Quantum Cramér–Rao–McCarthy Bound

• Quantum Cramér–Rao–McCarthy bound is a noncanonical extension of standard QCRB, representing bias-aware corrections or generalized moment inequalities in quantum estimation theory.
• It extends traditional quantum estimation methods by incorporating biased-estimator formulations and higher-order moment bounds derived via Hölder’s inequality.
• This framework provides practical insights into multiparameter and non-asymptotic regimes where standard bounds face measurement incompatibility challenges.

The expression “Quantum Cramér–Rao–McCarthy bound” does not denote a single universally recognized bound in quantum metrology. Standard literature uses Quantum Cramér–Rao bound (QCRB) for the SLD–QFI inequality, and several works explicitly state that no recognized “McCarthy” variant is used in that context (Toscano et al., 2017, Nurdin, 2024). In the literature surveyed here, the label appears in two specific, nonstandard senses: first, as the straightforward quantum generalization of the classical biased Cramér–Rao bound, introduced “for clarity in this discussion” as QCRMB (Liu et al., 2016); second, as a family of generalized McCarthy-type inequalities based on Hölder-conjugate moments and generalized Fisher informations (Cimini et al., 2020). The topic therefore belongs to the intersection of the standard QCRB, biased-estimator corrections, and generalized moment bounds.

1. Terminological status and scope

In the standard vocabulary of quantum estimation theory, the relevant bounds are the classical Cramér–Rao bound, the quantum Cramér–Rao bound, the Holevo bound, the RLD bound, and related Bayesian or non-asymptotic bounds. Multiple papers on pure-state attainability, waveform estimation, transmission estimation, and multiparameter saturability explicitly note that there is no recognized “McCarthy” variant in their nomenclature (Tsang et al., 2010, Woodworth et al., 2022). One multiparameter saturability study states directly that it “focuses on the standard multiparameter QCRB; no alternative ‘Cramér–Rao–McCarthy’ nomenclature is introduced” (Nurdin, 2024).

A distinct usage appears in the biased-estimation literature. One work states that the classical biased Cramér–Rao bound is “often attributed in the literature to McCarthy,” and uses “Quantum Cramér–Rao–McCarthy Bound” to denote the straightforward quantum generalization obtained by replacing classical Fisher information with quantum Fisher information (Liu et al., 2016). Another work develops generalized classical Cramér–Rao bounds from Hölder’s inequality and explicitly describes them as McCarthy-type inequalities, then extends them to the quantum setting through generalized $\beta$-norm quantum Fisher informations (Cimini et al., 2020).

This suggests that “Quantum Cramér–Rao–McCarthy bound” is best treated as an umbrella label for nonstandard extensions of the QCRB rather than as a canonical bound with a unique definition.

2. Standard quantum Cramér–Rao framework

The standard QCRB begins with a differentiable family of density operators $\rho_\theta$ and the symmetric logarithmic derivative $L_\theta$, defined by

$$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$

The corresponding quantum Fisher information is

$$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$

For $\nu$ independent repetitions and an unbiased estimator $\hat{\theta}$, the bound reads

$$\operatorname{Var}(\hat{\theta}) \ge \frac{1}{\nu F_Q(\theta)}.$$

For pure states under unitary encoding $|\psi(\theta)\rangle=e^{-i\theta G}|\psi\rangle$, the QFI reduces to

$F_Q = 4\,\operatorname{Var}(G),$

which is the standard relation between metrological sensitivity and generator variance (Toscano et al., 2017, Rubio et al., 2017).

In the multiparameter setting, with parameters $\rho_\theta$0 and SLDs $\rho_\theta$1 defined by

$\rho_\theta$2

the SLD quantum Fisher information matrix is

$\rho_\theta$3

and the matrix QCRB takes the form

$\rho_\theta$4

Single-parameter saturation is always possible by an appropriate measurement, but in the multiparameter case saturation generally fails because the optimal measurements for different parameters may be incompatible (Nurdin, 2024).

The standard QCRB is therefore the baseline object relative to which “McCarthy”-type variants are defined: either as biased-estimator modifications or as generalized-moment extensions.

3. Biased-estimator formulation and the QCRMB usage

A direct quantum analogue of the classical biased Cramér–Rao bound appears in the formulation called QCRMB in (Liu et al., 2016). For a scalar parameter $\rho_\theta$5 with estimator bias

$\rho_\theta$6

the pointwise quantum biased bound is

$\rho_\theta$7

This is the simplest sense in which “Quantum Cramér–Rao–McCarthy bound” is used in the literature surveyed here.

The same work develops a stronger Optimal Biased Bound (OBB) in a Bayesian setting. For estimation of a function $\rho_\theta$8 from a prior $\rho_\theta$9, with bias function $L_\theta$0 and $L_\theta$1, it proves

$L_\theta$2

For direct parameter estimation $L_\theta$3, this becomes

$L_\theta$4

The right-hand side is then minimized over differentiable bias functions by an Euler–Lagrange equation. In the special case of constant prior and constant QFI on $L_\theta$5, the resulting closed-form lower bound is

$L_\theta$6

and for $L_\theta$7 repetitions one replaces $L_\theta$8 by $L_\theta$9 (Liu et al., 2016).

In multiparameter form, the same work gives

$$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$0

with $$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$1. This is the natural matrix generalization of the biased QCRB.

