Evolution of spin entanglement and an entanglement witness in multiple-quantum NMR experiments
aa r X i v : . [ qu a n t - ph ] A p r Evolution of spin entanglement and an entanglement witness in multiple-quantumNMR experiments
E. B. Fel’dman, A. N. Pyrkov ∗ Institute of Problems of Chemical Physics of Russian Academy of Sciences, Chernogolovka, Moscow Region, Russia, 142432 (Dated: October 27, 2018)We investigate the evolution of entanglement in multiple–quantum (MQ) NMR experiments incrystals with pairs of close nuclear spins-1/2. The initial thermodynamic equilibrium state of thesystem in a strong external magnetic field evolves under the non-secular part of the dipolar Hamil-tonian. As a result, MQ coherences of the zeroth and plus/minus second orders appear. A simplecondition for the emergence of entanglement is obtained. We show that the measure of the spinpair entanglement, concurrence, coincides qualitatively with the intensity of MQ coherences of theplus/minus second order and hence the entanglement can be studied with MQ NMR methods. Weintroduce an Entanglement Witness using MQ NMR coherences of the plus/minus second order.
PACS numbers: 03.67.Mn, 03.67.-a, 75.10.Pq, 82.56.-b
I. INTRODUCTION
Entanglement [1] is the key concept in Quantum Information Theory. It has played a crucial role in experimentson quantum computing and quantum teleportation. This resulted in intensive interest to the physics of entanglementfrom both theorists and experimentalists [2, 3, 4, 5].Entanglement is detected with the help of a so-called Entanglement Witness (EW). By definition, EW is anobservable which has a positive expectation value for separable states and negative for some entangled states [6]. Inparticular, internal energy [7] and magnetic susceptibility [8] were used as EW in some cases. In this paper we proposea new type of an Entanglement Witness, the intensity of multiple quantum coherences in spin systems. This quantityis accessible in NMR experiments and thus opens a new approach to probing entanglement with highly advancedNMR techniques.In the present work we focus on the simplest relevant system, a pair of spins s = 1 / II. MQ DYNAMICS OF A DIPOLAR COUPLED SPIN PAIR AT LOW TEMPERATURES
We consider a two-spin system in a strong external magnetic field ~H . The thermodynamic equilibrium densitymatrix, ρ , of the system is ρ = exp( ~ ω kT I z ) Z (1)where ω = γH ( γ is the gyromagnetic ratio), T is the temperature, I α = I α + I α , and I jα ( j = 1 , α = x, y, z ) isthe projection of the angular spin momentum operator of spin j on the axis α , and Z is the partition function.The MQ NMR experiment consists of four distinct periods of time: preparation, evolution, mixing, and detection [9].MQ coherences are created by the multipulse sequence consisting of eight-pulse cycles on the preparation period [9]. ∗ Email address: [email protected]
In the rotating reference frame [12], the average Hamiltonian, H MQ , for the two-spin system describing the MQdynamics at the preparation period can be written as [9] H MQ = b (cid:0) I +1 I +2 + I − I − (cid:1) (2)where b = ( γ ~ / { r } )(1 − θ ) is the coupling constant between spins 1 and 2, r is the distance betweenspins 1 and 2, and θ is the angle between the internuclear vector ~r and the external magnetic field ~H ; I + j and I − j ( j = 1 ,
