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arXiv:1802.00140

Minimal Determination of a Pure State through Adaptive Tomography

Liang-Liang Sun, Yingqiu, Mao, Fei-Lei Xiong, Sixia, Yu, Zeng‐Bing Chen

Published Feb 1, 2018• 0 citations
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Abstract

Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. In the fixed measurement scheme four orthonormal bases are required even though there are far less parameters in a pure state. Peres conjectured that two unbiased bases suffice to determine a pure state up to some finite ambiguities. Here we shall at first prove Peres conjecture in the case of $d=3,4$, namely, two unbiased measurements determine a pure state up to to 6 and 16 candidates for a qutrit and ququad, respectively. And then, taking Peres' conjecture for established, we propose an adaptive 3-measurement scheme involving the minimal number of measurements, based on the observation that the ambiguities can be removed by an adaptive two-outcome projective measurement. With the help of this observation, we simplify a recent five-basis protocol $[Phys. Rev. Lett. 115, 090401 (2015)]$ to a three-basis one at the cost of the extra dichotomic measurement.

DOI
10.48550/arxiv.1802.00140

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Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. In the fixed measurement scheme four orthonormal bases are required even though there are far less parameters in a pure state. Peres conjectured that two unbiased bases suffice to determine a pure state up to some finite ambiguities. Here we shall at first prove Peres conjecture in the case of d=3,4d=3,4d=3,4, namely, two unbiased measurements determine a pure state up to to 6 and 16 candidates for a qutrit and ququad, respectively. And then, taking Peres' conjecture for established, we propose an adaptive 3-measurement scheme involving the minimal number of measurements, based on the observation that the ambiguities can be removed by an adaptive two-outcome projective measurement. With the help of this observation, we simplify a recent five-basis protocol [Phys.Rev.Lett.115,090401(2015)][Phys. Rev. Lett. 115, 090401 (2015)][Phys.Rev.Lett.115,090401(2015)] to a three-basis one at the cost of the extra dichotomic measurement.
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.
Efficiently extracting information from pure quantum states using minimal observables on the main system is a longstanding and fundamental issue in quantum information theory. Despite the inability of probability distributions of position and momentum to uniquely specify a wavefunction, Peres conjectured a discrete version wherein two complementary observables, analogous to position and momentum and realized as projective measurements onto orthogonal bases, can determine all pure qudits up to a finite set of ambiguities. Subsequent findings revealed the impossibility of uniquely determining ddd-dimenisonal pure states even when neglecting a measure-zero set with any two orthogonal bases, and Peres's conjecture is also correct for d=3d=3d=3 but not for d=4d=4d=4. In this study, we show that two orthogonal bases are capable of effectively filtering up to 2d−12^{d-1}2d−1 finite candidates by disregarding a measure-zero set, without involving complex numbers in the bases' coefficients. Additionally, drawing inspiration from sequential measurements to directly calculate the target coefficients of the wavefunction using two complementary observables, we show that almost all pure qudits can be uniquely determined by adaptively incorporating a POVM in the middle, followed by measuring the complementary observable.
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.
In this paper, we focus on two-qubit pure state tomography. For an arbitrary unknown two-qubit pure state, separable or entangled, it has been found that the measurement probabilities of 16 projections onto the tensor products of Pauli eigenstates are enough to uniquely determine the state. Moreover, these corresponding product states are arranged into five orthonormal bases. We design five quantum circuits, which are decomposed into the common gates in universal quantum computation, to simulate the five projective measurements onto these bases. At the end of each circuit, we measure each qubit with the projective measurement {|0⟩⟨0|,|1⟩⟨1|}. Then, we consider the open problem whether three orthonormal bases are enough to distinguish all two-qubit pure states. A necessary condition is given. Suppose that there are three orthonormal bases { B 1 , B 2 , B 3 } . Denote the unitary transition matrices from B 1 to { B 2 , B 3 } as U 1 and U 2 . All 32 elements of matrices U 1 and U 2 should not be zero. If not, these three bases cannot distinguish all two-qubit pure states.
We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary nnn-qubit pure state among all quantum states. We show that only 11 Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with 16 measurements, and only 31 Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with 64 measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. We experimentally test the protocol on two- and three-qubit systems with nuclear magnetic resonance techniques. We show that the pure-state tomography protocol saves us a number of measurements without considerable loss of fidelity. We compare our protocol with same-size sets of randomly selected Pauli operators and find that our selected set of Pauli measurements significantly outperforms those random sampling sets. As a direct application, our scheme can also be used to reduce the number of settings needed for pure-state tomography in quantum optical systems.
Full Pauli-basis tomography of an arbitrary nnn-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 nnn-qubit systems with n≥3n\geq 3n≥3, every pure state is UDA whenever the omitted Pauli observables contain no four distinct operators that commute pairwise and multiply to ±I\pm I±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.