Quantum information with programmable optical devices: Quantum state reconstruction of qudits Carlos Saavedra Center for Optics and Photonics, Universidad de Concepción

II Quantum Information Workshop, Paraty Table of contents I. Quantum state reconstruction using MUBs II. Manipulating single-photon qudits with programmable optical devices III. Quantum state reconstruction of qudits

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Standard tomographyc reconstruction Standard Procedure Standard Procedure The standard procedure applied for quantum state reconstruction consists in projecting The standard procedure applied for quantum state reconstruction consists in projecting the density operator onto 3 N, completely factorized, bases in the corresponding the density operator onto 3 N, completely factorized, bases in the corresponding Hilbert space All these measurements are obtained by applying rotations on single qubits Hilbert space All these measurements are obtained by applying rotations on single qubits followed by projective measurements. followed by projective measurements. U. Fano, Rev. Mod. Phys. 29, 74 (1957) U. Fano, Rev. Mod. Phys. 29, 74 (1957) Experimental Realization This was achieved at the Innsbruck group for the case of eight qubits, where trapped ions are used for defining the two level systems (H. Häffner et al., Nature 438, 643, 2005). This experiment was done by following the quantum computer architecture based on ions in a linear trap proposed by Cirac and Zoller (Phys. Rev. Lett. 74, 4091, 1995).

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Standard tomographyc reconstruction 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.

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Standard tomographyc reconstruction Error propagation In the case of a N-qubit accumulated errors are not uniform; these errors depend on the number of single logic gates used for determining a given element, so that larger errors appear when single logic gates act on all the particles. Assuming that there is an error  in the measurement of ion populations, then the error for anti-diagonal elements is of the order of  (2 N-1 +2 N-2 (2 N -1)) 1/2. MLE method 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 (Z. Hradil, Phys. Rev. A 55, R1561, 1997) has been used for the improvement of the density operators in several experiments.

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Mutually unbiased bases These bases possess the property of being maximally incompatible. This means that a state producing precise measurement results in one set produces maximally random results in all the others. The set of mutually unbiased projectors given by where W.K. Wooters and B.D. Fields, Ann. Phys. 191, 363 (1989) for p m

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Mutually unbiased bases Advantages of the MUBs The number of MUBs for N-qubits is (2 N +1), which is essentially less than 3 N. The use of MUBs can represent a considerable reduction in the time needed for performing the full state characterization. Strong reduction of the time needed for the reconstruction In the experiment with eight ions of the Innsbruck group, the reconstruction process takes more than 10 hours, because of the measurement in 6561 different bases and a hundred of times for each one. In the case of using MUB-tomography, the number of measurement bases is only 257 for determining all the elements of the density operator associated with this state. This could reduce the experimental time, roughly speaking, to 25 minutes only.

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Mutually unbiased bases Error propagation In the case of MUB-tomography each coefficient p n (  ) has an error associated with the measurement of only one projector P n (  ). Hence, the error in each coefficient is essentially determined by the ability of projecting the system onto P n (  ). Implementation In practice, such measurements are implemented by projecting the system onto the logical basis after performing a set of unitary transformations. Such transformations, due to their nonlocal features, can be decomposed into a sequence of single and nonlocal gates, so that the error in this reconstruction is mainly associated with the quality of these logic gates. Nonlocal operations The experimental implementation of MUB projectors is related to the fact that any set of MUBs contains non-factorizable bases. This non-factorizable bases (MUB) can be obtained by using logical rotations (local operations) and nonlocal operations.

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Mutually unbiased bases MUB Tomography To approach this problem, we consider a complete set of MUBs where the basis factorization is denoted by the following set of natural numbers: (k 1,k 2,...,k  (n) ), where  (n) is the number of possible decompositions of 2 N +1 as a sum of positive numbers, such that  j k j =2 N +1. The case of N=3 For 3 qubits, there are 4 different sets of MUBs, which are denoted as (k 1,k 2,k  ): (3,0,6), (1,6,2), (2,3,4) and (0,9,0). (k 1,k 2,k  ): (3,0,6), (1,6,2), (2,3,4) and (0,9,0). J. L. Romero et al., Phys. Rev. A 72, (2005)

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Mutually unbiased bases The optimal decomposition on single qubits operations and two-qubit CNOT operations up to 5 qubits is already known. A. B. Klimov, C. Muñoz, A. Fernández and C. Saavedra, Phys. Rev. A 77, (R) (2008) For entangled states state reconstruction using MUBs appear to be more efficient. R.B.A. Adamson, L.K. Shalm, M.W. Mitchell, and A.M. Steinberg, Phys. Rev. Lett. 98, (2007) R.B.A. Adamson and A.M. Steinberg, arXiv: v3

