QuantumInfo.Finite.Entropy.Relative

Let $w_j$ be a set of non-negative weights such that $\sum_j w_j = 1$, and let $b_j$ be non-negative real numbers. For any real number $q \geq 1$, the following weighted Jensen inequality holds: $$\left( \sum_j w_j b_j \right)^q \le \sum_j w_j b_j^q$$

Let $a_i$ and $b_j$ be non-negative real numbers for $i, j \in \{1, \dots, d\}$. Let $w_{ij}$ be a doubly stochastic matrix, such that $w_{ij} \geq 0$ for all $i, j$, and its rows and columns sum to 1 ($\sum_{j} w_{ij} = 1$ for all $i$, and $\sum_{i} w_{ij} = 1$ for all $j$). For any conjugate exponents $p, q > 1$ satisfying $\frac{1}{p} + \frac{1}{q} = 1$, the following inequality holds: $$\sum_{i,j} a_i b_j w_{ij} \le \left( \sum_{i} a_i^p \right)^{1/p} \left( \sum_{j} b_j^q \right)^{1/q}$$

Let $A$ and $B$ be complex Hermitian matrices of dimension $d$ that are positive semi-definite ($A \ge 0, B \ge 0$). Let $p$ and $q$ be real conjugate exponents such that $p > 1$ and $\frac{1}{p} + \frac{1}{q} = 1$. Then the real-valued Hilbert-Schmidt inner product of $A$ and $B$, denoted $\langle A, B \rangle_{\mathbb{R}}$, satisfies: $$\langle A, B \rangle_{\mathbb{R}} \le (\text{Tr}[A^p])^{1/p} (\text{Tr}[B^q])^{1/q}$$

Let $\sigma$ be a quantum mixed state with associated matrix $M$. For any real number $r > 0$, the $r$-th power of the matrix $M$ satisfies $M^r \le \mathbb{I}$, where $\mathbb{I}$ denotes the identity matrix.

Let $A$ be a Hermitian matrix in $\mathbb{C}^{d \times d}$. If $A \le I$ (where $I$ is the identity matrix), then for every index $i$ in the dimension type $d$, the $i$-th eigenvalue of $A$ satisfies $\lambda_i(A) \le 1$. Combined with the condition $A \ge 0$, this implies that all eigenvalues of $A$ lie in the interval $[0, 1]$.

Let $A$ be a complex Hermitian matrix of dimension $d$ such that $0 \le A \le I$, where $I$ is the identity matrix. For any real number $p \ge 1$, the trace of the $p$-th power of $A$ is less than or equal to the trace of $A$, i.e., $\text{Tr}(A^p) \le \text{Tr}(A)$.

Let $A$ be a positive semi-definite (PSD) Hermitian matrix of dimension $d \times d$ over the complex numbers $\mathbb{C}$. For any non-zero real number $p$, the product of the matrix power $A^{-p}$ and $A^p$ is equal to the orthogonal projection onto the support of $A$, denoted as $P_{\text{supp}(A)}$.

For any complex Hermitian matrix $A \in \mathbb{C}^{d \times d}$, the product of its support projection $P_A$ and the matrix itself is equal to $A$, i.e., $P_A A = A$.

For any complex Hermitian matrix $A$, the inner product of $A$ with its support projection $P_A$ is equal to the trace of $A$. Here, $\langle A, P_A \rangle$ denotes the Frobenius inner product (or Hilbert-Schmidt inner product), and $A.trace$ denotes the sum of the diagonal elements of $A$.

Let $A$ and $B$ be Hermitian matrices of dimension $d$ over the complex numbers $\mathbb{C}$. If the kernel of the linear map associated with $B$ is a subspace of the kernel of the linear map associated with $A$ (i.e., $\ker B \subseteq \ker A$), then the product of $A$ and the support projection of $B$ (denoted $\Pi_B$) satisfies $A \Pi_B = A$.

Let $A$ and $B$ be complex Hermitian matrices of dimension $d$. If the kernel of $B$ is contained in the kernel of $A$ (i.e., $\ker B \subseteq \ker A$), then the Frobenius inner product of $A$ and the support projection of $B$ (denoted $\Pi_B$) is equal to the trace of $A$, such that $\langle A, \Pi_B \rangle = \text{Tr}(A)$.

