Wigner–Yanase–Dyson Skew Information

• Wigner–Yanase–Dyson skew information is a measure that quantifies quantum asymmetry by evaluating the non-commutativity between a quantum state and an observable.
• Its metric-adjusted formulation connects it to quantum Fisher information and resource theories, enabling refined uncertainty and coherence analysis.
• Generalizations, including two- and three-parameter extensions, broaden its applicability in quantum metrology and asymmetric resource quantification.

Wigner–Yanase–Dyson Skew Information

The Wigner–Yanase–Dyson (WYD) skew information is a fundamental quantum information theoretic quantity quantifying the non-commutativity, or “quantum asymmetry,” of a state relative to an observable. Originally introduced to analyze quantum measurement and uncertainty in connection with conserved quantities, WYD skew information has become central in resource theories of asymmetry, quantum coherence, metrology, and generalized quantum uncertainty relations. The WYD family interpolates between the original Wigner–Yanase skew information and a range of monotone Riemannian metrics, linking it to the broader class of metric-adjusted skew informations and quantum Fisher information.

1. Definition and Basic Properties

Let $\rho$ be a density operator on a Hilbert space $\mathcal{H}$ and $H$ a self-adjoint observable (generator). For $0<\alpha<1$, the WYD skew information is defined as

$$I_\alpha(\rho, H) = -\frac{1}{2}\mathrm{Tr}\left([\rho^\alpha, H][\rho^{1-\alpha}, H]\right) = \mathrm{Tr}(\rho H^2) - \mathrm{Tr}(\rho^\alpha H \rho^{1-\alpha} H)$$

where $[X,Y]=XY-YX$. For $\alpha=1/2$, this reduces to the original Wigner–Yanase skew information, $$I_{1/2}(\rho, H) = -\frac{1}{2}\mathrm{Tr}([\sqrt{\rho}, H]^2)$$. Crucially, $I_\alpha(\rho,H)\ge 0$ with equality if and only if $[\rho,H]=0$ (Zhang et al., 2022, Takagi, 2018). On pure states, $I_\alpha(\rho,H)$ recovers the variance $\mathrm{Var}_\rho(H)$.

Key properties include:

• Convexity: $I_\alpha(\rho, H)$ is convex in $\rho$ for each fixed $H$ and $\alpha$ (Lieb's theorem).
• Symmetry: $I_\alpha = I_{1-\alpha}$.
• Unitary invariance: $$I_\alpha(U\rho U^\dag, UHU^\dag) = I_\alpha(\rho, H)$$.
• Monotonicity: $$I_\alpha(\mathcal{E}(\rho), H) \leq I_\alpha(\rho, H)$$ under CPTP maps $\mathcal{E}$ provided $\mathcal{E}$ is $H$-covariant (Takagi, 2018).
• Bounds: $0 \le I_\alpha(\rho,H) \le \mathrm{Var}_\rho(H)$.

2. Metric-adjusted Skew Information and Morozova–Chentsov Framework

The WYD family is a particular case of the metric-adjusted skew information formalism, in which a Riemannian metric on the quantum state space is characterized by an operator-monotone Morozova–Chentsov function $f$ (0803.1056, Zhang et al., 2022). For an operator-monotone $f$ with symmetry $f(t) = t f(1/t)$: $$I^f_\rho(H) = \frac{f(0)}{2} \mathrm{Tr}\big( (i[\rho,H]) c_f(L_\rho, R_\rho) (i[\rho,H]) \big)$$ where $c_f(x,y) = [y f(x/y)]^{-1}$. For the WYD family, $$f_\alpha(t) = \alpha(1-\alpha) (1-t)^2 / \big[(1-t^\alpha)(1-t^{1-\alpha})\big]$$ and $I^f_\rho(H)$ coincides with $I_\alpha(\rho, H)$ when $H$ is Hermitian.

Within this framework:

• The Wigner–Yanase skew information ($\alpha=1/2$) is the unique maximal element in the partial order on the space of operator-monotone functions generating skew informations (0803.1056).
• The metric-adjusted skew informations provide a basis for general uncertainty and correlation inequalities and their determinant (Robertson-type) variants (0803.1056, Furuichi et al., 2010).

3. Generalizations and Multi-parameter Extensions

Beyond the standard (single-parameter) WYD family, several multi-parameter generalizations exist.

Two-parameter extensions:

• The GWYD (generalized Wigner–Yanase–Dyson) skew information uses independent exponents $\alpha,\beta\ge0$, $\alpha+\beta\le 1$: $$I_\rho^{\alpha,\beta}(A) = -\frac{1}{2}\mathrm{Tr}\left( [\rho^\alpha,A][\rho^\beta,A]\rho^{1-\alpha-\beta} \right)$$ which reduces to the original WYD for $\beta=1-\alpha$ (Huang et al., 2021, Yanagi, 2010).
• Modified extensions like the MGWYD and MWGWYD allow generalized types of commutators, application to non-Hermitian operators, and to quantum channels via sums over Kraus operators (Wu et al., 2020, Wu et al., 2020, Pires et al., 2020).

