Semi-SIC POVM: Relaxed SIC Measurements

• Semi-SIC POVMs are rank-one quantum measurements that maintain constant Hilbert–Schmidt overlaps while relaxing the equal-trace condition.
• They bridge the gap between fully symmetric SIC-POVMs and general informationally complete measurements, enabling tailored quantum state reconstruction.
• Experimental realizations using photonic quantum walks and self-testing protocols validate their feasibility and potential in quantum certification.

Searching arXiv for relevant papers on semisymmetric informationally complete POVMs and closely related frameworks. A semisymmetric informationally complete POVM, usually abbreviated semi-SIC POVM, is a rank-one informationally complete POVM with constant pairwise Hilbert–Schmidt overlaps, but without the equal-trace requirement imposed in the standard definition of a SIC-POVM. In the now-standard formulation, the SIC axioms are split into three parts—rank one, constant overlap, and constant trace—and a semi-SIC is obtained by keeping the first two while dropping the third. This relaxation preserves a strong operator-level symmetry while allowing nonuniform weights, so SIC-POVMs appear as a special case of semi-SICs when all traces coincide (Geng et al., 2020, Xu et al., 2 Mar 2026).

1. Definition and minimal axioms

Let $\mathcal{H}_d$ be a $d$-dimensional Hilbert space. A POVM is a set of positive semidefinite operators $\{E_x\}$ such that

$E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$

It is informationally complete if the probabilities $p(x|\rho)=\mathrm{Tr}(\rho E_x)$ uniquely determine the state $\rho$. For a $d$-dimensional system, informational completeness requires at least $d^2$ linearly independent POVM elements (Xu et al., 2 Mar 2026).

A standard SIC-POVM is an informationally complete POVM with $d^2$ rank-one elements satisfying three conditions: each $E_x$ is rank one, all off-diagonal overlaps $d$0 for $d$1 are equal, and all traces $d$2 are equal. The usual SIC normalization gives $d$3, and the SIC overlap value is

$d$4

(Geng et al., 2020).

A semi-SIC POVM keeps the rank-one condition and the constant-overlap condition but drops the equal-trace condition. In the formulation emphasized in the later literature, one writes

$d$5

with

$d$6

for some fixed $d$7, while the traces $d$8 need not all be equal (Xu et al., 2 Mar 2026).

A central clarification is that constant overlap does not imply constant trace. If one assumes only rank one and $d$9 for all $\{E_x\}$0, then each trace $\{E_x\}$1 satisfies

$\{E_x\}$2

so the traces can take at most two values,

$\{E_x\}$3

This is the structural reason semi-SIC POVMs are genuinely broader than SIC-POVMs rather than a merely redundant reformulation (Geng et al., 2020).

Historically, the 2020 analysis fully established this mechanism and completely classified the qubit case while leaving higher-dimensional existence open (Geng et al., 2020). Subsequent work, as summarized in the 2026 experimental study following Geng, Golubeva, and Gour, treats semi-SIC POVMs as existing in every finite dimension (Xu et al., 2 Mar 2026). This shift is important in the modern literature.

2. The qubit family and its geometry

For $\{E_x\}$4, semi-SIC POVMs have four outcomes, which is the minimal number for qubit informational completeness. The family studied in both the self-testing and experimental works is parameterized by

$\{E_x\}$5

Its four elements can be written as

$\{E_x\}$6

with

$\{E_x\}$7

and

$\{E_x\}$8

where

$\{E_x\}$9

These vectors are unit Bloch vectors, so each effect is rank one (Drótos et al., 2023).

The traces occur in two pairs,

$E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$0

while all pairwise overlaps remain constant,

$E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$1

At the endpoint $E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$2, one has $E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$3, and the family reduces to the usual qubit SIC-POVM, namely the tetrahedral measurement (Geng et al., 2020).

The geometry interpolates continuously between the tetrahedral SIC and a degenerate limit. In the Bloch representation, the semi-SIC vectors form a digonal disphenoid, with four equal edges and two orthogonal opposite edges; when $E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$4, this becomes the regular tetrahedron of the qubit SIC (Drótos et al., 2023). At the excluded endpoint $E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$5, one gets $E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$6 and $E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$7, so informational completeness is lost (Geng et al., 2020).

The qubit case is also the only case that has been completely classified in explicit closed form in the cited literature. Every qubit semi-SIC POVM is unitarily equivalent to a member of this one-parameter family (Geng et al., 2020). This makes the qubit family the canonical testbed for realization, certification, and geometric analysis.