Let $\rho$ and $\sigma$ be density matrices. If the kernel of the linear map associated with $\sigma$ is contained within the kernel of the linear map associated with $\rho$ (denoted $\ker \sigma \subseteq \ker \rho$), then the real inner product of the support projection of $\sigma$ and the density matrix $\rho$ is equal to 1, i.e., $\langle \Pi_{\sigma}, \rho \rangle_{\mathbb{R}} = 1$.

Let $\rho$ and $\sigma$ be density operators (represented by their matrices $\rho.M$ and $\sigma.M$). If the kernel of $\sigma.M$ is contained in the kernel of $\rho.M$ ($\ker(\sigma.M) \subseteq \ker(\rho.M)$), then the quantity $\langle \rho.M, \log(\rho.M) - \log(\sigma.M) \rangle$ is non-negative, i.e., $0 \leq \langle \rho.M, \log \rho.M - \log \sigma.M \rangle$. Here $\langle A, B \rangle$ denotes the Hilbert-Schmidt inner product $\text{Tr}(A^\dagger B)$.

Let $\rho$ and $\sigma$ be density states with associated matrices $\rho.M$ and $\sigma.M$. Suppose the kernel of $\sigma.M$ is contained in the kernel of $\rho.M$ ($\ker \sigma.M \subseteq \ker \rho.M$). For any $\alpha \in \mathbb{R}$ such that $\alpha > 0$, the sandwiched Rényi relative entropy $D_{\alpha}(\rho \| \sigma)$ is non-negative ($0 \le D_{\alpha}(\rho \| \sigma)$), where $D_{\alpha}(\rho \| \sigma)$ is defined as: - If $\alpha = 1$, the relative entropy is given by the inner product $\langle \rho.M, \log \rho.M - \log \sigma.M \rangle$. - If $\alpha \neq 1$, it is given by $\frac{1}{\alpha - 1} \log \text{Tr}\left[ \left( \sigma.M^{\frac{1-\alpha}{2\alpha}} \rho.M \sigma.M^{\frac{1-\alpha}{2\alpha}} \right)^\alpha \right]$.

Let $A, C$ be square matrices of size $d_1 \times d_1$ over $\mathbb{C}$, and let $B, D$ be square matrices of size $d_2 \times d_2$ over $\mathbb{C}$. If the kernel of $A$ is contained in the kernel of $C$ ($\ker A \subseteq \ker C$) and the kernel of $B$ is contained in the kernel of $D$ ($\ker B \subseteq \ker D$), then the kernel of the Kronecker product $A \otimes B$ is contained in the kernel of the Kronecker product $C \otimes D$ ($\ker(A \otimes B) \subseteq \ker(C \otimes D)$). Here, $\ker M$ refers to the kernel of the linear map associated with a matrix $M$.

Let $\rho_1, \sigma_1$ be quantum mixed states of dimension $d_1$, and $\rho_2, \sigma_2$ be quantum mixed states of dimension $d_2$. If the kernel of the Hermitian operator associated with the product state $\sigma_1 \otimes \sigma_2$ is contained in the kernel of the Hermitian operator associated with the product state $\rho_1 \otimes \rho_2$ (i.e., $\ker(\sigma_1 \otimes \sigma_2).M \subseteq \ker(\rho_1 \otimes \rho_2).M$), then the kernel of $\sigma_1$ is contained in the kernel of $\rho_1$ ($\ker \sigma_1.M \subseteq \ker \rho_1.M$).

Let $\rho_1, \sigma_1$ be quantum mixed states of dimension $d_1$ and $\rho_2, \sigma_2$ be quantum mixed states of dimension $d_2$. If the kernel of the Hermitian operator of the product state $\sigma_1 \otimes \sigma_2$ is contained in the kernel of the Hermitian operator of the product state $\rho_1 \otimes \rho_2$ (i.e., $\ker(M_{\sigma_1 \otimes \sigma_2}) \subseteq \ker(M_{\rho_1 \otimes \rho_2})$), then the kernel of the Hermitian operator of $\sigma_2$ is contained in the kernel of the Hermitian operator of $\rho_2$ (i.e., $\ker M_{\sigma_2} \subseteq \ker M_{\rho_2}$).