II Quantum Information Workshop, Paraty I. Quantum state reconstruction using MUBs Summary Standard QS-Tomography Pyramidal QS-tomography MUBs QS-Tomography N two-d QS N M1M1 M2M2 M3M3 MNMN N two-d QS N M1M1 M2M2 M3M3 MNMN U N two-d QS N N-particle correlation U. Fano, Rev. Mod. Phys. 29, 74 (1957). - J. Řeháček, B. G. Englert, and D. Kaszlikowski, Phys. Rev. A 70, (2004). Phys. Rev. A 70, (2004). - A. Ling, K.P. Soh, A. Lamas-Linares and C. Kurtsiefer Phys. Rev. A 74, (2006) Phys. Rev. A 74, (2006) - W.K. Wooters and B.D. Fields, Ann. Phys. 191, 363 (1989). Ann. Phys. 191, 363 (1989). - A. B. Klimov, C. Muñoz, A. Fernández and C. Saavedra, Phys. Rev. A 77, (R) (2008) C. Saavedra, Phys. Rev. A 77, (R) (2008)

II Quantum Information Workshop, Paraty Manipulating single-photon qudits with programmable optical devices Spatial qudits in quantum information using two-photon states from SPCD L. Neves, G. Lima, J. G. A. Gómez, C. H. Monken, C. Saavedra and S. Pádua Phys. Rev. Lett. 94, (2005). M. N. O’Sullivan-Hale, I. A. Khan, R. W. Boyd, et al., Phys. Rev. Lett (2005).

II Quantum Information Workshop, Paraty Manipulating single-photon qudits with programmable optical devices Conditional quantum gates on qudit states using an ancilla

II Quantum Information Workshop, Paraty Manipulating single-photon qudits with programmable optical devices Spatial light modulators as conditional quantum gates on single-photon states LP LCD SLM

II Quantum Information Workshop, Paraty Manipulating single-photon qudits with programmable optical devices Manipulation of single qudits with SLM

II Quantum Information Workshop, Paraty Manipulating single-photon qudits with programmable optical devices Experimental setup

II Quantum Information Workshop, Paraty Manipulating single-photon qudits with programmable optical devices Experimental results for different transmission coefficients: Coincidences counts at the image plane

II Quantum Information Workshop, Paraty Manipulating single-photon qudits with programmable optical devices Experimental results for different transmission coefficients: Interference pattern after propagation

II Quantum Information Workshop, Paraty III. Quantum state reconstruction of qudits Reconstruction of spatial qubits A.G. White, D.F.V James, P.H. Eberhard, and P.G Kwiat, Phys. Rev. Lett. 83, 3103 (1999) G. Lima, F.A. Torres-Ruiz, L. Neves, A Delgado, C Saavedra and S. Pádua, J. Phys. B: (2008).

II Quantum Information Workshop, Paraty III. Quantum state reconstruction of qudits Experimental setup Input state Amp modulation Amp-Ph modulation

II Quantum Information Workshop, Paraty III. Quantum state reconstruction of qudits a. Input state Image Interferencepattern

II Quantum Information Workshop, Paraty III. Quantum state reconstruction of qudits b. Output state: MUB with  =1 Alternative scheme: Quantum state endoscopy -S. Wallentowitz and W. Vogel Phys. Rev. Lett. 75, 2932 (1996). - P.J. Bardroff, E. Mayr, and W.P. Schleich Phys. Rev. A 51, 4963 (1995). - P.J. Bardroff, C. Leichtle, G. Schrade and W.P. Schleich Phys. Rev. Lett. 77, 2198 (1996).

II Quantum Information Workshop, Paraty III. Quantum state reconstruction of qudits c. Reconstruction of the state “Optimal state tomography of qudits via mutually unbiased bases”, G. Lima, R. Guzmán, L. Neves, E. Valenzuela, A. Vargas, A. Delgado and C. Saavedra, in preparation.

II Quantum Information Workshop, Paraty III. Quantum state reconstruction of qudits c. Reconstruction of the state

II Quantum Information Workshop, Paraty III. Quantum state reconstruction of qudits Applications: - Higher dimensional qudits, preliminary results for D=16 and 32 - Controlled generation of mixed states of qudits Concepción: Gustavo Lima, Leonardo Neves, Wallon Nogueira Estebán Sepúlveda, Miguel A. Solis, A. Delgado C. Saavedra Estebán Sepúlveda, Miguel A. Solis, A. Delgado C. Saavedra Belo Horizonte: Sebastião de Pádua, Reinaldo Vianna, Carlos Monken

II Quantum Information Workshop, Paraty Announcement: Applications: The Center for Optics and Photonics, Universidad de Concepción, invites applications from candidates both for associated researchers and for postdoctoral researchers. This initiative also includes an enhancement of experimental facilities. Positions for associated researchers are for an initial period of five years. Postdoctoral positions are initially for a period of three years, which can be extended for two more years. The current search is open to candidates with experience in one of the following areas: Quantum Optics; Quantum Information; Non-linear Optics. From October 2009 in