3. Informational completeness and reconstruction

By definition, a semi-SIC POVM is informationally complete, so its outcome probabilities determine the state. The structural difference from a SIC-POVM appears not at the level of informational completeness, but in the reconstruction algebra. Because the traces split into two classes, the dual basis is no longer given by a single affine formula $E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$8 valid for all $E_x \ge 0,\qquad \sum_x E_x = \mathbb{I}_d.$9; instead it depends on which trace class the effect belongs to (Geng et al., 2020).

If $p(x|\rho)=\mathrm{Tr}(\rho E_x)$0 of the $p(x|\rho)=\mathrm{Tr}(\rho E_x)$1 effects have trace $p(x|\rho)=\mathrm{Tr}(\rho E_x)$2 and the remaining $p(x|\rho)=\mathrm{Tr}(\rho E_x)$3 have trace $p(x|\rho)=\mathrm{Tr}(\rho E_x)$4, define

$p(x|\rho)=\mathrm{Tr}(\rho E_x)$5

Then one dual basis $p(x|\rho)=\mathrm{Tr}(\rho E_x)$6 satisfying $p(x|\rho)=\mathrm{Tr}(\rho E_x)$7 is

$p(x|\rho)=\mathrm{Tr}(\rho E_x)$8

State reconstruction then takes the linear form

$p(x|\rho)=\mathrm{Tr}(\rho E_x)$9

(Geng et al., 2020).

This is a useful point of contrast with both ordinary SIC-POVMs and broader symmetric measurement frameworks. In a SIC-POVM, the full symmetry yields a notably simpler dual frame. In more general symmetric families, such as the $\rho$0-POVMs introduced as informationally complete symmetric measurements, reconstruction again takes a uniform closed form,

$\rho$1

with

$\rho$2

provided the symmetry parameters satisfy the $\rho$3-POVM constraints (Siudzińska, 2021). The semi-SIC case occupies an intermediate position: more symmetric than a generic IC POVM, but less uniform than a SIC.

A recurring misconception is that semi-SICs are merely “approximate SICs.” In the exact Geng-type definition they are not approximate objects: the rank-one condition and the off-diagonal overlap condition are exact, while only the equal-trace condition is relaxed. Approximate symmetry belongs instead to the ASIC literature, where overlaps are bounded near the SIC value rather than fixed exactly (Cao et al., 2023).

4. Photonic quantum-walk realization

The first reported experimental realization of qubit semi-SIC POVMs uses a one-dimensional discrete-time quantum walk of a single photon (Xu et al., 2 Mar 2026). In that architecture, the qubit to be measured is the coin space, encoded in photon polarization,

$\rho$4

while the walker position $\rho$5 is encoded in spatial modes.

A single step is

$\rho$6

where $\rho$7 is a coin operator and

$\rho$8

is the conditional translation. With site-dependent coin operations $\rho$9 at step $d$0,

$d$1

and each coin is implemented with half-wave plates and quarter-wave plates (Xu et al., 2 Mar 2026).

The experiment uses a five-step quantum walk to realize the four-outcome semi-SIC POVM on the polarization qubit. The apparatus contains 5 beam displacers implementing the conditional translations, 10 wave plates implementing the site-dependent coin operations, and four output ports corresponding to the four POVM outcomes $d$2. By choosing the wave-plate angles appropriately, the setup realizes the qubit semi-SIC family for $d$3 (Xu et al., 2 Mar 2026).

For $d$4, the paper reports a concrete comparison between the ideal and measured outcome probabilities for a test input state. The theoretical values were

$d$5

while the experimental values were

$d$6

$d$7

The maximum deviation was reported as $d$8, indicating close agreement with the ideal semi-SIC target (Xu et al., 2 Mar 2026).

Operationally, the quantum-walk realization is significant because the walker-position readout directly induces a nonprojective POVM on the polarization qubit. The same architecture also implements the tetrahedral SIC when $d$9, so the semi-SIC family is realized as a continuous deformation of a standard benchmark measurement rather than as an unrelated device-specific construction (Xu et al., 2 Mar 2026).

5. Self-testing and semi-device-independent certification

Semi-SIC POVMs have also been studied from the viewpoint of self-testing in a semi-device-independent prepare-and-measure scenario. The basic setting uses a qubit dimension bound and otherwise treats the preparation and measurement devices as black boxes. The simplest scenario identified so far for self-testing the semi-SIC family involves four preparations and four measurements, where the first three measurements are dichotomic and the fourth is a four-outcome POVM to be certified (Drótos et al., 2023).