Let $\rho_1, \sigma_1$ be mixed states on a Hilbert space of dimension $d_1$, and let $\rho_2, \sigma_2$ be mixed states on a Hilbert space of dimension $d_2$. The kernel of the density matrix of the product state $\sigma_1 \otimes \sigma_2$ is contained in the kernel of the density matrix of the product state $\rho_1 \otimes \rho_2$ (i.e., $\ker(\sigma_1 \otimes \sigma_2) \subseteq \ker(\rho_1 \otimes \rho_2)$) if and only if $\ker \sigma_1 \subseteq \ker \rho_1$ and $\ker \sigma_2 \subseteq \ker \rho_2$.

Let $A$ and $C$ be complex Hermitian matrices of dimension $d_1$, and let $B$ and $D$ be complex Hermitian matrices of dimension $d_2$. Then the Frobenius inner product of the Kronecker products $A \otimes B$ and $C \otimes D$ is equal to the product of the individual inner products, namely: $$\langle A \otimes B, C \otimes D \rangle = \langle A, C \rangle \langle B, D \rangle$$

Let $K \subset \mathbb{R}$ be a compact set. The function that maps a real exponent $r$ to the power function $t \mapsto t^r$ (viewed as an element of the space of functions on $K$ endowed with the topology of uniform convergence) is continuous on the interval $(0, \infty)$.

Let $\rho_1, \sigma_1$ be mixed states of dimension $d_1$ and $\rho_2, \sigma_2$ be mixed states of dimension $d_2$. Let $M_{\rho_1}, M_{\sigma_1}, M_{\rho_2}, M_{\sigma_2}$ denote their respective density matrices, and let $\otimes^M$ denote the tensor product of mixed states. Suppose that the kernel of $M_{\sigma_1}$ is a subspace of the kernel of $M_{\rho_1}$ ($\ker M_{\sigma_1} \subseteq \ker M_{\rho_1}$) and the kernel of $M_{\sigma_2}$ is a subspace of the kernel of $M_{\rho_2}$ ($\ker M_{\sigma_2} \subseteq \ker M_{\rho_2}$). Then the following additivity property for the Frobenius inner product of the logarithm of the states holds: $$\langle M_{\rho_1 \otimes^M \rho_2}, \log(M_{\rho_1 \otimes^M \rho_2}) - \log(M_{\sigma_1 \otimes^M \sigma_2}) \rangle = \langle M_{\rho_1}, \log M_{\rho_1} - \log M_{\sigma_1} \rangle_{\mathbb{R}} + \langle M_{\rho_2}, \log M_{\rho_2} - \log M_{\sigma_2} \rangle$$ where $\log$ denotes the functional calculus for Hermitian matrices.

Let $\rho$ and $\sigma$ be quantum states (represented by the type `MState d`) with associated density matrices $M_\rho$ and $M_\sigma$. Given a parameter $\alpha \in \mathbb{R}$, the Sandwiched Rényi Relative Entropy $D_\alpha(\rho \| \sigma)$ is defined as an extended non-negative real number as follows: 1. If $\alpha \le 0$, the value is $0$. 2. If $\alpha > 0$ and the kernel condition $\ker M_\sigma \le \ker M_\rho$ is not satisfied, the value is $\infty$. 3. If $\alpha = 1$ and $\ker M_\sigma \le \ker M_\rho$, it returns the standard relative entropy (Kullback-Leibler divergence) calculated using the natural logarithm: $$D_1(\rho \| \sigma) = \text{Tr}(M_\rho (\log M_\rho - \log M_\sigma))$$ 4. If $\alpha \in (0, 1) \cup (1, \infty)$ and $\ker M_\sigma \le \ker M_\rho$, it is given by: $$D_\alpha(\rho \| \sigma) = \frac{1}{\alpha - 1} \log \text{Tr} \left[ \left( M_\sigma^{\frac{1-\alpha}{2\alpha}} M_\rho M_\sigma^{\frac{1-\alpha}{2\alpha}} \right)^\alpha \right]$$ where $\text{Tr}$ denotes the trace and the power of the matrices is defined via functional calculus.

This notation defines the sandwiched Rényi relative entropy of order $\alpha$ between two mixed states $\rho$ and $\sigma$, denoted as $\tilde{D}_\alpha(\rho \| \sigma)$. It is defined for $\alpha \in \mathbb{R}$ and maps a pair of mixed states $\rho, \sigma$ to an extended non-negative real number $[0, \infty]$.

