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arXiv:2511.09505

Pure-State Quantum Tomography with Minimal Rank-One POVMs

Dan Edidin, Ivan Gonzalez, Itzhak Tamo

quant-phcs.ITmath.COSubmitted Nov 12, 2025• Announced Nov 13, 2025• 0 citations
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Abstract

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 \(n\).

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15 pages

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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).
Abstract. We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d − 4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log2(d) smaller than this upper bound. 1.
While quantum state tomography (QST) remains the gold standard for benchmarking and verifying quantum devices, it requires an exponentially large number of measurements and classical computational resources for generic quantum many-body systems, making it impractical even for intermediate-size quantum devices. Fortunately, many physical quantum states often exhibit certain low-dimensional structures that enable the development of efficient QST. A notable example is the class of states represented by matrix product operators (MPOs) with a finite matrix/bond dimension, which include most physical states in one dimension and where the number of independent parameters describing the states only grows linearly with the number of qubits. Whether a sample efficient quantum state tomography protocol, where the number of required state copies scales only linearly as the number of parameters describing the state, exists for a generic MPO state still remains an important open question. In this paper, we answer this fundamental question affirmatively by using a class of informationally complete positive operator-valued measures (IC-POVMs) -- including symmetric IC-POVMs (SIC-POVMs) and spherical ttt-designs -- focusing on sample complexity while not accounting for the implementation complexity of the measurement settings. For SIC-POVMs and (approximate) spherical 2-designs, we show that the number of state copies to guarantee bounded recovery error of an MPO state with a constrained least-squares estimator depends on the probability distribution of the MPO under the POVM but scales only linearly with nnn when the distribution is approximately uniform. For spherical ttt-designs with t≥3t\geq 3t≥3, we prove that only a number of state copies proportional to the number of independent parameters in the MPO is sufficient for a guaranteed recovery of any state represented by an MPO.
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most ttt samples. For sufficiently small ε\varepsilonε, estimating an unknown state on Cd\mathbb{C}^dCd of rank at most rrr to trace norm error ε\varepsilonε with constant success probability requires, and is achievable with, Θ(drε2max{1,rt})Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)Θ(ε2dr​max{1,t​r​}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most ttt samples improve the complexity of algorithms making single-sample measurements by at most a factor t\sqrt{t}t​. Further, measuring order r2r^2r2 samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on ttt samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
For any finite-dimensional Hilbert space, we construct explicitly five orthonormal bases such that the corresponding measurements allow for efficient tomography of an arbitrary pure quantum state. This means that such measurements can be used to distinguish an arbitrary pure state from any other state, pure or mixed, and the pure state can be reconstructed from the outcome distribution in a feasible way. The set of measurements we construct is independent of the unknown state, and therefore our results provide a fixed scheme for pure state tomography, as opposed to the adaptive (state-dependent) scheme proposed by Goyeneche et al. ( Phys. Rev. Lett. , 115 (2015) 090401). We show that our scheme is robust with respect to noise, in the sense that any measurement scheme which approximates these measurements well enough is equally suitable for pure state tomography. Finally, we present two convex programs which can be used to reconstruct the unknown pure state from the measurement outcome distributions.
We consider the problem of quantum-state tomography under the assumption that the state is pure, and more generally that its rank is bounded by a given value rrr. In this scenario two notions of informationally complete measurements emerge: rank-rrr-complete measurements and rank-rrr strictly-complete measurements. Whereas in the first notion, a rank-rrr state is uniquely identified from within the set of rank-rrr states, in the second notion the same state is uniquely identified from within the set of all physical states, of any rank. We argue, therefore, that strictly-complete measurements are compatible with convex optimization, and we prove that they allow robust quantum-state estimation in the presence of experimental noise. We also show that rank-rrr strictly-complete measurements are as efficient as rank-rrr-complete measurements. We construct examples of strictly-complete measurements and give a complete description of their structure in the context of matrix completion. Moreover, we numerically show that a few random bases form such measurements. We demonstrate the efficiency-robustness property for different strictly-complete measurements with numerical experiments. We thus conclude that only strictly-complete measurements are useful for practical tomography.
Quantum state tomography estimates an unknown density operator from measurement data. Dense reconstruction however, can be impractical for many-qubit systems because the Hilbert-space dimension grows exponentially. This contribution develops a rank-adaptive matrix-free approach to low-rank quantum state tomography based on rank-one atomic coordinates. The density operator is represented as a convex combination of pure-state atoms, which preserves positivity and unit trace while avoiding dense density, measurement, and gradient matrices. The resultant algorithm combines atom updates, simplex-constrained coefficient reweighting, and periodic spectral refactorization using only predicted probability vectors, atom vectors, and descriptor-level measurement actions. These computations decompose across measurement outcomes, and admit a master--worker implementation with the measurements partitioned across workers. A rank penalty adapts the representation size during optimization. Provable feasibility, prediction consistency, monotone descent of the penalized objective, and an exact spectral proximal characterization of the refactorization step are established. Simulations with Pauli measurements show favorable accuracy--runtime--memory tradeoffs relative to competing alternatives.