Lieb concavity, as $-\gamma \in (0, 1/2]$). Consequently, the second term of $f_\alpha$, which includes a factor of $-(\alpha - 1)$, is convex in $\sigma$.

Let $\alpha > 1$ be a real number and let $H$ be a $d \times d$ positive semidefinite Hermitian matrix. For any finite collection of weights $w_i \ge 0$ such that $\sum_i w_i = 1$, and any sequences of density matrices (quantum states) $\rho_i$ and $\sigma_i$, let $\rho_{\text{mix}} = \sum_i w_i \rho_i$ and $\sigma_{\text{mix}} = \sum_i w_i \sigma_i$ be their respective convex combinations. Then the variational function $f_\alpha$ satisfies the joint convexity inequality: $$f_\alpha(H, \rho_{\text{mix}}, \sigma_{\text{mix}}) \le \sum_i w_i f_\alpha(H, \rho_i, \sigma_i)$$ where $f_\alpha(H, \rho, \sigma) = \alpha \text{Tr}(\rho H) - (\alpha - 1) \text{Tr} \left[ \left( \sigma^{-\gamma} H \sigma^{-\gamma} \right)^{\frac{\alpha}{\alpha - 1}} \right]$ with $\gamma = \frac{1 - \alpha}{2\alpha}$.

Let $\alpha > 1$ be a real number and let $\rho$ and $\sigma$ be quantum states (density matrices) represented as $d \times d$ complex Hermitian matrices. If the kernel of $\sigma$ is contained in the kernel of $\rho$ (i.e., $\ker \sigma \subseteq \ker \rho$), then the variational function $f_\alpha$ defined as $$f_\alpha(H, \rho, \sigma) = \alpha \text{Tr}(\rho H) - (\alpha - 1) \text{Tr} \left[ \left( \sigma^{-\gamma} H \sigma^{-\gamma} \right)^{\frac{\alpha}{\alpha - 1}} \right]$$ is bounded above for all positive semidefinite $d \times d$ complex Hermitian matrices $H \ge 0$, where $\gamma = \frac{1 - \alpha}{2\alpha}$ and the matrix powers are defined via functional calculus.

Let $\alpha$ be a real number such that $\alpha > 1$. Let $\iota$ be a finite index set, and let $\{w_i\}_{i \in \iota}$ be a collection of non-negative weights such that $\sum_{i \in \iota} w_i = 1$. Let $\{\rho_i\}_{i \in \iota}$ and $\{\sigma_i\}_{i \in \iota}$ be families of quantum states (represented by density matrices of dimension $d$), and let $\rho_{\text{mix}} = \sum_{i \in \iota} w_i \rho_i$ and $\sigma_{\text{mix}} = \sum_{i \in \iota} w_i \sigma_i$ be their respective convex combinations. If for every $i \in \iota$, the kernel of $\sigma_i$ is contained in the kernel of $\rho_i$ ($\ker \sigma_i \subseteq \ker \rho_i$), then the supremum of the variational function $f_\alpha$ over positive semidefinite matrices $H$ satisfies: $$\sup_{H \ge 0} f_\alpha(H, \rho_{\text{mix}}, \sigma_{\text{mix}}) \le \sum_{i \in \iota} w_i \left( \sup_{H \ge 0} f_\alpha(H, \rho_i, \sigma_i) \right)$$ where $H$ ranges over all $d \times d$ complex positive semidefinite Hermitian matrices.

Let $\{w_i\}_{i \in \iota}$ be a collection of non-negative weights indexed by a finite set $\iota$. Let $\{\rho_i\}_{i \in \iota}$ and $\{\sigma_i\}_{i \in \iota}$ be families of $d \times d$ complex positive semidefinite Hermitian matrices. If for every $i \in \iota$, the kernel of $\sigma_i$ is contained in the kernel of $\rho_i$ ($\ker \sigma_i \subseteq \ker \rho_i$), then the kernel of the weighted sum of $\sigma_i$ is contained in the kernel of the weighted sum of $\rho_i$: $$\ker \left( \sum_{i \in \iota} w_i \sigma_i \right) \subseteq \ker \left( \sum_{i \in \iota} w_i \rho_i \right)$$

