--- title: 'Semi-SIC POVM: Relaxed SIC Measurements' url: https://www.emergentmind.com/topics/semisymmetric-informationally-complete-povm-semi-sic-povm type: topic --- # Semi-SIC POVM: Relaxed SIC Measurements 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 [2007.10483; 2603.01802]. ## 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 [2603.01802]. 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 \(\mathrm{Tr}(E_xE_y)\) for \(x\neq y\) are equal, and all traces \(\mathrm{Tr}(E_x)\) are equal. The usual SIC normalization gives \(\mathrm{Tr}(E_x)=1/d\), and the SIC overlap value is \[ \mathrm{Tr}(E_xE_y)=\frac{1}{d^2(d+1)}\qquad (x\neq y) \] [2007.10483]. 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 \[ E_x=a_x\,|\psi_x\rangle\langle\psi_x|,\qquad a_x>0,\qquad \sum_x E_x=\mathbb{I}_d, \] with \[ \mathrm{Tr}(E_xE_y)=B\qquad \forall x\neq y, \] for some fixed \(B>0\), while the traces \(a_x=\mathrm{Tr}(E_x)\) need not all be equal [2603.01802]. A central clarification is that **constant overlap does not imply constant trace**. If one assumes only rank one and \(\mathrm{Tr}(E_xE_y)=b\) for all \(x\neq y\), then each trace \(a_x=\mathrm{Tr}(E_x)\) satisfies \[ a_x^2-a_x+(d^2-1)b=0, \] so the traces can take at most two values, \[ a_\pm=\frac{1\pm\sqrt{1-4b(d^2-1)}}{2}. \] This is the structural reason semi-SIC POVMs are genuinely broader than SIC-POVMs rather than a merely redundant reformulation [2007.10483]. Historically, the 2020 analysis fully established this mechanism and completely classified the qubit case while leaving higher-dimensional existence open [2007.10483]. Subsequent work, as summarized in the 2026 experimental study following Geng, Golubeva, and Gour, treats semi-SIC POVMs as existing in every finite dimension [2603.01802]. This shift is important in the modern literature. ## 2. The qubit family and its geometry For \(d=2\), 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 \[ B\in\left(\frac{1}{16},\frac{1}{12}\right]. \] Its four elements can be written as \[ \begin{aligned} E_1&=\frac{a_-}{2}\bigl(I+\vec h_1\cdot\vec\sigma\bigr),& E_2&=\frac{a_-}{2}\bigl(I+\vec h_2\cdot\vec\sigma\bigr),\\ E_3&=\frac{a_+}{2}\bigl(I+\vec h_3\cdot\vec\sigma\bigr),& E_4&=\frac{a_+}{2}\bigl(I+\vec h_4\cdot\vec\sigma\bigr), \end{aligned} \] with \[ a_\pm=\frac{1\pm\sqrt{1-12B}}{2}, \] and \[ \begin{aligned} \vec h_1&=(r_-,r_-,q_-),& \vec h_2&=(-r_-,-r_-,q_-),\\ \vec h_3&=(-r_+,r_+,-q_+),& \vec h_4&=(r_+,-r_+,-q_+), \end{aligned} \] where \[ q_\pm=\frac{\sqrt{B}}{a_\pm},\qquad r_\pm=\sqrt{\frac{1-q_\pm^2}{2}}. \] These vectors are unit Bloch vectors, so each effect is rank one [2306.07248]. The traces occur in two pairs, \[ \mathrm{Tr}(E_1)=\mathrm{Tr}(E_2)=a_-,\qquad \mathrm{Tr}(E_3)=\mathrm{Tr}(E_4)=a_+, \] while all pairwise overlaps remain constant, \[ \mathrm{Tr}(E_iE_j)=B\qquad \forall i\neq j. \] At the endpoint \(B=1/12\), one has \(a_+=a_-=1/2\), and the family reduces to the usual qubit SIC-POVM, namely the tetrahedral measurement [2007.10483]. 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 \(B\to 1/12\), this becomes the regular tetrahedron of the qubit SIC [2306.07248]. At the excluded endpoint \(B=1/16\), one gets \(E_1=E_2\) and \(E_3=E_4\), so informational completeness is lost [2007.10483]. 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 [2007.10483]. 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 \(F_x=\alpha E_x+\beta I\) valid for all \(x\); instead it depends on which trace class the effect belongs to [2007.10483]. If \(k\) of the \(d^2\) effects have trace \(a_-\) and the remaining \(d^2-k\) have trace \(a_+\), define \[ S=\sum_{y=1}^{k}E_y,\qquad T=\sum_{y=k+1}^{d^2}E_y. \] Then one dual basis \(\{F_y\}\) satisfying \(\mathrm{Tr}(E_xF_y)=\delta_{xy}\) is \[ F_y= \begin{cases} \displaystyle \frac{1}{a_-^2-b}E_y +\frac{a_+^2-a_-^2}{(a_-^2-b)(1-d)}\,S +\frac{1}{1-d}I_d, & 1\le y\le k,\\[2ex] \displaystyle \frac{1}{a_+^2-b}E_y +\frac{a_-^2-a_+^2}{(a_+^2-b)(1-d)}\,T +\frac{1}{1-d}I_d, & k+1\le y\le d^2. \end{cases} \] State reconstruction then takes the linear form \[ \rho=\sum_{y=1}^{d^2} p_y F_y,\qquad p_y=\mathrm{Tr}(E_y\rho) \] [2007.10483]. 