Abstract In this paper, we establish a dynamical quantum state tomography framework. Under this framework, it is feasible to obtain complete knowledge of any unknown state of a d-level system via only one operator of a special type of positive operator-valued measure (POVM) in dimension d. We define a new channel, referred to as the time-dependent average channel. Utilizing this channel, we show that we can acquire a collection of projective operators that is informationally complete (IC) and thus obtain the corresponding informationally complete POVMs (IC-POVMs). Zauner conjectured that for any dimension d there exists a fiducial vector, such that all remaining d 2 − 1 elements of the desired symmetric informationally complete POVM (SIC-POVM) can be obtained by acting on said vector with unitary matrices representing elements of the Weyl-Heisenberg group. We show that under certain condition, it is possible to obtain infinite families of projective operators that are IC, and obtain infinite families of corresponding IC-POVMs; otherwise, Zauner’s conjecture is incorrect. We also show how to simulate a SIC-POVM on any unknown quantum state by using the time-dependent average channel.
Abstract The article establishes a framework for dynamic generation of informationally complete POVMs in quantum state tomography. Assuming that the evolution of a quantum system is given by a dynamical map in the Kraus representation, one can switch to the Heisenberg picture and define the measurements in the time domain. Consequently, starting with an incomplete set of positive operators, one can obtain sufficient information for quantum state reconstruction by multiple measurements. The framework has been demonstrated on qubits and qutrits. For some types of dynamical maps, it suffices to initially have one measurement operator. The results demonstrate that quantum state tomography is feasible even with limited measurement potential.
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
t-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
n when the distribution is approximately uniform. For spherical
t-designs with
t≥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.
It is a well-known fact that the optimal positive operator valued measure (POVM) for quantum-state tomography is the symmetric, informationally complete POVM (SIC-POVM). We investigate the same problem only in the case when there is some a priori information about the state, specifically when some parameters are known. In this paper, we mainly focus on solving a three-dimensional optimization problem, which gives us a non-trivial example for the so-called conditional SIC-POVMs, a straightforward generalization of the concept of SIC-POVMs. We also present other special cases to show further applications of the proposed numerical methods and to illustrate the complexity of this topic.
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
(2n+1) over
R and at most
n2 over
C, and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever
w∣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.
This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in
Cn whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such
n2 vectors in
Cn for any
n .But the existence is only proved when
n≤15 and
n=19 , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in
Cn .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1
ρ∈Mn(C) is called a density matrix (or state) if
ρ≥0 and
′b(ρ)=1.For any density matrix
ρ , we can define a state
ρ^ by
ρ^(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t
(≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators
{Pi}i=1k⊂Mn(C) is called a positive operator valued measure (POVM) if
Pi≥0(1≤i≤k) and
∑i=1kP=I.If
P1 is a projection, then
{Pi}i=1k is called a projection valued measure
(PVM) .