The quantum relative entropy, also known as the Umegaki relative entropy, is defined for two quantum states $\rho$ and $\sigma$ (elements of the space of mixed states $\text{MState } d$) as: $$D(\rho \parallel \sigma) = \text{Tr}[\rho (\log \rho - \log \sigma)]$$ The value is represented as an extended non-negative real number ($\text{ENNReal}$). In this library, it is implemented as a special case of the sandwiched Rényi relative entropy $\tilde{D}_\alpha(\rho \parallel \sigma)$ evaluated at the limit $\alpha \to 1$.

This represents the mathematical notation $\mathbf{D}(\rho \parallel \sigma)$, which denotes the quantum relative entropy between two quantum states $\rho$ and $\sigma$. It is defined formally as the value of the function `qRelativeEnt ρ σ`, which calculates the Umegaki relative entropy $D(\rho \parallel \sigma) = \text{Tr}[\rho (\log \rho - \log \sigma)]$.

Let $\rho_1, \sigma_1$ be mixed states of dimension $d_1$ and $\rho_2, \sigma_2$ be mixed states of dimension $d_2$. For any real numbers $\alpha$ and $\beta$, the trace of the sandwiched term of the combined states satisfies the following factorization property: $$\operatorname{Tr}\left[ \left( (\rho_1 \otimes \rho_2) \cdot (\sigma_1 \otimes \sigma_2)^\beta \cdot (\rho_1 \otimes \rho_2)^\dagger \right)^\alpha \right] = \operatorname{Tr}\left[ \left( \rho_1 \cdot \sigma_1^\beta \cdot \rho_1^\dagger \right)^\alpha \right] \cdot \operatorname{Tr}\left[ \left( \rho_2 \cdot \sigma_2^\beta \cdot \rho_2^\dagger \right)^\alpha \right]$$ where $M.conj$ denotes the conjugation operation $A \cdot B \cdot A^\dagger$, $\otimes$ is the tensor product of mixed states, and $M$ is the density matrix associated with a mixed state.

For any real parameter $\alpha \neq 1$ and any quantum mixed states $\rho_1, \sigma_1$ on a Hilbert space of dimension $d_1$ and $\rho_2, \sigma_2$ on a Hilbert space of dimension $d_2$, the sandwiched Rényi relative entropy $\tilde{D}_\alpha$ is additive under the tensor product. That is, $$\tilde{D}_\alpha(\rho_1 \otimes \rho_2 \parallel \sigma_1 \otimes \sigma_2) = \tilde{D}_\alpha(\rho_1 \parallel \sigma_1) + \tilde{D}_\alpha(\rho_2 \parallel \sigma_2),$$ where $\otimes$ denotes the tensor product of the mixed states.

For any Renyi parameter $\alpha$ and mixed states $\rho_1, \sigma_1$ on a system of dimension $d_1$ and $\rho_2, \sigma_2$ on a system of dimension $d_2$, the sandwiched Rényi relative entropy $\tilde{D}_\alpha$ is additive under the tensor product of states: $$\tilde{D}_\alpha(\rho_1 \otimes \rho_2 \,\|\, \sigma_1 \otimes \sigma_2) = \tilde{D}_\alpha(\rho_1 \,\|\, \sigma_1) + \tilde{D}_\alpha(\rho_2 \,\|\, \sigma_2)$$ where $\otimes$ denotes the tensor product of mixed states.

For any quantum mixed states $\rho_1, \sigma_1$ on a Hilbert space of dimension $d_1$, and any quantum mixed states $\rho_2, \sigma_2$ on a Hilbert space of dimension $d_2$, the quantum relative entropy $D$ is additive under the tensor product of states. Specifically, $$D(\rho_1 \otimes \rho_2 \parallel \sigma_1 \otimes \sigma_2) = D(\rho_1 \parallel \sigma_1) + D(\rho_2 \parallel \sigma_2)$$ where $\otimes$ denotes the tensor product of the mixed states.

