Informational completeness of qubit measurements and IC preservability of qubit channels: Characterization and Quantification

Abstract: Informationally complete (IC) measurements are a useful class of measurements, as their outcome statistics uniquely determine an unknown quantum state. Hence, they are important for certain tasks such as quantum state tomography, quantum process tomography, etc. In this work, we study the quantification of informational completeness for arbitrary quantum measurements by introducing and characterizing a faithful measure for it. We explicitly evaluate the informational completeness of qubit symmetric informationally complete (SIC) measurements and show that it is an upper bound for all qubit minimal informationally complete measurements. Furthermore, by introducing a faithful measure, we try to quantify and characterize the ability of an arbitrary quantum channel to preserve informational completeness of any IC measurement when the channel acts on it in the Heisenberg picture. We call this measure informational completeness-preservability (IC preservability) of quantum channels. After studying its properties, we finally establish its relation to another quantity, namely, the absolute output coherence of a quantum channel, which quantifies the minimum amount of coherence (w.r.t. an arbitrary incoherent basis) that can always be obtained from the output of that channel. Thus, in this work, not only do we try to provide a quantitative framework for studying both the informational completeness of quantum measurements and the ability of quantum channels to preserve it, but we also try to offer key insight into the conceptual relation between informational completeness and quantum coherence.

• The paper introduces a fidelity-based quantifier that measures the informational completeness of qubit measurements, achieving an optimal value of 1/√6 for SIC-POVMs.
• The study characterizes IC-preservability of quantum channels through analytical bounds derived from singular values and Bloch sphere parameters.
• It establishes a quantitative link between informational completeness and quantum coherence, offering a tool for benchmarking tomography and device certification.

Informational Completeness of Qubit Measurements and IC Preservability of Qubit Channels: Characterization and Quantification

Introduction and Context

Informationally complete (IC) measurements are essential for quantum information science, enabling the unique determination of quantum states from measurement statistics. This property is critical in tasks such as quantum state and process tomography, as well as in foundational studies. Of particular interest are symmetric informationally complete positive operator-valued measures (SIC-POVMs), minimal IC-POVMs with optimal symmetry and established practical applications in tomography, entanglement detection, and device certification.

Despite their centrality, a precise and quantitative understanding of the degree of informational completeness exhibited by a given measurement, especially under physically realistic noise and channel evolutions, remains underexplored. Similarly, the behavior of IC under quantum channels—formalized as the IC-preservability of a channel—requires detailed characterization, especially to understand the robustness of quantum protocols in realistic settings. This work provides a systematic analytical account of these questions for qubit systems, linking informational completeness with quantum resource measures such as coherence.

Quantification of Informational Completeness

The primary technical contribution is the definition and analysis of a fidelity-based quantifier for IC, applicable to arbitrary quantum measurements. For a quantum measurement $A$ (POVM), the proposed measure $\mathfrak{D}(A)$ is defined as

$$\mathfrak{D}(A) := \inf_{\rho \neq \rho'} \frac{\sum_x |\operatorname{Tr}[A(x)\rho]-\operatorname{Tr}[A(x)\rho']|}{\|\rho-\rho'\|_1},$$

where the infimum is over all pairs of distinct quantum states on the relevant Hilbert space. This measure captures the minimal normalized statistical distinguishability induced by the measurement across all pairs of quantum states.

The paper rigorously establishes that $\mathfrak{D}$ is unitary-invariant and faithful, i.e., it vanishes if and only if the measurement is informationally incomplete. Furthermore, it provides an explicit computation for qubit SIC-POVMs: for any qubit SIC, $\mathfrak{D}(A) = 1/\sqrt{6}$. This value is shown to be an optimal upper bound for all minimal (i.e., four-outcome) informationally complete qubit measurements, with equality precisely for SIC-POVMs.

These results not only quantify the advantage of SICs among minimal IC measurements but also furnish a numerical benchmark for experimental realizations and the analysis of noisy measurement constructions.

IC-Preservability of Quantum Channels

Recognizing that quantum channels may degrade the informational power of a measurement, the authors introduce IC-preservability for quantum channels. Explicitly, for a channel $\Lambda$, the IC-preservability is

$$\tilde{\mathfrak{D}}(\Lambda) := \sup_{A} \mathfrak{D}(\Lambda^\dagger(A)),$$

where the supremum is over all measurements $A$ and $\Lambda^\dagger$ is the dual (Heisenberg-picture) map.

The analysis thoroughly characterizes the mathematical properties of $\tilde{\mathfrak{D}}$:

• It is invariant under unitary pre/post-processing of the channel.
• It is monotonic under quantum channel post-processing and statistical morphisms.
• Faithfulness is established: $\mathfrak{D}(A)$0 iff every measurement image under $\mathfrak{D}(A)$1 is informationally incomplete.

Tight bounds are derived for qubit channels in terms of the minimal singular value of the affine matrix representation of the channel and the translation vector in the Bloch sphere parametrization:

$\mathfrak{D}(A)$2

where $\mathfrak{D}(A)$3 is the smallest singular value and $\mathfrak{D}(A)$4 is the length of the translation vector. These results offer an analytic method to assess the robustness of IC under arbitrary qubit quantum operations.

Relation with Quantum Coherence

A key conceptual result is the establishment of a lower bound on the channel's absolute coherence output in terms of its IC-preservability. The absolute coherence output

$\mathfrak{D}(A)$5

(where the minimization is over all bases $\mathfrak{D}(A)$6 and $\mathfrak{D}(A)$7 denotes a valid coherence monotone) quantifies the minimal attainable coherence for any choice of incoherent basis and any output state.

For qubit channels, the authors show

$\mathfrak{D}(A)$8

establishing that the ability of a channel to preserve IC is always subordinate to its ability to generate coherence resources. Explicit lower and upper bounds are formulated in terms of singular values, and the precise value is computed for unital channels, where $\mathfrak{D}(A)$9 equals the intermediate singular value.

Implications and Future Directions

The formal framework developed here for IC quantification and preservability is immediately applicable in quantum tomography analysis, device certification, and analysis of quantum communication protocols in noisy environments. Quantitative links between IC and coherence suggest new pathways for resource-theoretic analysis, where measurement informativeness and coherence can be jointly optimized.

Significantly, this work provides:

• A faithful, computable quantifier for informational completeness suitable for both theoretical analysis and experimental benchmarking.
• Analytic bounds and explicit values for standard measurements and channels, enabling systematic robustness studies.
• A rigorous operational bridge between informational completeness and quantum coherence, positioning these as interlinked quantum resources.

Extending these results to higher-dimensional systems, examining their optimization in metrological and channel discrimination settings, and linking the IC quantifier with quantum Fisher information or other estimation-theoretic resources are promising directions for subsequent work.

Conclusion

This paper achieves a thorough quantification of informational completeness for qubit measurements and develops a well-founded notion of IC-preservability for quantum channels, connecting these analytic measures to quantum coherence output. The technical results deliver both concrete numerical bounds and conceptual clarity on the interplay of measurement informativeness and quantum resources, furnishing a rigorous structure for further developments in quantum information science (2606.03964).