Experimentally Witnessing the Quantumness of Correlations

Home , Quantum discord

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I. hsosrainmtvtstesac o alternatives for search the errors. motivates experimental observation unavoidable This the propagate additional and demand- by are ing out procedures carried These the is manipulations. one numerical laboratory, tomographic second state the the quantum and by methods In implemented is task followedfirst procedures. state extremization system’s the by of knowledge complete volve entanglement a little as with com- proposed 5]. or some [4, was in without advantage it models quantum And the putational for merit 14]. of [6–8, figure attention of popular deal most the is the non-classicality of of discord kind One this for theory. measures ver- information quantum and of classical sions between departure the a beyond by quan- tified non-classicality be can a Un- which paradigm, lies (1) entanglement-separability states equation correlations. such in non-classical given derneath present terms those in therefore, as cast basis and, be local cannot orthogonal that of states separable are There ussessaespaces state subsystems where P ρ { n lsia iswoemr eea omis form general gen- more be communicat- bits—whose can by classical coordinated that ing operations states local —those via states erated separable of set p i a ngnrl esrso o-lsia orltosin- correlations non-classical of measures general, In ( h ls fsae neuto 1 scnandi the in contained is (1) equation in states of class The i,j i ρ p i b } i savlddniyoeao o h subsystem the for operator density valid a is ) ρ sapoaiiydistribution. probability a is i a {| ρ ⊗ 1] hsqatfirhsbe eevn great a receiving been has quantifier This [13]. aueo orltos nti Letter this In correlations. of nature α cc tt oorpi ehd.Tu it Thus methods. tomographic state h ins eut ihtoefrthe for those with results witness the reain srvae yperforming by revealed is orrelations ρ i esi omtmeauenuclear temperature room a in ness i} ement. = i b geet sal nal laboured entails usually nglement) where , 4 dim and n .M Serra M. 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also a necessary condition for the absence of quantumness (a) Correlation in the correlations of the composite system (in this case Witness 1 hO4iρ = 0 [15]). We can easily verify that the observ- H (θ ) bd k n k ables in equation (2) can be written in terms of one com- x state ponent of the magnetization in one subsystem as hOiiρ = preparation a Ib † hσ1 ⊗ iξi , with ξi = Ua→b Rni (θi)ρRni (θi) Ua→b, 13 C a b a(b) (θ ) where Rni (θi) = Rni (θi) ⊗ Rni (θi), and Rni (θi) is a k nk local rotation by an angle θi around direction ni, with θ1 = 0, θ2 = θ3 = π/2, n2 = y, and n3 = z. Ua→b is the (b) |11> State Preparation pseudo (θ ) C Read-out controlled-NOT gate with the subsystem a as control. k n k NOT EPR x xy y x x yxy ρ ρ (a) ρQC CC T 1H Acq.

x y x y x x y y xyxy n ,θ33 1 1 1 1 3 13 C 4J 4J 2J 2J 2J

n ,θ22 Gradient

n ,θ11

200 1000 -100 -200 200 1000 -100 -200 200 1000 -100 -200 Hz Hz Hz FIG. 1. (Color online) (a) Schematic representation of the op- (b) eration sequence used to witness the non-classical nature of 3 correlations. (b) Equivalent pulse sequence employed in our experiment. The thicker filled bars represent π/2 pulses, the 2 thinner bars indicate π/4 pulses, and the grey bars indicate Witness π/6 pulses with the phases as shown (negative pulse phases 1 are described by a bar over the symbol). The pulses represented as unfilled dashed bars are modified to achieve the 0 ρ ρ ρ different rotations necessary for the witness protocol. The QC CC T dashed gradient pulse is applied to obtain the classically cor- (c) 1 3 1 related Bell-diagonal state. The time periods 2J , 2J , and 4J 6 represent free evolutions under the J coupling [18].

regarding the classification of correlations in quantum Correlations 2 states. Once the nature of these correlations somehow de- 0 termines what can and cannot be done with a given sys- ρ ρ ρ tem, it is sometimes enough to know whether the correla- QC CC T tions in that system have a classical or a quantum nature. 1 FIG. 2. (Color online) (a) The H spectra (normalized by To accomplish this last task it is convenient to have an the thermal equilibrium state spectrum) obtained after the observable witness for the quantumness of correlations in witness circuit execution (with rotations Rn1 , Rn2 , and Rn3 ), the system. However, as the state space of classical cor- (b) witness expectation value, (c) quantum discord (light blue related systems is not convex, a linear witness cannot be columns) and classical correlation (dark blue columns) mea- used in general, and we have to take advantage of a non- sured in three different initial states, ρQC quantum correlated, linear witness. For a wide class of two-qubit systems, ρ = ρCC classically correlated, and ρT thermal equilibrium state. ab 3 a b a b a b The dashed line represents the experimental error bound for I + (Aiσ ⊗ I + BiI ⊗ σ + Ciσ ⊗ σ ) /4, a Pi=1 i i i i determination of classically correlated (zero discord) states. sufficient condition for classicality of correlations is [15] The witness was measured directly performing the circuit depicted in Fig 1 (a), while the classical correlation and the 3 4 symmetric quantum discord was computed after full QST and Wρ = |hOiiρhOj iρ| =0, (2) numerical extremization procedures. The correlations are dis- X X 2 i=1 j=i+1 played in units of (ε / ln 2)bit [18].

