Optimal quantum-state reconstruction for cold trapped ions

Abstract

We study the physical implementation of an optimal tomographic reconstruction scheme for the case of determining the state of a multi-qubit system, where trapped ions are used for defining qubits. The protocol is based on the use of mutually unbiased measurements and on the physical information described in H. Häffner et. al [Nature 438, 643-646 (2005)]. We introduce the concept of physical complexity for different types of unbiased measurements and analyze their generation in terms of one and two qubit gates for trapped ions.PACS numbers: 03.67. Lx, 03.65.Wj, A main task in any experimental physical setup for implementing quantum computation is the ability to determine the output state of any given quantum algorithm [1]. The standard procedure applied for quantum state reconstruction of a density operator lying in a 2 N dimensional quantum system, in the case of N qubits, consists in projecting the density operator onto 3 N , completely factorized, bases in the corresponding Hilbert space [2]. All these measurements are obtained by applying rotations on single qubits (which are referred to as local operations) followed by projective measurements onto the logical basis. This was recently achieved for the case of eight qubits, with trapped ions [3]. The experiment was done by following the quantum computer architecture based on ions in a linear trap proposed by Cirac and Zoller [4]. Besides, the experimental implementations of several quantum protocols have also been reported by using trapped ions [5]. In all these cases the quality of the protocols is tested using standard tomography for quantum state determination. This scheme has also been used in the cases of considering optical setups [6] and NMR [7].As it was mentioned above, in the standard measurement scheme only local operations are required to generate all the necessary projections. In each basis (setup) 2 N − 1 independent measurements can be performed, so that not all the experimental outcomes obtained in different bases are linearly independent, that is, there are redundant measurements. In the case of a N -qubit system the anti-diagonal elements have the larger errors. Actually, accumulated errors are not uniform; these errors depend on the number of single logic gates used for determining given elements, so that larger errors appear when single logic gates act on all the particles. Assuming that there is an error ε in the measurement of ion populations, then the accumulated error for anti-diagonal elements is of the order of ε 2 N −1 + 2 N −2 (2 N − 1). These errors may lead to a density operator which does not satisfy the positiveness condition and so the information from the experimental data must be optimized. For this purpose the maximum likelihood estimation (MLE) method [8] has been used for the improvement of the density operators in experiments with light qubits [9] as well as in experiments with matter qubits [5].It is well known that the optimal quantum state determination is related to the concept of measurements on Mutually Unbiased Bas...

Simple adaption of measurements for qudit estimation

“…There it was possible to come quite close to the theoretical limit given in [5]. For higher dimensional states single measurement schemes based on maximum likelihood estimation [1,13] and mutually unbiased measurements [14] have already been used in experiments [15][16][17]. However the application of adaptive methods to higher dimensions is not straightforward.…”

A new method of construction of all sets of mutually unbiased bases for two-qubit systems

“…The three separable bases given by Table 1 are obtained as the common eigenvectors of tensor product of the operatorsσ z andÎ (all the three possibilities: see the first row of Table 1), tensor product ofσ x andÎ (second row), and tensor product ofσ y andÎ (third row), respectively. The table of operators with this property are called the standard set of MU operators [7]. The last two rows of Table 1 generate entangled bases which are called belle and beau [12].…”

“…The MUBs constitute now a basic ingredient in many applications of quantum information processing: quantum tomography, quantum key distribution required in cryptography [3], discrete Wigner function [4], quantum teleportation [5], or quantum error correction codes [6]. If the measurements used for the state reconstruction are built with the help of the mutually unbiased bases, then this scheme is called MUB tomography [7]. The simplest example of MUBs can be given for qubits.…”

Optimal design of measurement settings for quantum-state-tomography experiments

“…[12] used hundreds of thousands of measurements and weeks of post-processing to get a maximum likelihood estimate of an entangled state of 8 trapped-ion qubits. Later, this experiment was simplified as there was put forward a much more economic tomographic scheme, which is based on the concept of mutually unbiased bases and promises to reduce about 95% of the number of measurements required [13]. To give another example, in Ref.…”