Full Pauli-basis tomography of an arbitrary
n-qubit state uses the expectation values of all nonidentity Pauli observables. For multiqubit systems, prior knowledge that the state is pure allows some of these observables to be omitted while retaining uniqueness. We study two resulting notions of uniqueness: whether every pure state is uniquely determined among pure states (UDP), and whether every pure state is uniquely determined among all states, including mixed states (UDA). We derive quartic uniqueness certificates from the fourth elementary symmetric polynomial of operators in the real span of the omitted Pauli observables. An exact quartic identity shows that, for
n-qubit systems with
n≥3, every pure state is UDA whenever the omitted Pauli observables contain no four distinct operators that commute pairwise and multiply to
±I. For three qubits, this condition is also necessary for UDP, yielding an exact characterization of both UDP and UDA. We further prove that UDP and UDA coincide for any number of qubits whenever the omitted Pauli observables commute pairwise. These results provide directly checkable structural criteria for certifying pure-state tomography from restricted Pauli measurements.
Abstract Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. Peres conjectured that two unbiased bases may be sufficient to determine a pure state up to some finite ambiguities. Here we find that the Peres conjecture is correct in the case d = 3 where a pure state is determined up to at most six candidates states and is incorrect in the case of d = 4 for which a counterexample is constructed. We observe that the target state can be picked out from candidates states by an adaptive two-outcome measurement. We thus provided a minimal qutrit tomography protocol constituted of three measurements in contrast to the four measurements required in previous non-adaptive method. We also reduce a five-measurement protocol for a pure qudit state into three full-dimensional measurements and one two-outcome measurement protocol, which exhibits robustness to the usual white noise thus also applies to nearly pure state.
In experiments it is often required that we determine an unknown quantum state produced by a source. Quantum Tomography accomplishes this through repeated measurements on the state. This process requires 4^n measurements for an n-qubit system. However, if it is known that the state is pure, fewer measurements are required. Many such results already exist, but they either use non-local measurements or they have some chance of failing. Our goal is to obtain the smallest set of Pauli operators which can uniquely determine any pure state among all states. This involves ensuring the complement of the span of the measurements must have two positive and two negative eigenvalues. This is nontrivial since there are few relationships between the eigenvalues of a set of operators and the eigenvalues their real linear combinations. We obtain the lower bound of 30 Paulis necessary for 3-qubit tomography which improves on any known result.
We describe a simple method for certifying that an experimental device prepares a desired quantum state ρ. Our method is applicable to any pure state ρ, and it provides an estimate of the fidelity between ρ and the actual (arbitrary) state in the lab, up to a constant additive error. The method requires measuring only a constant number of Pauli expectation values, selected at random according to an importance-weighting rule. Our method is faster than full tomography by a factor of d, the dimension of the state space, and extends easily and naturally to quantum channels.
Abstract We present a three-qubit quantum state tomography scheme requiring a set of 17 measurements, significantly reducing the experimental overhead compared to the conventional 63 Pauli measurements. Using IBM's 127-qubit open-access quantum processor \texttt{ibm\textunderscore{osaka}}, we prepare the three-qubit W state and employ our tomography scheme to reconstruct it. Additionally, we implement a two-qubit tomography protocol, involving a set of 7 measurements, on \texttt{ibm\textunderscore{osaka}} to reconstruct {\em two} of the two-qubit marginals of the W state. This serves as a {\em proof-of-principle} demonstration of the well-known theoretical result that any two of the two-qubit reduced density matrices can uniquely determine most of the whole three-qubit pure states. We show that the fidelity of the W-state reconstructed from its two-qubit subsystems is consistently larger than that obtained from the full three-qubit tomography, highlighting the practical advantage of the subsystem-based tomography approach.
Pauli Measurements are the most important measurements in both theoretical and experimental aspects of quantum information science. In this paper, we explore the power of Pauli measurements in the state tomography related problems. Firstly, we show that the \textit{quantum state tomography} problem of
n-qubit system can be accomplished with
O(ε210n) copies of the unknown state using Pauli measurements. As a direct application, we studied the \textit{quantum overlapping tomography} problem introduced by Cotler and Wilczek in Ref. \cite{Cotler_2020}. We show that the sample complexity is
O(ε210k⋅log((kn)/δ))) for quantum overlapping tomography of
k-qubit reduced density matrices among
n is quantum system, where
1−δ is the confidential level, and
ε is the trace distance error. This can be achieved using Pauli measurements. Moreover, we prove that
Ω(ε2log(n/δ)) copies are needed. In other words, for constant
k, joint, highly entangled, measurements are not asymptotically more efficient than Pauli measurements.