a b 3 a with Oi = σi ⊗ σi for i =1, 2, 3 and O4 = i=1(ziσi ⊗ a b P We experimentally implemented the aforementioned Ib + w Ia ⊗ σb). The σ ( ) is the ith component of the i i i witness using the room temperature nuclear magnetic Pauli operator in subsystem a(b). A , B ,z , w ∈ ℜ i i i i resonance (NMR) system. In this scenario the qubits with z , w randomly chosen and constrained such that i i (quantum bits) are encoded in nuclear spins and they z2 = w2 = 1. For the so-called Bell-diagonal class Pi i Pi i are manipulated by radio-frequency (rf) pulses. Unitary Iab 3 a b of states, ρbd = + i=1 Ciσi ⊗ σi /4, Wρbd = 0 is operations are achieved by suitable choice of pulse am- P 3

(a) Re(Δρ ) Im(Δρ ) after the aforementioned pulse sequence. The witnessing 1.0 1.0 of the thermal equilibrium state was also performed as a

0.0 0.0 reference. The experimental procedure depicted in Fig. 1 was ran three times for each initial state in order to −1.0 −1.0 measure the magnetization hσai in the states ξ that 00 00 1 ξi i 01 01 a b 11 11 leads to the two-point correlation functions hσi ⊗ σi iρ. 10 10 10 10 01 01 11 00 11 00 So, the witness given in equation (2) is directly measured

(b) Re(Δρ ) Im(Δρ ) (Fig. 2). 1.0 1.0

0.0 0.0 6 3

−1.0 −1.0 5 1.0 00 00 01 01 0.0 11 11 4 2 10 10 10 10 01 01 −1.0 11 00 11 00 00 3 01 11 10 10

Re(Δρ ) Im(Δρ ) 01 Witness (c) 11 00 1.0 1.0 2 1 Correlations

0.0 0.0 1

−1.0 −1.0 0 0 00 00 0.0 0.10.20.3 0.4 0.5 0.6 01 01 11 11 10 10 10 10 (T*) ( *) 2 13 T2 1 01 11 01 C H 11 00 00 t (s)

FIG. 3. (Color online) Real (left) and imaginary (right) parts FIG. 4. (Color online) Witness and correlations decoherence of the deviation matrix elements reconstructed by QST for the dynamics. The panel displays the measured witness and com- two initial prepared Bell-diagonal states: (a) ρQC quantum puted correlations for ρQC [Fig. 3 (a)] relaxed during a time correlated (equivalent to C1 = 2ε, C2 = 2ε, and C3 = −2ε); (b) interval, tn = nδt (δt = 55.7 ms, n = 0, 2, ..., 11), before per- ρCC classically correlated (equivalent to C1 = 0, C2 = 0, and forming the witness measurement protocol. The red tick bars C3 = −4ε); and also for (c) ρT the thermal equilibrium state. represent the witness expectation value (scale on the right), The deviation matrix elements are displayed in the usual com- the dark grey section represents the amount of classical corre- putational basis, where |0i and |1i represent the eigenstates lation, the light grey section represents the symmetric quan- of σz for each qubit. The accuracy of prepared initial states tum discord, the entire grey bars (light and dark sections can be estimated by the normalized trace distance from the summed) display the quantum mutual information (scale on ideal ones, δ(ρideal,ρprep.)/ε = tr |∆ρideal − ∆ρprep.| /2 ≈ 0.1 the left). The classicality bound is represented by the blue (for both ρQC and ρCC ). dotted line. The inset image shows the real part of the deviation matrix elements reconstructed by QST for an inter- mediate classically correlated state. The effective transversal ∗ ∗ 1 plitudes, phases and durations, and the transverse mag- relaxation times are T2 = 0.31 s and T2 = 0.12 s, for H and 13C nuclei, respectively. The correlations are displayed netizations are obtained directly from the NMR signal 2 [16]. The state of the two-qubit system is described in units of (ε / ln 2)bit. by a density matrix in the high temperature expansion (where entanglement was ruled out), which takes the For the sake of comparison, we performed a full quan- ab −5 form ρ = I /4 + ε∆ρ, with ε = ~ωL/4kBT ∼ 10 tum state tomography (QST) [19] of the initial states as the ratio between the magnetic and thermal energies (displayed in Fig. 3) and computed, from these data, the ab ab and ∆ρ as the deviation matrix [12, 16]. A carbon-13 symmetric quantum discord [6, 20] —Q(ρ )= I(ρ ) − ab ab ◦ max a a I(χ ), where I(χ ) is the measurement- enriched chloroform (CHCl3) solution at 25 C was used {Πi ,Πj } in the experiments, with the two qubits being encoded induced mutual information —and its classical counter- in the 1H and 13C spin-1/2 nuclei. In order to experi- part [18] present in each state. The results are shown in mentally demonstrate the witnessing protocol, two initial Fig. 2. The correlation quantifiers are computed from states were prepared by mapping them into the deviation the experimentally reconstructed deviation matrix in the matrix using the general pulse sequence scheme as shown leading order in ε, following the approach introduced in in Fig. 1. The first state corresponds to a quantum cor- Refs. [6, 18]. Since the error in the witness expectation related Bell-diagonal state, which is obtained from the value depends on many parameters (i.e., signal-to-noise thermal equilibrium by applying the pulse sequence for ratio and residual rf pulse sequence imperfections), we producing the pseudo-pure state |11i, followed by the used as a reference the thermal equilibrium state, which pulses that implement a pseudo-EPR gate [17], see Fig. is supposed to have no correlations at room temperature