2) are the rasing and lowering operators of spin j. The two-spin Hamiltonian H MQ can be diagonalized with the transformation (in the standard basis {| i , | i , | i , | i} ) U = √ √ √ − √ , (3)and the density matrix, ρ ( τ ) , at the end of the preparation period is ρ ( τ ) = e − iH MQ τ ρ e iH MQ τ = 12(1 + cosh β ) cosh β + cos(2 bτ ) sinh β i sin(2 bτ ) sinh β − i sin(2 bτ ) sinh β β − cos(2 bτ ) sinh β (4)where β = ~ ω / ( kT ) . The diagonal part of the density matrix of Eq. (4), ρ (0) ( τ ) , is responsible for the MQ coherenceof the zeroth order, and the non-diagonal parts, ρ (2) ( τ ) , ρ ( − ( τ ) , are responsible for the MQ coherences of theplus/minus second orders[9, 10]. The intensities of the MQ coherences of the zeroth, G ( τ ) , and plus/minus second, G ± ( τ ) , orders are[13] G ( τ ) = Tr (cid:16) ρ (0) ( τ ) ρ ht (0) ( τ ) (cid:17) , G ± ( τ ) = Tr (cid:16) ρ (2) ( τ ) ρ ht ( − ( τ ) (cid:17) , (5)where ρ ht (0) ( τ ) is the diagonal part of ρ ht ( τ ) = e − iH MQ τ I z e iH MQ τ (6)and ρ ht (2) ( τ ) , ρ ht ( − ( τ ) are the non-diagonal parts of the density matrix ρ ht ( τ ) . Using Eqs. (4)–(6) one can find that G ( τ ) = tanh β (2 bτ ) , G ± ( τ ) = 12 tanh β (2 bτ ) . (7)It is worth to emphasize that intensities of MQ coherences are observables in MQ NMR experiments. Eq. (7) showsthat the intensities of the MQ coherences of the second order, G ( τ ) , and the minus second order, G − ( τ ) , are equal.However, in real experiment, certain errors are present and the experimental results for G ( τ ) and G − ( τ ) are notthe same. Some of the errors can be compensated and the accuracy can be improved if one detects the sum of thesecoherences[14]. It is also worth to notice that the accuracy of the measurement of G ( τ ) + G − ( τ ) is higher than for G ( τ ) [14]. Below we will use the sum of the MQ coherences of the plus/minus second order in order to introduce theentanglement witness. III. CONCURRENCE AND ENTANGLEMENT WITNESS IN MQ NMR EXPERIMENTS
The initial state of the system determined by Eq. (1) is separable. Entanglement appears in the course of thepreparation period of the MQ NMR experiment when the MQ coherence of the second order has a sufficiently largeintensity. In order to estimate the entanglement quantitatively we apply the Wootters criterion [11]. According to [11],one needs to construct the spin-flip density matrix˜ ρ ( τ ) = ( σ y ⊗ σ y ) ρ ∗ ( τ )( σ y ⊗ σ y ) (8)where the asterisk denotes complex conjugation in the standard basis {| i , | i , | i , | i} and the Pauli matrix σ y = 2 I y . The concurrence of the two–spin system with the density matrix ρ ( τ ) is equal to [11] C = max { , λ − λ − λ − λ − λ } , λ = max { λ , λ , λ , λ } (9)where λ , λ , λ , and λ are the square roots of the eigenvalues of the product ρ ( τ )˜ ρ ( τ ) . Using Eqs. (4), (8), (9) oneobtains λ , = q (2 bτ ) sinh β ± | sin(2 bτ ) | sinh β β , λ , = 14 cosh β . (10)As a result, the concurrence, C , is C = | sin(2 bτ ) | sinh β −
12 cosh β . (11)The entangled state can appear only at sinh β > T < ~ ω k ln(1 + √ . (12)If one takes ω = 2 π · s − the entangled state emerges at the temperature T E ≈ mK. It is interesting tonotice that in a linear chain of dipolar coupled nuclear spins in the thermodynamic equilibrium state, entanglementappears only at microkelvin temperatures [15].The simple connection between the concurrence, C, and the intensities of the MQ coherences of the plus/minussecond orders, G ± ( τ ) , can be found from Eqs. (7), (9): C = r tanh β G ( τ ) + G − ( τ )] −
12 cosh β . (13)Thus, entanglement is possible only when G ( τ ) + G − ( τ ) >
12 sinh β cosh β , (14)and Entanglement Witness (EW) can be introduced as the following EW = 12 sinh β cosh β − { G ( τ ) + G − ( τ ) } . (15)In the initial moment of time G (0) + G − (0) = 0 , EW > EW, changes its sign. It means that an entangled state appears. According to Eq. (7) the intensities of theMQ coherences periodically change in time. The sign of EW changes also periodically. Thus separable and entangledstates change periodically in the considered system. The time evolutions of the MQ coherences of the zeroth andsecond orders together with the corresponding concurrence are represented in Fig. 1 at β = 3 . One can see thatthe concurrence is close to the sum of the MQ coherences of the plus/minus second orders, G ( τ ) + G − ( τ ) , atalmost all durations of the preparation period of the MQ NMR experiment. At large β (small temperatures) theexpression [2 sinh β cosh β ] − tends to zero and the maximal value of G ( τ ) + G − ( τ ) tends to one. This meansthat the concurrence coincides with the maximal value of G ( τ ) + G − ( τ ) at small temperatures. The correspondingdependencies of the concurrence and the maximal value of the sum G ( τ ) + G − ( τ ) on β are given in Fig. 2. We canconclude that the entangled states appear in MQ NMR experiments at sufficiently small temperatures. In contrastto Ref. [5] we study entanglement in a system of nuclear spins(not electron ones). Such systems are more robustto decoherence which causes the loss of the quantum information which was obtained during quantum computation.A problem, related to ours, was studied in Ref. [16]. That work focuses on the high temperature regime in whichentanglement is absent. The prediction of the existence of an entangled state at high temperatures [16] is an artifactof an incorrect choice of the initial density matrix. FIG. 1: The dependencies of the MQ coherences and the concurrence on the time of the preparation period, τ, of the MQ NMRexperiment at β = 3 . The coupling constant is equal to b = 2 π − ; solid line – concurrence; dashed line – intensity of theMQ coherence of the zeroth order; dash-point line – G ( τ ) + G − ( τ ) (see the text).FIG. 2: The dependence of the concurrence (dashed line) and the maximal value of G ( τ ) + G − ( τ ) (dash-point line) on theparameter β . The solid line describes the function [2 sinh β cosh β ] − . Here the coupling constant is equal to b = 2 π − . Theentangled state emerges at temperatures less than T E = ~ ω / ( kβ E ) . IV. CONCLUSION
MQ NMR experiments can be used for the analysis of entangled states in spin systems. We have introduced anEntanglement Witness using observable intensities of the MQ NMR coherences of the plus/minus second order andanalyzed entanglement in term of the Wootters criterion. Entangled states emerge when the sum of intensities of theMQ coherences of the plus/minus second order exceeds an exactly calculated threshold depending on the externalmagnetic field and the temperature. MQ NMR can be considered as a new method for obtaining entangled states inspin systems.
V. ACKNOWLEDGMENTS
The authors wish to express their gratitude to S.I. Doronin and M. A. Yurishchev for many helpful discussions.This work is supported by the Russian Foundation for Basic Research through the grant 07-07-00048. [1] M. A. Nielsen I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).[2] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, W. K. Wootters, Phys. Rev. A , 3824 (1996).[3] L. Amico, R. Fazio, A. Osterloh, V. Vedral, arxiv: quant-ph/0703044;[4] S. Ghosh, T. F. Rosenbaum, G. Aeppll, S. N. Coppersmith, Nature , 48 (2003).[5] A. M. Souza, M. S. Reis, D. O. Soares-Pinto, I. S. Oliveira, R. S. Sarthour, Phys. Rev. B , 104402 (2008).[6] M. Horodeski, P. Horodeski, R. Horodeski, Phys. Lett. A , 1 (1996).[7] X. Wang, Phys. Rev. A , 034302 (2002).[8] M. Weisniak, V. Vedral, C. Brukner, New. J. Phys 7, 258 (2005).[9] J. Baum, M. Munowitz, A. N. Garroway, A. Pines, J. Chem. Phys. , 2015 (1985).[10] E. B. Fel’dman, S. Lacelle, J. Chem. Phys. , 7067 (1997).[11] W. K. Wootters, Phys. Rev. Lett. , 2245 (1998).[12] M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids (Clarendon, Oxford, 1970).[13] E. B. Fel’dman, I. I. Maximov, J. Magn. Reson. , 106 (2002).[14] G. Cho J. P. Yesinowski, J. Phys. Chem. , 15716 (1996).[15] Doronin S. I., Pyrkov A. N. Fel’dman E. B., JETP Letters , 519 (2007).[16] S. I. Doronin, Phys. Rev. A68