For any quantum mixed states $\rho, \sigma$ defined on a finite-dimensional Hilbert space with basis $d$, and for any bijection $e: d_2 \simeq d$ between basis sets, the sandwiched Rényi relative entropy of order $\alpha$ is invariant under relabeling, such that $\tilde{D}_\alpha(\rho' \| \sigma') = \tilde{D}_\alpha(\rho \| \sigma)$, where $\rho'$ and $\sigma'$ are the states relabeled by $e$.

For any quantum state $\rho$ and any real parameter $\alpha > 0$, the sandwiched Rényi relative entropy of the state with itself, denoted as $\tilde{D}_{\alpha}(\rho \parallel \rho)$, is equal to 0.

Let $\rho$ and $\sigma$ be quantum mixed states on a finite-dimensional Hilbert space. If the density matrix of $\sigma$ (denoted $\sigma.M$) is non-singular (invertible), then for any real order $\alpha$, the sandwiched Rényi relative entropy $\tilde{D}_{\alpha}(\rho \| \sigma)$ is not equal to positive infinity ($\infty$).

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(\alpha) = \frac{1 - \alpha}{2\alpha}$ is continuous on the interval $(0, \infty)$.

For any complex number $z \in \mathbb{C}$, the function $f: \mathbb{R} \to \mathbb{C}$ defined by $f(r) = z^r$ is continuous on the interval $(0, \infty)$. Here, $z^r$ denotes the complex power function, and $(0, \infty)$ is the set of positive real numbers.

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $\alpha \mapsto \frac{1 - \alpha}{2\alpha}$ maps the interval $(1, \infty)$ into the interval $(-\infty, 0)$.

Let $X$ be a topological space that satisfies the $T_1$ separation axiom and is a perfect space (meaning it has no isolated points). For any point $p \in X$, the frontier (boundary) of the singleton set $\{p\}$ is equal to the set $\{p\}$ itself.

Let $\rho$ and $\sigma$ be mixed states of a $d$-dimensional Hilbert space ($MState\, d$). The sandwiched Rényi relative entropy $\tilde{D}_\alpha(\rho\|\sigma)$, considered as a function of the order parameter $\alpha$, is continuous on the interval $(0, \infty)$.

For any two quantum states $\rho$ and $\sigma$ in a $d$-dimensional Hilbert space, if the kernel of the density operator $\sigma.M$ is contained within the kernel of the density operator $\rho.M$ (denoted $\ker \sigma \subseteq \ker \rho$), then the quantum relative entropy $D(\rho \parallel \sigma)$ is given by the inner product $\langle \rho, \log \rho - \log \sigma \rangle$. In the context of operators, this corresponds to the trace formula $\text{Tr}[\rho(\log \rho - \log \sigma)]$.

For any two quantum mixed states $\rho$ and $\sigma$ of dimension $d$, the quantum relative entropy $D(\rho \parallel \sigma)$ relates to the von Neumann entropy $S_{vn}(\rho)$ as follows: $$D(\rho \parallel \sigma) = -S_{vn}(\rho) + \begin{cases} -\langle \rho, \log \sigma \rangle_{HS} & \text{if } \ker \sigma \subseteq \ker \rho \\ \infty & \text{otherwise} \end{cases}$$ where $\langle \cdot, \cdot \rangle_{HS}$ denotes the Hilbert-Schmidt inner product and $\ker \sigma \subseteq \ker \rho$ indicates that the kernel of the density operator $\sigma$ is contained within the kernel of $\rho$.

Let $\rho$ and $\sigma$ be quantum mixed states on a system with basis $d$. For any bijection (equivalence) $e : d_2 \simeq d$, let $\rho.\text{relabel } e$ and $\sigma.\text{relabel } e$ be the mixed states on the system with basis $d_2$ obtained by reindexing the density matrices. Then the quantum relative entropy $D(\rho \parallel \sigma)$ is invariant under such relabeling, i.e., $D(\rho.\text{relabel } e \parallel \sigma.\text{relabel } e) = D(\rho \parallel \sigma)$.

Let $d$ be a type with exactly one element (representing a 1-dimensional Hilbert space). For any two mixed states $\rho$ and $\sigma$ in $\text{MState } d$, the sandwiched Rényi relative entropy of order $\alpha$, denoted $D̃_{\alpha}(\rho‖\sigma)$, is equal to 0.

Let $d$ be a type with a unique element (i.e., a singleton type). For any two mixed states $\rho$ and $\sigma$ defined over $d$, the quantum relative entropy $\mathbf{D}(\rho\|\sigma)$ is equal to $0$.

