Wigner–Yanase Skew Information

• Wigner–Yanase skew information is a measure quantifying the noncommutativity between a quantum state’s square root and a conserved observable, revealing intrinsic quantum uncertainty.
• It underpins quantum coherence, Fisher information, and resource theories by establishing optimal bounds on quantum variance and parameter estimation.
• Its lattice structure among operator monotone functions facilitates rigorous derivation of multi-operator uncertainty relations and enhances metrological precision.

The Wigner–Yanase skew information is a foundational concept in quantum information theory characterizing the noncommutativity of a quantum state with respect to a conserved observable. It formalizes quantum uncertainty originating from incompatibility between the state and the observable and serves as a core ingredient in understanding quantum coherence, quantum Fisher information, resource theories of asymmetry, information-based uncertainty relations, and quantification of nonclassical correlations.

1. Definition and Mathematical Structure

The Wigner–Yanase skew information for a quantum state $\rho$ (a positive definite density matrix) and a (possibly non-selfadjoint) observable $A$ is defined as: $$I_\rho(A) = -\frac{1}{2} \operatorname{Tr}\left([\sqrt{\rho}, A]^2\right),$$ where $[\cdot,\cdot]$ denotes the commutator. For self-adjoint $A$ and $\rho$, this expression quantifies the degree of noncommutativity of $\sqrt{\rho}$ and $A$.

This notion is generalized within the framework of metric-adjusted skew informations: $$I^c_\rho(A) = \frac{m(c)}{2} \, i[\rho,A^*] \, c(L_\rho,R_\rho) \, i[\rho,A],$$ where $c(x,y)$ is a Morozova–Chentsov function associated to an operator monotone function $f$ such that $c(x,y) = 1/(y f(x/y))$, $f(1)=1$, $f(t) = t f(1/t)$, and $m(c) = f(0)>0$ (0803.1056). The Wigner–Yanase–Dyson family arises when $$f_p(t) = p(1-p)\,(t-1)^2/\left[(t^p-1)(t^{1-p}-1)\right]$$ with the special Wigner–Yanase case at $p=1/2$.

In terms of quantum variance or symmetrized covariance,

$\_\,_{\rho}(A) = \frac{1}{2} \left[\operatorname{Tr}(\rho AA^* + \rho A^* A)\right] - \left|\operatorname{Tr}(\rho A) \right|^2.$

The metric-adjusted skew information satisfies the basic quantum information inequality: $0 \le I^c_\rho(A) \le \_\,_{\rho}(A)$ for positive definite $\rho$ and arbitrary $A$, which generalizes to multi-operator matrix inequalities and determinant inequalities of "Robertson–type" (0803.1056).

2. Lattice Structure and Order Optimality

A key structural result concerns the ordering of quantum Fisher information measures and their associated skew informations. The space $\mathcal{F}_{\text{op}}$ of operator-monotone functions $f$ (with $f(1)=1$, $f(t)=t f(1/t)$) is equipped with the partial order $f \preceq g$ if

$\varphi(t) := \frac{t+1}{2} \frac{f(t)}{g(t)}$

is operator monotone. This forms a lattice on $\mathcal{F}_{\text{op}}$, with extremal elements $f_{\min}(t) = 2t/(t+1)$ and $f_{\max}(t) = (t+1)/2$.

Ordering in $\mathcal{F}_{\text{op}}$ "pushes through" to the corresponding metric-adjusted skew informations: if $f \preceq g$, then $I^f_\rho(A)\le I^g_\rho(A)$ for all $\rho$, $A$. Within the Wigner–Yanase–Dyson one-parameter family $(0 < p < 1)$,

$$I_\rho(p, A) = -\frac{1}{2}[\rho^p, A][\rho^{1-p}, A],$$

it is shown that the standard Wigner–Yanase ($p=1/2$) skew information is maximal with respect to this order structure (0803.1056). In particular, for any $p \in (0,1)$, the function $f_p$ satisfies

$f_p \preceq f_{1/2} \ \text{for all} \ p,$

so $I_\rho(1/2, A)$ is the largest within the family for all input data.

3. Relation to Quantum Uncertainty and Variance

The Wigner–Yanase skew information provides an information-theoretic form of the uncertainty principle. For a single observable, it is squeezed between zero and the quantum variance; for multiple observables $A_1,\ldots,A_k$, the matrix inequality

$\left( I^c_\rho(A_i, A_j) \right) \le \left( \_\,_{\rho}(A_i, A_j) \right)$

leads to determinant ("Robertson-type") uncertainty relations: $0 \le \det \left( I^c_\rho(A_i, A_j) \right) \le \det \left( \_\,_{\rho}(A_i, A_j) \right),$ which generalize and unify the known dynamic uncertainty principles for collections of observables (0803.1056).

Because quantum Fisher information mediates the local statistical distinguishability of states, the position of the Wigner–Yanase skew information as maximal among the WYD family makes it "optimal" for bounding variances and parameter estimation uncertainties (quantum Cramér–Rao bounds) in tasks where the underlying symmetry constraints are present.

4. Implications for Quantum Information Theory

The maximality and lattice order of Wigner–Yanase skew information have several consequences:

• Optimality for quantum uncertainty: It defines the sharpest lower bound for quantum uncertainty in the presence of symmetries—relevant for quantum estimation and metrological precision limits.
• Framework for deriving inequalities: The order structure yields new, often stricter, matrix inequalities and determinant-based uncertainty relations for multiple operators, extending beyond standard particle or variance-based formulations.
• Resource theory perspective: When viewed as a monotone under covariant channels (those respecting conserved quantities), the Wigner–Yanase skew information quantifies the "resourcefulness" of asymmetry in a state for quantum metrology, time-keeping, or reference frame alignment tasks (Takagi, 2018).
• Robust lower bounds: As the largest member of the WYD skew informations, it is robust for assessing the minimum quantum information content under various mixing and measurement schemes, informing both theoretical and experimental strategies in quantum parameter estimation.

5. Relations to Other Quantum Information Quantities

The Wigner–Yanase skew information connects deeply to several other quantum informational quantities:

• Quantum Fisher information: Via the metric-adjusted framework, Wigner–Yanase skew information is closely related to the quantum Fisher information, with both governed by operator monotone functions. In the order lattice, quantum Fisher information can be minimal or maximal depending on the chosen operator monotone function.
• Quantum variance: When the density matrix $\rho$ reduces to a pure state, the skew information coincides with the variance. For mixed states, it is always less than or equal to the variance, quantifying the genuinely quantum portion of the uncertainty.
• Convexity and additivity: The Wigner–Yanase skew information is convex in the state argument and additive under tensor products of independent systems, crucial for its operational interpretation in distributed or composite quantum systems.

6. Extensions and Generalizations

Recent works have focused on generalized forms of skew information, such as the metric-adjusted and WYD skew informations, and have analyzed their mathematical and operational properties—including order optimality, convexity, and monotonicity under symmetry-respecting operations. The order-theoretic analysis enables the comparison and optimization of quantum information measures over large functional classes (0803.1056, Takagi, 2018). This structural understanding supports their application in modern quantum information science, including quantum metrology, uncertainty relations, coherence theory, and the resource theory of asymmetry.

The maximality of the Wigner–Yanase skew information within the WYD family under the operator monotone order renders it a fundamental quantity for quantifying nonclassicality and quantum uncertainty constrained by symmetries. Its derivations underpin key results in multi-operator uncertainty quantification, and its properties make it indispensable in analyzing the fundamental informational limits in quantum theory.