Maximal coherence of quantum measurement and the resource theory of sharpness

Abstract: A resource theory of quantum measurement can be addressed in terms of quantum coherence and measurement sharpness, respectively. The former analyzes the off-diagonal structure of POVM elements in a predetermined basis while the latter analyzes the deviation from trivial, state-independent, measurements. We establish a direct connection between the two resource theories by identifying measurement sharpness as the maximal coherence that is achievable under all possible unitary changes of the reference basis. For a broad class of POVMs whose elements share a common eigenbasis, we show that the maximal distance-based coherence of measurement coincides exactly with the corresponding distance-based sharpness monotone. We further extend this equivalence, with element-additive distances, to POVMs whose elements admit a common mutually unbiased basis structure. These results provide a measurement-theoretic analogue of the maximal-coherence & purity correspondence for quantum states. We also show that the maximal coherence of measurement is faithful with respect to trivial measurements and is monotonic under fuzzifying operations for dichotomic measurements, as well as under mixed-unitary and unitarily covariant preprocessing channels. Finally, we illustrate the operational meaning and limitations of the equivalence through qubit POVMs, single-photon phase sensing, and noisy photon-number resolving detection. In particular, the maximal Fisher information in a Mach-Zehnder interferometer is shown to be determined by the squared maximal coherence of the measurement, while in an imperfect photon-number resolving detector the maximal coherence behaves as a proper sharpness monotone, unlike conventional PVM-based unsharpness measures.

• The paper establishes that maximal measurement coherence equals distance-based sharpness for POVMs with a common eigenbasis or, for additive distances, a common mutually unbiased basis, while generic POVMs satisfy only an inequality.
• The paper proves maximal coherence is faithful to nontrivial measurements and remains monotonic for dichotomic POVMs, mixed-unitary channels, and selected covariant channels, leaving general non-dichotomic preprocessing unresolved.
• The paper connects maximal coherence to applications by showing that interferometric phase sensitivity obeys \(\mathcal{I}_{\max}=(2C_{D_1}^{\max})^2\) and that the measure tracks detector sharpness more reliably than older PVM-based unsharpness measures.

Overview

The paper establishes a quantitative bridge between two resource theories of quantum measurement: the coherence of a POVM, which captures the off-diagonal structure of its elements relative to a fixed reference basis [Baek et al., NJP 2020], and the sharpness of a POVM, which quantifies deviation from trivial, state-independent measurements within the operational resource theory of Buscemi, Kobayashi, and Minagawa. The central object is the maximal coherence of measurement,

$$C_D^{\max}(\mathsf{A})=\max_U C_D(U^\dagger \mathsf{A}U),$$

i.e., the coherence optimized over all unitary choices of reference basis. The main result is that $C_D^{\max}$ coincides exactly with the corresponding distance-based sharpness monotone $S_D$ for POVMs whose elements share a common eigenbasis, and—for element-additive distances—for POVMs admitting a common mutually unbiased basis (MUB) structure. This is the measurement-theoretic analogue of the maximal-coherence/purity correspondence for quantum states established by Streltsov et al.

Background: coherence and sharpness as measurement resources

A POVM $\mathsf{A}=\{A_x\}_{x=1}^n$ on a $d$-dimensional Hilbert space is incoherent if all elements are diagonal in the reference basis; free operations are maximally incoherent operations for measurements (MIO-M), i.e., channels whose Heisenberg duals preserve $\mathcal{I}(d,n)$. Distance-based coherence monotones are defined as $$C_D(\mathsf{A})=\min_{\mathsf{B}\in\mathcal{I}(d,n)}D(\mathsf{A},\mathsf{B})$$, where $D$ is faithful, contractive under CP unital maps, and jointly convex—properties inherited by the induced statistical distance $$D(\mathsf{A},\mathsf{B})=\sup_\rho D(\mathbf{p}_{\rho,\mathsf{A}},\mathbf{p}_{\rho,\mathsf{B}})$$ and by the diamond distance between measure-and-prepare channels.

On the sharpness side, the paper adopts the generalized notion of sharpness: a POVM is generalized-sharp if every effect has eigenvalue 1, and trivial measurements $\{p_x I\}$ form the free set, with fuzzifying operations $C_D^{\max}$0 as free operations. Distance-based sharpness is $C_D^{\max}$1. The paper emphasizes that earlier PVM-based unsharpness measures—the entropic measure $C_D^{\max}$2, the variance-based $C_D^{\max}$3, and the operator-norm $C_D^{\max}$4—are not monotones under fuzzifying operations because they cannot distinguish projective from trivial measurements.

Main results

Faithfulness. Theorem 1 shows $C_D^{\max}$5 if and only if $C_D^{\max}$6 is trivial. Necessity follows by conjugating any non-scalar effect into an MUB via a discrete Fourier transform argument: since the eigenvalues are not all equal, at least one off-diagonal entry survives, so some basis renders the POVM coherent. This means maximal coherence is faithful with respect to the correct free set (trivial, not merely diagonal, measurements)—a nontrivial property given that basis-optimization could in principle erase basis-dependence.

Equivalence with sharpness. Theorem 2 proves that if all POVM elements share a common eigenbasis, then

$C_D^{\max}$7

for distances invariant under simultaneous unitary conjugation. The proof is clean: the upper bound holds because trivial measurements are incoherent; the lower bound uses the unitary mapping the common eigenbasis to an MUB, followed by dephasing in that MUB, which maps any incoherent measurement to a trivial one while leaving the rotated POVM invariant. Corollary 1 extends this to POVMs whose elements have distinct eigenbases sharing a common MUB, provided the distance is element-additive—a condition satisfied by the Schatten-$C_D^{\max}$8 norm distance $C_D^{\max}$9 of Tendick et al.; for $S_D$0 the equivalence $S_D$1 holds throughout this class.