Let $d_1$ and $d_2$ be finite types with decidable equality, representing the index sets of two quantum systems. Let $\rho_1, \sigma_1$ be mixed states on the system indexed by $d_1$, and $\rho_2, \sigma_2$ be mixed states on the system indexed by $d_2$. If the index sets are equal ($d_1 = d_2$) and the states are heterogeneously equal (denoted by $\rho_1 \cong \rho_2$ and $\sigma_1 \cong \sigma_2$), then their sandwiched Rényi relative entropies of order $\alpha$ are equal, i.e., $\tilde{D}_\alpha(\rho_1 \| \sigma_1) = \tilde{D}_\alpha(\rho_2 \| \sigma_2)$.

Given two finite dimensional quantum systems with basis types $d_1$ and $d_2$ and states $\rho_1, \sigma_1$ on $d_1$ and $\rho_2, \sigma_2$ on $d_2$, if there is an equality between the basis types $d_1 = d_2$ such that the states are related by the corresponding relabeling ($\rho_1 = \text{relabel}(\rho_2)$ and $\sigma_1 = \text{relabel}(\sigma_2)$), then the sandwiched Rényi relative entropies are equal: $\tilde{D}_\alpha(\rho_1 \| \sigma_1) = \tilde{D}_\alpha(\rho_2 \| \sigma_2)$.

Let $d_1$ and $d_2$ be finite types. Let $\rho_1, \sigma_1$ be mixed states (of type `MState`) acting on a Hilbert space indexed by $d_1$, and $\rho_2, \sigma_2$ be mixed states acting on a Hilbert space indexed by $d_2$. If the underlying types are equal ($d_1 = d_2$) and the states are heterogeneous equal (denoted by $\rho_1 \cong \rho_2$ and $\sigma_1 \cong \sigma_2$), then the quantum relative entropies are equal: $D(\rho_1 \Vert \sigma_1) = D(\rho_2 \Vert \sigma_2)$.

Let $\rho$ and $\sigma$ be quantum states (represented as density matrices in $MState \ d$). If $\sigma$ is non-singular (i.e., it has full rank), then the quantum relative entropy $D(\rho \| \sigma)$ is given by the inner product $$\langle \rho, \log \rho - \log \sigma \rangle$$ where $\log$ denotes the matrix logarithm and $\langle A, B \rangle = \text{Tr}(A^* B)$ is the Hilbert-Schmidt inner product.

Let $\rho$ and $\sigma$ be mixed states in $\text{MState } d$, where $M$ denotes the underlying density matrix. Suppose the kernel of $\sigma$ is contained in the kernel of $\rho$ ($\ker \sigma \subseteq \ker \rho$). If the Hilbert-Schmidt inner product of $\rho$ and the matrix logarithm of $\sigma$ is strictly less than some real value $y$ (i.e., $\langle \rho, \log \sigma \rangle < y$), then for all mixed states $x$ in a sufficiently small neighborhood of $\sigma$, if $\ker x \subseteq \ker \rho$, then $\langle \rho, \log x \rangle < y$.

Let $\rho$ and $x$ be mixed states in $\text{MState } d$ such that the kernel of the matrix $x$ is not contained in the kernel of the matrix $\rho$ ($\ker x \not\le \ker \rho$). For any value $y \in [0, \infty]$, if $y < \infty$, then there exists a neighborhood of $x$ such that for all mixed states $x'$ in that neighborhood, the value $y$ is strictly less than the quantity defined by: $$\begin{cases} \langle \rho, \log \rho - \log x' \rangle & \text{if } \ker x' \le \ker \rho \\ \infty & \text{otherwise} \end{cases}$$ where the expression is treated as an element of the extended real numbers $\overline{\mathbb{R}}$.

For any fixed mixed state $\rho$ of dimension $d$, the quantum relative entropy $\mathbf{D}(\rho \| \sigma)$ is lower semicontinuous as a function of the mixed state $\sigma$. That is, for every $\sigma \in \text{MState } d$, and for any $y < \mathbf{D}(\rho \| \sigma)$, there exists a neighborhood of $\sigma$ such that for all $\sigma'$ in that neighborhood, $y < \mathbf{D}(\rho \| \sigma')$.