Let $\alpha$ be a real number such that $\alpha > 1$. Let $\{w_i\}_{i \in \iota}$ be a finite collection of non-negative weights that sum to 1. For each $i \in \iota$, let $\rho_i$ and $\sigma_i$ be quantum states (density matrices) such that the kernel of $\sigma_i$ is contained in the kernel of $\rho_i$ ($\ker \sigma_i \subseteq \ker \rho_i$). Let $\rho_{\text{mix}} = \sum_{i \in \iota} w_i \rho_i$ and $\sigma_{\text{mix}} = \sum_{i \in \iota} w_i \sigma_i$ be the weighted averages of these states. Then the sandwiched trace functional $\tilde{Q}_\alpha$, defined as $\tilde{Q}_\alpha(\rho\|\sigma) = \text{Tr}\left[\left(\sigma^\gamma \rho \sigma^\gamma\right)^\alpha\right]$ with $\gamma = \frac{1 - \alpha}{2\alpha}$, satisfies the joint convexity property: $$\tilde{Q}_\alpha(\rho_{\text{mix}} \| \sigma_{\text{mix}}) \le \sum_{i \in \iota} w_i \tilde{Q}_\alpha(\rho_i \| \sigma_i)$$

Let $\rho_1, \sigma_1$ be quantum mixed states on a Hilbert space $\mathcal{H}_A$ and $\rho_2, \sigma_2$ be quantum mixed states on a Hilbert space $\mathcal{H}_B$. The sandwiched trace functional $\tilde{Q}_\alpha$ satisfies the following multiplicativity property with respect to the tensor product: $$\tilde{Q}_\alpha(\rho_1 \otimes \rho_2 \| \sigma_1 \otimes \sigma_2) = \tilde{Q}_\alpha(\rho_1 \| \sigma_1) \cdot \tilde{Q}_\alpha(\rho_2 \| \sigma_2)$$ where $\tilde{Q}_\alpha(\rho \| \sigma) = \text{Tr}\left[\left(\sigma^\gamma \rho \sigma^\gamma\right)^\alpha\right]$ and $\gamma = \frac{1 - \alpha}{2\alpha}$.

For any real number $\alpha > 0$ and any quantum state $\rho$, the sandwiched trace functional $\tilde{Q}_\alpha(\rho\|\rho)$ is equal to $1$.

For any real number $\alpha > 0$, mixed states $\rho$ and $\sigma$ on a system $A$, and a mixed state $\tau$ on a system $B$, the sandwiched trace functional satisfies $$\tilde{Q}_\alpha(\rho \otimes \tau \| \sigma \otimes \tau) = \tilde{Q}_\alpha(\rho \| \sigma)$$ where $\otimes$ denotes the tensor product of mixed states.

Given a bipartite mixed state $\rho$ on the composite system $\mathcal{H}_A \otimes \mathcal{H}_B$ and a unitary matrix $V \in \text{U}(d_B)$ acting on the second subsystem, this definition produces the mixed state obtained by conjugating $\rho$ with the tensor product of the identity operator $I_A$ and the unitary $V$. Specifically, the resulting state is $(I_A \otimes V) \rho (I_A \otimes V)^\dagger$.

For a bipartite mixed state $\rho$ on the composite system $\mathcal{H}_A \otimes \mathcal{H}_B$ and a unitary matrix $V \in \text{U}(d_B)$ acting on the second subsystem, the matrix representing the conjugated state $(I_A \otimes V) \rho (I_A \otimes V)^\dagger$ is equal to the congruence transformation $(I_A \otimes V) M_\rho (I_A \otimes V)^\dagger$, where $M_\rho$ is the Hermitian matrix associated with $\rho$ and $I_A$ is the identity matrix on $\mathcal{H}_A$.

For any bipartite mixed states $\rho$ and $\sigma$ on the composite system $\mathcal{H}_A \otimes \mathcal{H}_B$ and any unitary matrix $V \in \text{U}(d_B)$ acting on the second subsystem, the sandwiched trace functional $\tilde{Q}_\alpha$ is invariant under simultaneous conjugation by $I_A \otimes V$: $$\tilde{Q}_\alpha((I_A \otimes V) \rho (I_A \otimes V)^\dagger \| (I_A \otimes V) \sigma (I_A \otimes V)^\dagger) = \tilde{Q}_\alpha(\rho \| \sigma)$$ where $I_A$ is the identity matrix on $\mathcal{H}_A$.

