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Chinese Physics B

Two-qubit pure state tomography by five product orthonormal bases

Yu Wang, Yun Shang

Published Oct 1, 2018• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1674-1056/27/10/100306

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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.
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.
A long-standing problem in quantum mechanics is the minimum number of observables required for the characterization of unknown pure quantum states. The solution to this problem is especially important for the developing field of high-dimensional quantum information processing. In this work we demonstrate that any pure d-dimensional state is unambiguously reconstructed by measuring five observables, that is, via projective measurements onto the states of five orthonormal bases. Thus, in our method the total number of different measurement outcomes (5d) scales linearly with d. The state reconstruction is robust against experimental errors and requires simple postprocessing, regardless of d. We experimentally demonstrate the feasibility of our scheme through the reconstruction of eight-dimensional quantum states, encoded in the momentum of single photons.
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.
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.
The successful implementation of complex quantum algorithms depend crucially on our ability to determine unknown quantum states and operations. In this work, we present adaptive and non-adaptive methods to determine a 1-qubit unitary gate with 5 and 6 Pauli measurements. We demonstrate the method on the Bloch sphere to show that studying higher dimensional real space may help in finding tomography methods of multiple qubit unitary gates. Next, we show an adaptive method to uniquely determine a general d-dimensional pure state among all quantum states with at most 2d - 1 measurements. This method is then applied to determine a general d-dimensional unitary gate with at most d^2 + d - 1 measurements. These methods are applied in tomographing a 2-qubit universal gate set with five unitary gates. On the NMR experimental system, the lowest fidelity achieved was above 97% with 42 Pauli measurements, comparing to 99% using traditional method that requires 240 Pauli measurements.
Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.