The equivalence fails outside these regimes: since $S_D$2, one always has $S_D$3, but strict inequality can occur, e.g., for tetrahedral SIC qubit POVMs. This is an important structural limitation: maximal coherence recovers sharpness only when the POVM's spectral structure aligns with the incoherent structure through a common eigenbasis or common MUB.

Monotonicity. Faithfulness holds generally, but monotonicity under arbitrary fuzzifying operations is obstructed by the nested optimization in $S_D$4: the max over unitaries and min over incoherent measurements cannot be interchanged, and Sion-type minimax conditions fail because the unitary group is not convex and the objective is not concave. The paper nevertheless establishes monotonicity in three cases:

POVM / channel class	Monotonicity
Dichotomic, any fuzzifying operation	Yes
Non-dichotomic, mixed-unitary channel	Yes
Non-dichotomic, unitarily covariant channel	Yes
Non-dichotomic, general fuzzifying operation	Open

For dichotomic POVMs, both elements commute, so the equivalence with sharpness applies to both $S_D$5 and $S_D$6, transferring sharpness monotonicity directly. For non-dichotomic POVMs, joint convexity of $S_D$7 yields monotonicity under mixed-unitary duals $S_D$8, and a covariant version covers unitarily covariant channels including the Werner–Holevo channel. Whether $S_D$9 is a full sharpness monotone for general CP unital preprocessing remains open.

Physical illustrations

Qubit POVMs. Along the convex path $\mathsf{A}=\{A_x\}_{x=1}^n$0 between the joint $\mathsf{A}=\{A_x\}_{x=1}^n$1–$\mathsf{A}=\{A_x\}_{x=1}^n$2 POVM (whose rank-one effects' eigenbases lie in the equatorial plane, mutually unbiased to the $\mathsf{A}=\{A_x\}_{x=1}^n$3-basis) and the tetrahedral SIC POVM, the spectral-norm quantities $\mathsf{A}=\{A_x\}_{x=1}^n$4 and $\mathsf{A}=\{A_x\}_{x=1}^n$5 coincide exactly at $\mathsf{A}=\{A_x\}_{x=1}^n$6 and exhibit a strictly positive gap for $\mathsf{A}=\{A_x\}_{x=1}^n$7. This confirms numerically that the common-MUB condition is not merely sufficient in principle but sharply delineates where the equivalence breaks.

Phase sensing. For a single-photon Mach–Zehnder interferometer with visibility $\mathsf{A}=\{A_x\}_{x=1}^n$8, the output binary POVM has common eigenbasis for each $\mathsf{A}=\{A_x\}_{x=1}^n$9, so $d$0, independent of $d$1 as required by unitary invariance. The classical Fisher information attains $d$2 at $d$3, giving the exact identity

$d$4

using the fact that for dichotomic POVMs the trace-, diamond-, and Schatten-$d$5-based quantifications coincide. Maximal coherence thus has a direct metrological interpretation: it determines the achievable phase sensitivity of the device.

Imperfect photon-number resolving detection. For the PNRD with efficiency $d$6, whose elements are binomial mixtures diagonal in the Fock basis, Theorem 2 gives $d$7. At $d$8 the detector reduces to the trivial vacuum-reporting POVM—which is formally a PVM—and consequently the HS norm and the measures $d$9, $\mathcal{I}(d,n)$0, $\mathcal{I}(d,n)$1 are all maximized at both $\mathcal{I}(d,n)$2 and $\mathcal{I}(d,n)$3, hence non-monotonic in $\mathcal{I}(d,n)$4. Only $\mathcal{I}(d,n)$5 and the tunability measure of Buscemi et al. vanish on the trivial limit and increase monotonically to the ideal PNRD. This example demonstrates concretely why the generalized notion of sharpness, rather than PVM-unsharpness, is the operationally meaningful one in a resource theory.

Limitations and open questions

Three limitations bound the scope of the results. First, the equivalence $\mathcal{I}(d,n)$6 requires either a common eigenbasis or (with element-additive distances) a common MUB structure; for generic POVMs such as SIC measurements only the inequality $\mathcal{I}(d,n)$7 is known, and characterizing the gap is left open. Second, monotonicity of $\mathcal{I}(d,n)$8 under arbitrary fuzzifying operations on non-dichotomic POVMs is unresolved, due to the failure of minimax interchange in the nested optimization. Third, the covariant-channel result carries the technical assumption that the maximizing unitary can be chosen within the covariance group, which restricts its applicability. The authors also flag the broader open problem of unifying coherence, sharpness, and incompatibility into a single resource theory of measurements.

Conclusion

This paper identifies basis-optimized coherence as a basis-independent quantifier of measurement sharpness for a broad and physically relevant class of POVMs, mirroring the state-level maximal-coherence/purity correspondence. The exact identity $\mathcal{I}(d,n)$9 for common-eigenbasis and common-MUB POVMs, the faithfulness with respect to trivial measurements, the partial monotonicity results, and the saturated relation between maximal coherence and Fisher information in interferometric phase sensing together give maximal coherence both structural and operational standing as a sharpness monotone. The remaining gaps—for general POVM structures and general free operations—define the precise boundary of what is currently established.