Informationally Complete Quantum Instruments

• Informationally Complete Quantum Instruments are measurement setups whose statistics uniquely reconstruct any quantum state in a d-dimensional Hilbert space, typically via MIC-POVMs.
• They provide a minimal operator basis for state tomography by achieving exactly d² outcomes and minimizing reconstruction errors with optimal configurations like SIC-POVMs.
• Their applications span quantum tomography, device-independent certification, cryptography, and foundational studies, with sequential protocols enabling comprehensive state assessment even with limited outcomes.

Informationally complete quantum instruments are a fundamental toolset in quantum theory, enabling the complete reconstruction of quantum states via measurement statistics. These instruments operationalize the idea of informational completeness: a quantum measurement or instrument is informationally complete if its outcome statistics suffice to uniquely determine any quantum state on a given system. The relationship between such instruments, minimal positive operator-valued measures (MIC-POVMs), symmetric informationally complete measurements (SICs), and generalized measurement protocols has profound implications for quantum tomography, cryptography, and foundational studies.

1. Definitions and Structural Properties

Let $H$ be a $d$-dimensional Hilbert space, $L(H)$ the real vector space of Hermitian operators. A POVM $\{E_i\}_{i=1}^n$ is informationally complete (IC) if and only if the measurement statistics $p_i = \operatorname{Tr}[\rho E_i]$ are sufficient to reconstruct any quantum state $\rho$. Necessarily, IC requires that $\textrm{span}\{E_i\}$ is all of $L(H_d)$, so $n\geq d^2$.

A minimal informationally complete POVM (MIC-POVM or "MIC") is one where $n = d^2$; its elements are linearly independent, positive semidefinite operators summing to identity: $E_i \succeq 0,\,\, \sum_{i=1}^{d^2} E_i = I_d.$ Given outcome statistics, a dual basis $\{F_i\}$ exists such that $\operatorname{Tr}[F_i E_j] = \delta_{ij}$, permitting the inversion: $\rho = \sum_{i=1}^{d^2} p_i F_i.$ The MIC thus provides a direct operator basis mapping between probabilities and quantum states. If the POVM is also symmetric (a SIC-POVM), it is unbiased with $e_i := \operatorname{Tr}[E_i] = 1/d$ and minimal state-reconstruction error.

A quantum instrument is a collection $\{\mathcal{I}_x\}$ of completely positive trace-nonincreasing maps such that $\sum_x \mathcal{I}_x$ is CPTP. Its associated POVM is $E_x = \mathcal{I}_x^\dagger(I)$, and the instrument is IC if $\{E_x\}$ is IC.

2. MIC-POVMs as Standard Quantum Measurements

MIC-POVMs play the role of standard measurements in quantum information for several reasons:

• Tomographic Completeness: MICs provide all necessary outcome probabilities for full quantum state tomography; no ancillary measurements are needed.
• Minimality: MICs saturate the lower bound of $d^2$ outcomes for a $d$-dimensional system.
• Operator Basis: The elements of a MIC-POVM serve as an operator basis, analogous to a coordinate frame in real vector spaces, establishing a linear relationship between measured probabilities and density matrix elements.
• Optimality and Symmetry: Symmetric MICs (SIC-POVMs) minimize statistical errors in state estimation, and their structure is deeply linked with optimality in quantum cryptography and foundational reconstructions of quantum mechanics.

Fuchs et al. have argued that in an information-theoretic reconstruction of quantum theory, MIC-POVMs should be viewed as the canonical measurement (Smania et al., 2018).

3. Informational Completeness in Quantum Instruments and Sequential Schemes

A quantum instrument with $m<d^2$ outcomes can, under sequential application, nevertheless yield complete information about a quantum system. Given an instrument $\{\mathcal{J}_j\}_{j=1}^m$ on $H$, label joint sequential outcomes as $j_1,...,j_N$. The key structural object is the set of sequential effects

$$E_{j_1...j_N} := \mathcal{J}_{j_1}^* \circ ... \circ \mathcal{J}_{j_N}^*(I),$$

where $\mathcal{J}_j^*$ is the dual (Heisenberg picture) map. The instrument is informationally complete under $N$ sequential uses if and only if $\textrm{span}\{E_{j_1...j_N}\} = L(H)$. This generalizes the concept of informational completeness beyond single-shot POVMs (Zhuravlev et al., 2020).

Notably, projective measurements and two-outcome Lüders instruments are insufficient in this scenario: their sequential effects cannot span the full operator space for $N < d^2 - 1$, due to simultaneous diagonalizability and limited outcome support.