1 [18]. The second state is a classically correlated Bell- [6]. The witness measured for this state (WρT ) was about diagonal state, obtained by applying a z-gradient pulse 0.05, which is assumed to be the error margin for our ex- 4 periment. This introduces the bound shown in Fig. 2 (b) UFABC, CNPq, CAPES, FAPESP, and FAPERJ. This for a classically correlated (zero discord) state. work was performed as part of the Brazilian National In- The witness measured for the three initial states is dis- stitute of Science and Technology for Quantum Informa- played in Fig. 2 (b). For the quantum correlated Bell- tion (INCT-IQ). RMS thanks S. P. Walborn for insightful diagonal state the witness (WρQC ) is found to be about discussions. 3.13 (far above the 0.05 bound), while for the classical correlated Bell-diagonal state (WρCC ) it is about 0.04, i.e., within the classicality cut-off limit. In fact, the witness works perfectly in the present setup, in the sense that it easily sorts out quantum and classically correlated states. Figure 2 (c) also displays the quantum discord Note added.—After the submission of this Letter, a re- computed from the experimentally reconstructed devia- lated work has appeared [22], which employs other meth- tion matrices using the approach introduced in Ref. [6]. ods to witness nonclassicality in an NMR system. As can be seen, the result for the quantum discord is in agreement with the witness, but the former is obtained after full QST and numerical extremization procedures. Finally, we followed the decoherence dynamics of the wit- ∗ [email protected] ness, by letting the state ρQC evolves freely during a time [1] J. S. Bell, Speakable and Unspeakable in Quantum Me- period tn, after this decoherent evolution we performed chanics (Cambridge University Press, Cambridge, 1988). the witness circuit, and also a QST in order to com- [2] R. F. Werner, Phys. Rev. A 40, 4277 (1989). pare the witness results with those for correlation quan- [3] M. Piani, P. Horodecki, and R. Horodecki, Phys. Rev. tifiers. The noise spin environment causes loss of phase Lett. 100, 090502 (2008). relations among the energy eigenstates and exchange of [4] A. Datta, A. Shaji, and C. M. Caves, Phys. Rev. Lett. 100, 050502 (2008). energy between system and environment, resulting in re- [5] B. P. Lanyon et al., Phys. Rev. Lett. 101, 200501 (2008). laxation to a Gibbs ensemble. In Fig. 4 we observe, that, [6] D. O. Soares-Pinto et al., Phys. Rev. A 81, 062118 (2010). in the course of the witness and correlations evolution, [7] A. Brodutch and D. R. Terno, Phys. Rev. A 83, 010301(R) the non-classicality is diminished until reaching an only (2011). classically correlated state. This occurs near the 1H ef- [8] D. Cavalcanti et al., Phys. Rev. A 83, 032324 (2011); V. fective transversal relaxation time. After such evolution Madhok and A. Datta, Phys. Rev. A 83, 032323 (2011). period there are just reminiscent classical correlations, [9] J. Oppenheim et al., Phys. Rev. Lett. 89, 180402 (2002); W. H. Zurek, Phys. Rev. A 67, 012320 (2003). which are also diminished resulting in an uncorrelated [10] M. S. Sarandy, Phys. Rev. A 80, 022108 (2009); J. state (the room temperature thermal equilibrium state) Maziero et al., Phys. Rev. A 82, 012106 (2010); T. Wer- after the spin-lattice relaxation time. Again, we obtain lang et al., Phys. Rev. Lett. 105, 095702 (2010); J. a fairly good agreement between the witness expectation Maziero et al., arXiv:1012.5926 (2010). values and the correlation quantifiers. [11] K. Brádler et al., Phys. Rev. A 82, 062310 (2010). Summarizing, we presented a direct experimental im- [12] M. A. Nielsen and I. L. Chuang, Quantum Computation plementation of a witness for the quantumness of cor- and Quantum Information (Cambridge University Press, Cambridge, 2000). relations (other than entanglement) in a composite sys- [13] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 tem. Our work showed that it is possible to infer the na- (2001); L. Henderson and V. Vedral, J. Phys. A: Math. ture of the correlations in a bipartite system performing Gen. 34, 6899 (2001). only few local measurements over one of the subsystems [14] J. Maziero et al., Phys. Rev. A 80, 044102 (2009); J. (just three measurements for both ρQC and ρCC). The Maziero et al., Phys. Rev. A 81, 022116 (2010); L. Maz- witness presented in Eq. (2) was generalized to higher- zola, J. Piilo, and S. Maniscalco, Phys. Rev. Lett. 104, dimensional systems [21]. Therefore, the methods em- 200401 (2010); J.-S. Xu et al., Nat. Commun. 1, 7 (2010). [15] J. Maziero and R. M. Serra, arXiv:1012.3075. ployed here can also be easily applied for witnessing cor- [16] I. S. Oliveira et al., NMR Quantum Information Process- relations in systems with dimensions higher than two. ing (Elsevier, Amsterdam, 2007). Our strategy precludes the demanding tomographic state [17] I. L. Chuang et al., Proc. R. Soc. A 454, 447 (1998). reconstruction and the numerical extremization methods [18] See Supplemental Material for a detailed description of included in quantum correlation quantifiers (like quan- the experimental and quantum discord calculation proce- tum discord). This method offers a versatile test-bed for dures. the nature of a composite system that can be applied to [19] G. L. Long, H. Y. Yan, and Y. Sun, J. Opt. B: Quantum Semiclass. Opt. 3, 376 (2001); J. Teles et al., J. Chem. other experimental physical contexts. Moreover, in such Phys. 126, 154506 (2007). a proof of principle, we showed that non-classical correla- [20] J. Maziero, L. C. Céleri, and R. M. Serra, tions can be present even in highly mixed states as those arXiv:1004.2082. in room temperature magnetic resonance experiments. [21] Z.-H. Ma, Z.-H Chen, and J.-L. Chen, arXiv:1104.0299. The authors acknowledge financial support from [22] G. Passante et al., arXiv:1105.2262. 5