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 \((N,M)\)-POVMs introduced as informationally complete symmetric measurements, reconstruction again takes a uniform closed form, \[ \rho=\sum_{\alpha=1}^{N}\sum_{k=1}^{M} p_{\alpha,k}\,F_{\alpha,k}, \] with \[ F_{\alpha,k}=\frac{1}{x-y}\left(E_{\alpha,k}-\frac{(N-1)z+y}{Nw}\,\mathbb{I}_d\right), \] provided the symmetry parameters satisfy the \((N,M)\)-POVM constraints [2111.08101]. 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 [2310.06418]. ## 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 [2603.01802]. In that architecture, the qubit to be measured is the **coin space**, encoded in photon polarization, \[ |H\rangle\equiv|0\rangle,\qquad |V\rangle\equiv|1\rangle, \] while the walker position \(x\in\mathbb{Z}\) is encoded in spatial modes. A single step is \[ U=TC, \] where \(C\) is a coin operator and \[ T=\sum_x \bigl(|x+1,H\rangle\langle x,H|+|x-1,V\rangle\langle x,V|\bigr) \] is the conditional translation. With site-dependent coin operations \(C_{x,n}\) at step \(n\), \[ C=\sum_x |x\rangle\langle x|\otimes C_{x,n}, \] and each coin is implemented with half-wave plates and quarter-wave plates [2603.01802]. 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 \(E_1,\dots,E_4\). By choosing the wave-plate angles appropriately, the setup realizes the qubit semi-SIC family for \(B=1/12,1/13,1/14,1/15\) [2603.01802]. For \(B=1/13\), the paper reports a concrete comparison between the ideal and measured outcome probabilities for a test input state. The theoretical values were \[ P(E_1)=0.1807,\quad P(E_2)=0.3584,\quad P(E_3)=0.2305,\quad P(E_4)=0.2305, \] while the experimental values were \[ P(E_1)=0.1832\pm0.0023,\quad P(E_2)=0.3563\pm0.0021, \] \[ P(E_3)=0.2316\pm0.0022,\quad P(E_4)=0.2289\pm0.0015. \] The maximum deviation was reported as \(<0.005\), indicating close agreement with the ideal semi-SIC target [2603.01802]. 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 \(B=1/12\), so the semi-SIC family is realized as a continuous deformation of a standard benchmark measurement rather than as an unrelated device-specific construction [2603.01802]. ## 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 [2306.07248]. The analytical treatment starts with a linear witness \[ \mathcal{W}=\sum_{x,y,b} w_{xy}^{(b)}\,p(b|x,y), \] specialized for the first three measurements to a \(4\times 3\) witness matrix depending on two parameters \(c_1\) and \(c_2\). Optimizing this witness over qubit strategies self-tests four pure states whose Bloch vectors match those of the target semi-SIC family once \[ c_1=\pm \sqrt{\frac{2\bigl(1-6B+\sqrt{1-12B}\bigr)}{24B-1}},\qquad c_2=\pm \sqrt{\frac{2\bigl(1-6B-\sqrt{1-12B}\bigr)}{24B-1}} \] are chosen with the same sign [2306.07248]. For this choice, the maximal witness value is \[ Q(B)=24\sqrt{\frac{B}{24B-1}}. \] At \(B=1/12\), corresponding to the qubit SIC, this yields \(Q=4\sqrt{3}\) [2306.07248]. The four-outcome POVM is then self-tested by extending the witness to \[ W=W_1-k\sum_{x=1}^{4} P(b=x|x,y=4), \] 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 [2603.01802]. The 2026 experiment implements this certification scheme with single photons and linear optics. For \(B=1/13,1/14,1/15\), the measured witness values were \[ 7.2908\pm0.0304,\qquad 7.5615\pm0.0149,\qquad 7.9393\pm0.0434, \] while the corresponding theoretical maxima were \[ 7.2363,\qquad 7.5895,\qquad 8. \] The paper interprets this proximity, allowing for wave-plate inaccuracies and detector efficiencies, as semi-device-independent certification of the implemented semi-SIC POVMs [2603.01802]. A key structural reason this program works is that the qubit semi-SIC family is **extremal** for \(B\in(1/16,1/12]\): 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 [2306.07248]. ## 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 \(a=\mathrm{tr}(N_j^2)\), with \[ \frac{1}{d^3}