A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements on a pair of identically prepared quantum states that are Fisher symmetric for all pure states. Such measurements are optimal in achieving the minimal statistical error without adaptive measurements. We then determine all collective measurements on a pair that are Fisher symmetric for the completely mixed state and for all pure states simultaneously. For a qubit, these measurements are Fisher symmetric for all states. The minimal optimal measurements are tied to the elusive symmetric informationally complete measurements, which reflects a deep connection between local symmetry and global symmetry. In the study, we derive a fundamental constraint on the Fisher information matrix of any collective measurement on a pair, which offers a useful tool for characterizing the tomographic efficiency of collective measurements.A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements on a pair of identically prepared quantum states that are Fisher symmetric for all pure states. Such measurements are optimal in achieving the minimal statistical error without adaptive measurements. We then determine all collective measurements on a pair that are Fisher symmetric for the completely mixed state and for all pure states simultaneously. For a qubit, these measurements are Fisher symmetric for all states. The minimal optimal measurements are tied to the elusive symmetric informationally complete measurements, which reflects a deep connection between local symmetry and global symmetry. In the study, we derive a fundamental constraint on the Fisher information matrix of any collective measurement on a pair, which offers a useful tool for characterizing the tomographic efficiency of collective measurements.

arXiv:1709.06112Physical Review LettersJan 19, 201840 citations

Pure-State Quantum Tomography with Minimal Rank-One POVMs

Dan Edidin, Ivan Gonzalez, Itzhak Tamo

Quantum state tomography seeks to reconstruct an unknown state from measurement statistics. A finite measurement (POVM) is \emph{pure-state informationally complete} (PSI-Complete) if the outcome probabilities determine any pure state up to a global phase. We study \emph{rank-one} POVMs that are minimally sufficient for this task. We call such a POVM \emph{vital} if it is PSI-Complete but every proper subcollection is not PSI-Complete. We prove sharp upper bounds on the size of vital rank-one POVMs in dimension $n$ : the size is at most $\binom{n+1}{2}$ over $\mathbb{R}$ and at most $n^{2}$ over $\mathbb{C}$ , and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever $w \mid n(n-1)$ , an $(n,w,w-1)$ design produces a vital rank-one POVM with $n + n(n-1)/w$ outcomes. We provide explicit constructions for $w=2,n-1$ , and $n$ .Quantum state tomography seeks to reconstruct an unknown state from measurement statistics. A finite measurement (POVM) is \emph{pure-state informationally complete} (PSI-Complete) if the outcome probabilities determine any pure state up to a global phase. We study \emph{rank-one} POVMs that are minimally sufficient for this task. We call such a POVM \emph{vital} if it is PSI-Complete but every proper subcollection is not PSI-Complete. We prove sharp upper bounds on the size of vital rank-one POVMs in dimension $n$: the size is at most $\binom{n+1}{2}$ over $\mathbb{R}$ and at most $n^{2}$ over $\mathbb{C}$, and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever $w \mid n(n-1)$, an $(n,w,w-1)$ design produces a vital rank-one POVM with $n + n(n-1)/w$ outcomes. We provide explicit constructions for $w=2,n-1$, and $n$.

quant-pharXiv:2511.09505Nov 13, 20250 citations

Super-symmetric informationally complete measurements

Huangjun Zhu

arXiv:1412.1099Annals of PhysicsAug 10, 201540 citations

From minimal informationally complete measurements to orthocentric simplices and back again