The analytical treatment starts with a linear witness

$d^2$0

specialized for the first three measurements to a $d^2$1 witness matrix depending on two parameters $d^2$2 and $d^2$3. Optimizing this witness over qubit strategies self-tests four pure states whose Bloch vectors match those of the target semi-SIC family once

$d^2$4

are chosen with the same sign (Drótos et al., 2023).

For this choice, the maximal witness value is

$d^2$5

At $d^2$6, corresponding to the qubit SIC, this yields $d^2$7 (Drótos et al., 2023). The four-outcome POVM is then self-tested by extending the witness to

$d^2$8

so that saturation of the quantum maximum forces the fourth measurement to coincide, up to unitary or antiunitary equivalence, with the target semi-SIC POVM (Xu et al., 2 Mar 2026).

The 2026 experiment implements this certification scheme with single photons and linear optics. For $d^2$9, the measured witness values were

$d^2$0

while the corresponding theoretical maxima were

$d^2$1

The paper interprets this proximity, allowing for wave-plate inaccuracies and detector efficiencies, as semi-device-independent certification of the implemented semi-SIC POVMs (Xu et al., 2 Mar 2026).

A key structural reason this program works is that the qubit semi-SIC family is extremal for $d^2$2: the four unit Bloch vectors satisfy the geometric noncoplanarity requirement for extremal four-outcome qubit POVMs. Since non-extremal POVMs can be simulated as convex mixtures of extremal ones, extremality is essential for self-testing (Drótos et al., 2023).

6. Related frameworks, applications, and terminological variants

Semi-SIC POVMs sit within a broader hierarchy of symmetric or partially symmetric informationally complete measurements. One nearby class is that of general SIC-POVMs, where the effects still have equal trace and constant Hilbert–Schmidt overlaps but need not be rank one. These are parameterized by $d^2$3, with

$d^2$4

where $d^2$5 gives the usual rank-one SIC and intermediate values correspond to higher-rank symmetric IC measurements (Rastegin, 2013).

A more expansive framework is the class of informationally complete symmetric $d^2$6-POVMs, which generalizes both SIC-POVMs and mutually unbiased measurements. In that setting, the symmetry is distributed across multiple POVMs through uniform intra-POVM and inter-POVM overlaps. The authors explicitly remark that their construction can be generalized “in a manner similar to the introduction of semi-SIC POVMs,” placing semi-SICs within a wider program of partial-symmetry measurement design (Siudzińska, 2021).

Another adjacent notion is the ASIC-POVM, where one keeps rank one and informational completeness but replaces exact equiangularity by overlap bounds of the form

$d^2$7

The constructions in dimensions $d^2$8 and $d^2$9 discussed in that literature yield biangular frames or asymptotically optimal codebooks rather than exact semi-SICs. They are therefore best understood as approximate or asymptotic relatives, not as exact instances of the Geng-type semi-SIC definition (Cao et al., 2023).

There is also an important terminological variant. In the tomography literature with prior information, the phrase “conditional SIC-POVM” denotes a POVM that is SIC-like on the unknown-parameter subspace and orthogonal to the known part. Such POVMs satisfy scaled-projection and constant-overlap conditions together with

$E_x$0

for operators $E_x$1 corresponding to known parameters. This is a different generalization from the Geng-type semi-SIC, because the relaxation concerns the domain of informational completeness rather than the equal-trace condition [(Petz et al., 2012); (Petz et al., 2015)]. Conflating these two usages is a common source of ambiguity.

The applications attached to semi-SICs and nearby symmetric IC measurements reflect this structural position between full SIC symmetry and generic POVMs. The experimental and self-testing works emphasize quantum state tomography, quantum certification, and broader uses of nonprojective multi-outcome POVMs in quantum cryptography and randomness generation (Xu et al., 2 Mar 2026). The $E_x$2-POVM framework extends SIC- and MUB-style techniques to entropic uncertainty relations and entanglement detection (Siudzińska, 2021), while general SIC-POVMs admit exact formulas for the index of coincidence and associated Tsallis and Rényi entropic bounds (Rastegin, 2013). This suggests that semi-SIC POVMs are best viewed not as isolated curiosities, but as one exact and operationally accessible branch of a broader theory of symmetric informational completeness under relaxed constraints.