Three-parameter weighted extensions:

• The $(\alpha,\beta,\gamma)$-weighted WYD (WWYD) skew information further interpolates the left/right weighting and convex combination of $\rho^\alpha$ and $\rho^\beta$: $$I_\rho^{\alpha,\beta,\gamma}(A) = -\frac{1}{2} \mathrm{Tr}\left( [ (1-\gamma)\rho^\alpha + \gamma \rho^\beta, A ]^2 \rho^{1-\alpha-\beta} \right)$$ and unifies previous definitions as special cases (Xu et al., 2022, Xu et al., 2023, Xu et al., 2022).

4. Operational Role: Asymmetry Monotones and Resource Theory

The WYD skew information has a fundamental interpretation as a bona fide asymmetry monotone in the resource theory of quantum asymmetry (Takagi, 2018). Specifically:

• For symmetry group $U(1)$ generated by $H$, $I_\alpha(\rho, H)$ quantifies the “amount of asymmetry” of $\rho$—that is, how much it fails to be invariant under $e^{-i H t}$ for all $t$.
• Monotonicity under $H$-covariant quantum channels, convexity, and partial-trace monotonicity (local discard) all hold for $I_\alpha$ and its metric-adjusted generalizations.
• Skew information measures the potential of a state to function as a quantum reference frame, and thus directly quantifies coherence in the eigenbasis of $H$.
• In distributed quantum clock synchronization and quantum estimation, the non-superadditivity of skew information places optimality constraints on concentrating asymmetry among subsystems (Takagi, 2018).

5. Uncertainty Relations and Quantum Bounds

WYD skew information supports a range of refined quantum uncertainty relations extending and strengthening the Heisenberg and Robertson bounds.

Schrödinger-type uncertainty (purely quantum part):

For $\alpha\in[1/2,1]$: $$U_{\rho,\alpha}(A)U_{\rho,\alpha}(B) \ge 4\alpha(1-\alpha)|\text{Corr}_{\rho,\alpha}(A,B)|^2$$ where $$U_{\rho,\alpha}(A) = \sqrt{I_{\rho,\alpha}(A) J_{\rho,\alpha}(A)}$$ and the correlation term is constructed from differences between quantum and classical covariances, capturing “purely quantum” uncertainty (Furuichi et al., 2010).

Sum-uncertainty and determinant inequalities:

For $N$ observables, norm-based lower bounds on sum-skew informations have been constructed, strictly improving prior inequalities: $$\sum_{i=1}^N I_\alpha(\rho, A_i) \geq \max \left\{ \ldots \right\}$$ with optimal vector-norm expressions involving sums and differences of observables or, for channels, their Kraus operators (Zhang et al., 2022, Xu et al., 2022, Xu et al., 2023, Xu et al., 2022).

Channel and unitary bounds:

Equivalent sum-form bounds are available for collections of quantum channels and unitary operations, providing tight constraints on the combined “coherence” a state displays relative to a set of processing maps (Zhang et al., 2022, Xu et al., 2023, Xu et al., 2022, Xu et al., 10 Nov 2025).

6. Connections to Quantum Fisher Information and Quantum Metrology

The WYD skew information, especially at $\alpha=1/2$, is tightly linked to quantum Fisher information (QFI), a central quantity for quantum metrology (Pires et al., 2020). Specifically,

$$I_{1/2}(\rho, H) \le \frac{1}{2} \mathcal{F}_Q(\rho, H) \le \mathcal{F}_Q(\rho, H) \le 2 I_{1/2}(\rho, H)$$

with $I_{1/2}(\rho, H)$ a computable lower bound to QFI, which in turn governs ultimate limits for precision in quantum parameter estimation (Pires et al., 2020). The second moment of the generalized multiple quantum coherence (MQC) spectrum can be identified with $I_\alpha(\rho, H)$: this connects asymmetry resource quantification, quantum coherence spectroscopy, and Fisher-metric based analysis (Pires et al., 2020).

7. Extensions: Channels, Complementarity, Experimental Access

The framework of WYD skew information has been extended to general (non-Hermitian) operators, quantum channels, and measurement processes:

• Modified generalized skew informations (MGWYD, MWGWYD) and their channel analogs serve as resources for quantifying coherence with respect to incompatible dynamics, noisy channels, and general state transformations (Wu et al., 2020, Wu et al., 2020, Xu et al., 10 Nov 2025).
• Complementarity and conservation relations involving skew information and its anti-commutator analogs under channel actions quantify trade-offs between wave-particle duality and symmetry-asymmetry (Wu et al., 2020).
• Experimental access to skew information via weak-value measurements and interferometric protocols has been proposed, enabling direct quantification of quantum uncertainty and asymmetry properties in laboratory settings (Sahil, 5 Feb 2024, Xu et al., 2022).

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In summary, WYD skew information forms a bridge between quantum coherence, asymmetry resource theories, fine-grained uncertainty quantification, quantum metrology, and the structure of quantum information measures. Its metric-adjusted and generalized multi-parameter forms provide a unifying structure for extensions to channels, composable resource monotones, and tight quantum uncertainty bounds (Zhang et al., 2022, Takagi, 2018, Xu et al., 2023, Pires et al., 2020, Xu et al., 2022, 0803.1056, Furuichi et al., 2010, Huang et al., 2021, Xu et al., 10 Nov 2025, Wu et al., 2020).