Fisher Discord in Quantum Complexity

• Fisher discord is an information-theoretic measure that quantifies the quantum complexity of a state by comparing two quantum Fisher information versions.
• It is derived from the discrepancy between SLD-based quantum Fisher information and Wigner–Yanase skew information, capturing noncommutative effects and state mixing.
• Its practical implications in quantum metrology and dynamics help distinguish genuine quantum advantages from classical statistical uncertainty.

Fisher discord is an information-theoretic quantifier of quantum complexity for a quantum state, constructed by analogy with quantum discord but based on the formalism of quantum Fisher information. Given a quantum system and a fixed Hamiltonian that drives its evolution, Fisher discord evaluates the divergence between two quantum generalizations of the classical Fisher information: one defined via the symmetric logarithmic derivative (SLD)—the canonical quantum Fisher information—and another defined via the Wigner–Yanase skew information (which employs the square root of the density operator). This divergence, which does not occur in classical probability theory, serves as a measure of the nonclassical (hybrid) features of quantum states that intermingle classical and quantum characteristics relative to the observable of interest.

1. Conceptual Foundation and Definition

The classical Fisher information provides a unique measure of the sensitivity of a probability distribution $p(x;\theta)$ to infinitesimal changes in a parameter $\theta$. In the quantum regime, however, there are inequivalent generalizations—primarily due to noncommutativity. The two principal quantum versions are:

• Quantum Fisher Information (SLD-based):

For density operator $\rho$ and Hermitian operator $H$ associated with parameter shift, the symmetric logarithmic derivative (SLD) $L$ solves

$$\frac{\partial \rho}{\partial \theta} = \frac{1}{2}(L \rho + \rho L) \, .$$

The quantum Fisher information is then

$I_F(\rho, H) = \frac{1}{4}\mathrm{Tr}[\rho L^2].$

In the context of unitary evolution $\rho_\theta = e^{-i \theta H} \rho e^{i \theta H}$, $I_F(\rho, H)$ becomes parameter-independent.

• Wigner–Yanase Skew Information:

Defined by

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

This quantity captures the quantum uncertainty in $H$ due to noncommutativity between $\sqrt{\rho}$ and $H$.

Fisher discord is then defined as the difference:

$C(\rho, H) = I_F(\rho, H) - I_W(\rho, H) \, .$

This definition systematically quantifies the complexity—interpreted as the degree to which quantum characteristics manifest in a state with respect to $H$—by measuring the excess quantum contribution beyond classical analogues (Li et al., 17 Oct 2025).

2. Mathematical Formulation and Explicit Expressions

For a mixed quantum state with spectral decomposition $$\rho = \sum_m \lambda_m |\varphi_m\rangle\langle \varphi_m|$$, Fisher discord admits the following explicit formula:

$$C(\rho, H) = \sum_{m, n} \frac{ \lambda_m^{1/2} \lambda_n^{1/2} (\lambda_m^{1/2} - \lambda_n^{1/2})^2 }{ \lambda_m + \lambda_n } |\langle \varphi_m | H | \varphi_n \rangle |^2$$

This expression highlights its dependence on both the spectral composition of $\rho$ and the off-diagonal elements of $H$ in the eigenbasis. For pure states or for $\rho$ commuting with $H$, $C(\rho, H) = 0$; nonzero Fisher discord arises only from noncommutativity and/or mixedness.

The SLD Fisher information and Wigner–Yanase skew information reduce to the same value for pure states and for density operators that commute with the observable, supporting the interpretation that nonzero Fisher discord reflects genuine quantum complexity rather than classical statistical uncertainty.

3. Foundational Properties

Fisher discord satisfies several important properties:

• Nonnegativity: $C(\rho, H) \geq 0$ always, since $I_F(\rho, H) \geq I_W(\rho, H)$ by construction.
• Vanishing on Classical States: Fisher discord vanishes for pure states and for any $\rho$ such that $[\rho, H] = 0$ (i.e., equilibrium or stable states), signifying zero quantum complexity in this setting.
• Additivity: For product states and Hamiltonians decomposable across subsystems, Fisher discord is additive.
• Invariance: $C(\rho, H)$ is invariant under energy shifts $H \mapsto H + c I$ and unitary rotations of $H$.
• Bounds: $0 \leq C(\rho, H) \leq I_W(\rho, H)$.
• Comparison to Other Complexity Measures: Unlike measures such as the Fisher–Shannon or Fisher–Wehrl complexities, which rely on entropy powers, Fisher discord directly compares intrinsic quantum information quantities and does not single out thermal/Gaussian states as minimizers (Li et al., 17 Oct 2025).

4. Prototypical Quantum States: Illustrative Applications

Fisher discord is illustrated by its application to several well-known quantum states:

• Generic Qubit State: For a qubit $\rho = \frac{1}{2}(I + \mathbf{r} \cdot \sigma)$ and $H$ a Pauli operator, Fisher discord vanishes if and only if $\rho$ is pure or commutes with $H$.
• Fock-Diagonal States (Thermal Ensembles): For $\rho$ diagonal in photon number (e.g., thermal states), $C(\rho, a^\dagger a) = 0$.
• Displaced/Squeezed States: Displaced Fock-diagonal and squeezed states exhibit nontrivial Fisher discord relative to quadrature and photon number Hamiltonians. The complexity depends on parameters such as displacement amplitude and squeezing strength, revealing mixed classical/quantum features.
• Mixtures of Photon Number States: For states like $$\rho = p|0\rangle\langle 0| + (1-p)|n\rangle\langle n|$$, Fisher discord exhibits parametric dependence, capturing the interplay between coherence and mixing.

Representative figures quantify complexity as a function of parameters, revealing, for example, nonmonotonic behavior under squeezing, which reflects the competition between thermal noise and quantum fluctuations.

5. Operational Consequences for Quantum Estimation and Complexity

Fisher discord connects directly to quantum metrology:

• Parameter Estimation: In quantum parameter estimation (e.g., phase sensitivity), Fisher discord encodes the excess sensitivity available through nonclassical correlations; it quantifies the deviation from the Cramér–Rao bound achievable by classical states or measurements.
• Quantum Complexity: Fisher discord provides an operational measure of quantum complexity, identifying states whose estimation advantages (relative to classical baselines) arise from noncommutativity and genuine quantum mixtures.
• Relevance for Quantum Dynamics: By vanishing on pure states and equilibria (commuting observables), Fisher discord singles out intrinsically complex states—those where quantum effects can enhance dynamical or metrological phenomena.
• General Implications: Because Fisher discord is constructed from fundamental quantum Fisher information concepts, it interfaces with quantum entanglement, coherence, uncertainty relations, and non-Markovian dynamics. Its versatility supports its potential use in quantum information processing, quantum metrology, and the paper of quantum phase transitions.

6. Significance and Outlook

Fisher discord establishes a precise bridge between quantum estimation theory and quantum complexity—a major step in characterizing the hybrid nature of quantum states and their advantages in measurement and control scenarios. By formalizing the difference between two quantum generalizations of Fisher information, it provides an intrinsic, operational, and additive measure of nonclassical complexity applicable to both discrete and continuous-variable systems, as well as to practical scenarios in metrology and quantum computing. Its vanishing for pure and stable states establishes it as a robust discriminator of quantum complexity, distinct from classical statistical uncertainty or noise, and anchors future research on the structure, utility, and manipulation of complex quantum states (Li et al., 17 Oct 2025).