Within this usage, the “McCarthy” aspect is not a different information geometry; it is a bias-aware correction to the standard QCRB.

4. Generalized McCarthy-type moment bounds

A second, more general meaning of the term comes from higher-order Cramér–Rao inequalities derived by Hölder’s inequality. For conjugate exponents $$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$2 and $$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$3 satisfying

$$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$4

define the generalized classical Fisher information

$$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$5

Then the generalized McCarthy-type inequality is

$$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$6

For unbiased estimators this reduces to

$$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$7

The special case $$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$8, $$\partial_\theta \rho_\theta = \frac{1}{2}\bigl(\rho_\theta L_\theta + L_\theta \rho_\theta\bigr).$$9 recovers the variance-based Cramér–Rao bound, while $$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$0, $$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$1 yields an explicit third-order absolute-moment inequality (Cimini et al., 2020).

The corresponding quantum generalization introduces the SLD through

$$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$2

defines

$$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$3

and obtains

$$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$4

For $$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$5, this reduces to the usual QCRB; for $$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$6, the bound controls higher moments of the estimation error rather than just the variance (Cimini et al., 2020).

Operationally, this extension was used to show that the third-order absolute moment can give a superior capability in revealing biases in the estimation, compared to standard approaches in a phase-estimation experiment with quantum light (Cimini et al., 2020). In this sense, the “McCarthy” formulation is a generalized-moment refinement of the usual Fisher-information picture.

5. Multiparameter attainability and the hierarchy of commutativity conditions

The standard multiparameter SLD-QCRB is not generally saturable. One important line of work studies exactly when saturation is possible.

For general mixed states in the single-copy multiparameter setting, one 2024 result establishes necessary and sufficient conditions in terms of projected SLDs and a nonlinear PDE system. At a fixed parameter point, the conditions are: projected commutativity,

$$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$7

together with the existence of a unitary $$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$8 solving a coupled nonlinear PDE. These two conditions are necessary and sufficient for single-copy saturability of the multiparameter QCRB. The same work also provides practical sufficient conditions involving the off-support blocks $$F_Q(\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).$$9 and an auxiliary unitary $\nu$0, and shows that when these sufficient conditions hold the optimal measurement can be chosen to be projective and explicitly characterized (Nurdin, 2024).

A 2026 hierarchy result sharpens the logical structure of multiparameter attainability conditions under unitary encoding. It defines strong commutativity, one-sided commutativity, partial commutativity, and weak commutativity, and proves the chain

$\nu$1

It also shows that the converse implications fail in general, and, crucially, that commutativity of the parameter-encoding generators alone does not ensure the saturability of the QCR bound once realistic noise produces mixed probe states (Imai et al., 12 Feb 2026).

These results place the “McCarthy” variants in context. The biased and generalized-moment bounds do not remove the core multiparameter difficulty: even the standard SLD-QCRB itself may fail to be jointly attainable, and the relevant obstruction is measurement incompatibility rather than bias alone.

6. Local validity, global estimation, and broader precision bounds

A recurring theme in the recent literature is that the QCRB is primarily a local, asymptotic statement. One non-asymptotic analysis emphasizes that many conclusions drawn from standard CRB/QCRB methods “do not always hold when the analysis is more carefully performed,” and quantifies the number of observations and prior knowledge needed before the QCRB is a valid approximation to the Bayesian mean-squared error (Rubio et al., 2017). The OBB construction in the biased-estimation setting was motivated by the same issue: with a limited number of measurements, biased estimators can outperform the unbiased QCRB benchmark, so a lower bound valid for all estimators is required (Liu et al., 2016).

The limitation becomes sharper in global quantum estimation. In a non-IID bosonic setting with a single copy of a many-boson state, one 2024 study finds situations where the Cramér–Rao approach does and does not work for global estimation. For unitary estimation with certain binomial and Dicke-type probes, local CR predictions such as $\nu$2 or even local Heisenberg $\nu$3 behavior do not translate into vanishing global minimax error; by contrast, geometric probes in a D-invariant model do match the global scaling predicted by the CR approach (Hayashi et al., 2024). A plausible implication is that “McCarthy”-type bias corrections and higher-moment refinements do not eliminate the distinction between local and global estimation regimes.

In incompatible multiparameter models, the asymptotically attainable benchmark is instead the Holevo bound, with the SLD-QCRB serving only as a lower reference point. Geometric reconstructions of the QCRB through the quantum metric and mean Uhlmann curvature show explicitly how a nonzero incompatibility measure enlarges the gap between the SLD bound and the attainable scalar bound (Li et al., 2022).

A separate controversy concerns claims of beating the QCRB. In dissipative adiabatic measurements, it has been argued that the effective POVM depends explicitly on the unknown parameter, so the usual step $\nu$4 fails and the standard QCRB is not applicable (Zhang et al., 2020). This does not define a “McCarthy” bound, but it reinforces the broader lesson that the operational content of any Cramér–Rao statement depends on its assumptions: unbiasedness, locality, estimator class, measurement independence, and attainability conditions.

Taken together, the literature supports a precise interpretation. The Quantum Cramér–Rao–McCarthy bound is not a standard single bound; it is a noncanonical label used for either the biased quantum Cramér–Rao correction or the generalized McCarthy-type higher-moment inequalities. Both are best understood as extensions of the standard QCRB rather than replacements for it.