Piotr Bereza, Wojciech Słomczyński, Anna Szymusiak

The reconstruction of unknown quantum states via minimal informationally complete measurements (MICs) is a cornerstone of quantum tomography. Although the statistical properties of these measurements are well-understood, their geometric structure has remained elusive. In this work, we establish a correspondence between the class of minimal $s$ -tight informationally complete measurements, encompassing, among others, tight IC and morphophoric measurements, and the classical geometry of orthocentric simplices. In particular, we prove a three-way equivalence: a MIC is $s$ -tight if and only if its measurement vectors, upon suitable rescaling, form the vertices of an acute orthocentric simplex with the orthocentre at the origin, and such simplices are precisely the homothetically self-dual ones. This geometric manifestation of operational ''tightness'' provides a bridge between the physical world and Euclidean geometry. Furthermore, the $s$ -tight class is fully characterised by its measurement directions: the angles between them must be obtuse and satisfy a cross-ratio condition. We determine the space of admissible direction configurations: modulo rotations, every such configuration is encoded by a single probability vector, the ''skeleton'' of the measurement, together with an orientation class, so that the moduli space of $s$ -tight MIC directions is $Δ^{\circ}_{d+1}\times\{\pm 1\}$ . Conversely, every acute orthocentric simplex with the orthocentre at the origin can be anchored in the state space, generating a class of minimal $s$ -tight IC measurements that contains exactly one tight IC measurement up to overall rescaling.The reconstruction of unknown quantum states via minimal informationally complete measurements (MICs) is a cornerstone of quantum tomography. Although the statistical properties of these measurements are well-understood, their geometric structure has remained elusive. In this work, we establish a correspondence between the class of minimal $s$-tight informationally complete measurements, encompassing, among others, tight IC and morphophoric measurements, and the classical geometry of orthocentric simplices. In particular, we prove a three-way equivalence: a MIC is $s$-tight if and only if its measurement vectors, upon suitable rescaling, form the vertices of an acute orthocentric simplex with the orthocentre at the origin, and such simplices are precisely the homothetically self-dual ones. This geometric manifestation of operational ''tightness'' provides a bridge between the physical world and Euclidean geometry. Furthermore, the $s$-tight class is fully characterised by its measurement directions: the angles between them must be obtuse and satisfy a cross-ratio condition. We determine the space of admissible direction configurations: modulo rotations, every such configuration is encoded by a single probability vector, the ''skeleton'' of the measurement, together with an orientation class, so that the moduli space of $s$-tight MIC directions is $Δ^{\circ}_{d+1}\times\{\pm 1\}$. Conversely, every acute orthocentric simplex with the orthocentre at the origin can be anchored in the state space, generating a class of minimal $s$-tight IC measurements that contains exactly one tight IC measurement up to overall rescaling.

quant-pharXiv:2608.00809Aug 4, 20260 citations

Finitely Correlated Pure States

M. Fannes, Bruno Nachtergaele, Reinhard F. Werner

Journal of Functional AnalysisMar 1, 199474 citations

Finitely Correlated Pure States

Reinhard F. Werner

NATO ASI series. Series B : PhysicsJan 1, 199469 citations

n

: the size is at most

\binom{n+1}{2}

over

\mathbb{R}

and at most

n^{2}

over

\mathbb{C}

, and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever

w \mid n(n-1)

, an

(n,w,w-1)

design produces a vital rank-one POVM with

n + n(n-1)/w

outcomes. We provide explicit constructions for

w=2,n-1

, and

n

s

-tight informationally complete measurements, encompassing, among others, tight IC and morphophoric measurements, and the classical geometry of orthocentric simplices. In particular, we prove a three-way equivalence: a MIC is

s

-tight if and only if its measurement vectors, upon suitable rescaling, form the vertices of an acute orthocentric simplex with the orthocentre at the origin, and such simplices are precisely the homothetically self-dual ones. This geometric manifestation of operational ''tightness'' provides a bridge between the physical world and Euclidean geometry. Furthermore, the

s

-tight class is fully characterised by its measurement directions: the angles between them must be obtuse and satisfy a cross-ratio condition. We determine the space of admissible direction configurations: modulo rotations, every such configuration is encoded by a single probability vector, the ''skeleton'' of the measurement, together with an orientation class, so that the moduli space of

s

-tight MIC directions is

Δ^{\circ}_{d+1}\times\{\pm 1\}

. Conversely, every acute orthocentric simplex with the orthocentre at the origin can be anchored in the state space, generating a class of minimal

s

-tight IC measurements that contains exactly one tight IC measurement up to overall rescaling.