Let $M$ be a complex matrix of size $(d_A \cdot d_B) \times (d_A \cdot d_B)$ and $V$ be a complex matrix of size $d_B \times d_B$. For any indices $a_1, a_2 \in \{1, \dots, d_A\}$ and $b_1, b_2 \in \{1, \dots, d_B\}$, the $((a_1, b_1), (a_2, b_2))$-th entry of the matrix $(I_A \otimes V) M (I_A \otimes V)^\dagger$ is given by $$\left((I_A \otimes V) M (I_A \otimes V)^\dagger\right)_{(a_1, b_1), (a_2, b_2)} = \left(V M_{a_1, a_2} V^\dagger\right)_{b_1, b_2}$$ where $I_A$ is the identity matrix of size $d_A \times d_A$, $\otimes$ denotes the Kronecker product, $V^\dagger$ is the conjugate transpose of $V$, and $M_{a_1, a_2}$ is a $d_B \times d_B$ matrix defined by $(M_{a_1, a_2})_{b_1', b_2'} = M_{(a_1, b_1'), (a_2, b_2')}$.

Let $\kappa$ be a finite index set and $d_B$ be the dimension of a complex Hilbert space. Let $\{V_i\}_{i \in \kappa}$ be a collection of unitary matrices of size $d_B \times d_B$ such that for any Hermitian matrix $X$, the average over the unitaries satisfies $$\frac{1}{|\kappa|} \sum_{i \in \kappa} V_i X V_i^\dagger = \frac{\text{Tr}(X)}{d_B} I,$$ where $I$ is the identity matrix. Then, for any Hermitian matrix $X$ and indices $b_1, b_2 \in \{1, \dots, d_B\}$, the $(b_1, b_2)$-th entry of the sum of the transformed matrices is given by $$\sum_{i \in \kappa} (V_i X V_i^\dagger)_{b_1, b_2} = \frac{\text{Tr}(X)}{d_B} \cdot |\kappa| \cdot \delta_{b_1, b_2},$$ where $\text{Tr}(X)$ is the trace of $X$ and $\delta_{b_1, b_2}$ is the Kronecker delta (which is $1$ if $b_1 = b_2$ and $0$ otherwise).

Let $\kappa$ be a finite index set and $\{V_i\}_{i \in \kappa}$ be a collection of $d_B \times d_B$ unitary matrices. Suppose that for every Hermitian matrix $H \in \text{Herm}_{d_B}(\mathbb{C})$, the average over the group action satisfies: $$\frac{1}{|\kappa|} \sum_{i \in \kappa} V_i H V_i^\dagger = \frac{\text{Tr}(H)}{d_B} I,$$ where $I$ is the identity matrix and $d_B$ is the dimension of the system. Then, for any arbitrary complex matrix $X \in M_{d_B}(\mathbb{C})$, the $(b_1, b_2)$-th entry of the sum of the transformed matrices is given by: $$\sum_{i \in \kappa} \left( V_i X V_i^\dagger \right)_{b_1, b_2} = \frac{\text{Tr}(X)}{d_B} |\kappa| \delta_{b_1, b_2},$$ where $\delta_{b_1, b_2}$ is the Kronecker delta (equal to $1$ if $b_1 = b_2$ and $0$ otherwise).

Given a bipartite mixed state $\rho$ on a system with dimensions $d_A \times d_B$ and a unitary matrix $V$ in the unitary group $\mathbf{U}(d_B)$, this function returns the new mixed state obtained by conjugating $\rho$ by the operator $I_A \otimes V$. Mathematically, the resulting state is: $$(I_A \otimes V) \rho (I_A \otimes V)^\dagger$$ where $I_A$ is the identity matrix on the first subsystem, $\otimes$ denotes the Kronecker product, and $\dagger$ denotes the conjugate transpose.