4. Realization: Device-Independent and Experimental Certification

Experimental realization and certification of MIC-POVMs, especially in a device-independent paradigm, have become essential benchmarks. In the bipartite Bell scenario, device independence is ensured by modeling the apparatuses as black boxes and relying solely on measurement statistics. The approach satisfies freedom of choice, no inter-device communication, and fair sampling criteria.

Certification is performed by demonstrating violation of a modified Bell expression that is only achievable by an irreducible MIC-POVM. Specifically, in the qubit case, the use of a modified Bell operator

$$\beta_{\text{el}}^m = \beta_{\text{el}} - k \sum_{i=1}^4 P(a=i, b=+1 | x=4, y=i)$$

sets a bound: any three-outcome qubit measurement cannot exceed $\beta_{\text{el}}^m = 6.8782$, while a true four-outcome MIC-POVM can achieve the quantum optimum $4\sqrt{3} \approx 6.928$. Observing an experimental value exceeding $6.8782$ under black-box assumptions certifies the non-decomposability of the measurement and thus its informational completeness as a MIC-POVM (Smania et al., 2018).

Experimentally, in entangled photon systems, the tetrahedral MIC-POVM for qubits was realized with a Sagnac interferometer, and multi-outcome discriminations were achieved with precise polarization-path coupling and tomography. The observed performance ($\beta_{\text{el}}^m = 6.960 \pm 0.007$) validated device-independent certification.

5. Relationship to Symmetric ICs, Wigner Functions, and Quantum Channels

SIC-POVMs are a special class of MICs that possess maximal symmetry: all elements are rank-1 and form a complex projective 2-design. Their structure leads to uniformity in the Fisher information matrix and extremizes bounds in Wigner-function representability (Zhu et al., 2017, DeBrota et al., 2019).

Discrete Wigner functions arise from orthogonalizing the basis formed by a MIC, producing a Wigner basis $\{F_i\}$ via the canonical map

$F_i = S_E^{-1/2}(E_i),$

where $S_E$ is the rescaled frame operator associated with the MIC (DeBrota et al., 2019). SICs are, in this sense, extremal MICs, yielding the most "classical-like" Wigner representations (least negativity) and minimizing the Hilbert–Schmidt distance between probabilistic (MIC) and quasi-probabilistic (Wigner) representations.

From the view of quantum channels, the Lüders instrument associated with a SIC-POVM precisely realizes the depolarizing channel with $\lambda = 1/(d+1)$. The existence of a SIC is equivalent to the existence of a Lüders channel $E$ such that $E(\rho) = \frac{1}{d+1}(I + \rho)$ (DeBrota et al., 2019). The SIC-Lüders channel minimizes both output entropy and entropy exchange among all such channels, establishing its entropic optimality.

6. Joint and Sequential Measurements of Conjugate Observables

Informationally complete joint measurements of two canonically conjugate observables (e.g., computational basis and its Fourier dual) can be engineered via covariant phase-space POVMs parameterized by a "seed" state. For odd $d$, minimal-noise joint measurements are already informationally complete. For even $d$, additional uniform noise must be injected to restore completeness. These joint POVMs can also be realized by sequential measurement processes, such as implementing the Lüders instrument for the first observable followed by a sharp measurement for the second (Carmeli et al., 2011).

Minimal-noise uniqueness and the failure of informational completeness in even dimensions highlight the necessity of non-commutativity and sufficient outcome cardinality for realizing IC joint instruments.

7. Applications and Outlook

Informationally complete quantum instruments underpin multiple areas of quantum information science:

• Quantum state tomography: MIC-POVMs and associated instruments permit efficient, minimal-error reconstruction protocols. Sequential protocols allow IC measurements even with physically constrained detectors.
• Device-independent quantum information: Device-independent certification of MIC-POVMs enables robust state and resource verification under minimal trust assumptions.
• Quantum cryptography and randomness certification: The optimality and symmetry properties of MICs and SICs yield protocols with uniform security guarantees and are intimately linked to foundational tasks such as randomness generation.
• Quantum foundational studies: MIC-POVMs provide frameworks for reconstructing quantum theory from informational principles and bridge the gap between probabilistic and quasiprobabilistic descriptions.

Extensions to higher dimensional systems, efficient construction of collective Fisher-symmetric measurements, and the development of analogues for multi-partite or continuous-variable systems remain active areas of investigation. The mathematical structure of informationally complete instruments continues to inform both practical protocol development and the conceptual understanding of quantum mechanics.