Supplemental Material - Details on experimental 99.8 % CDCl3 in a 5 mm NMR tube. Both samples were and calculation procedures provided by the Cambridge Isotope Laboratories - Inc. NMR experiments were performed at 25◦ C using a Var- NMR experiments ian 500 MHz Premium Shielded (1H frequency), at the Brazilian Centre for Physics Research (CBPF). A Var- Nuclear magnetic resonance (NMR) experiments were ian 5mm double resonance probe-head equipped with a performed on a two-qubit system comprised by nuclear magnetic ﬁeld gradient coil was used. spins of 1H and 13C atoms in a carbon-13 enriched chloroform molecule (CHCl3). The sample was prepared by The rotating frame nuclear spin Hamiltonian account- 13 mixing 100 mg of 99 % C-labelled CHCl3 in 0.2 ml of ing for the relevant NMR interactions reads [1]

H H C C H C H = − ωH − ω I − ωC − ω I +2πJI I rf z rf z z z +ωH IH cos ϕH + IH sin ϕH + ωC IC cos ϕC + IC sin ϕC , (3) 1 x y 1 x y

IH IC pared. Finally, the witnessing of the thermal equilibrium where α β is the spin angular momentum operator in state was performed as a reference for error estimation of the α, β = x,y,z direction for 1H (13C); ϕH ϕC defines the whole procedure. The y rotation in the witness proto- the direction of the rf field and ωH ωC is the intensity 1 1 col was implemented directly by a single rf pulse, while of RF pulse for 1H (13C) nuclei. for the z rotation the pulse sequence π - π - π 1 2 −y 2 x 2 y The first two terms describe the free precession of H was applied in both nuclei. The controlled-NOT gate 13 and C nuclei about the external constant magnetic field π C 3 π C was achieved by the pulse sequence 2 y -U 2J - 2 −x- B0 with frequencies ωH /2π ≈ 500 MHz and ωC /2π ≈ C C C H H H π - π - π - π - π - π , where the super 125 MHz, respectively. The third term describes a scalar 2 −y 2 −x 2 y 2 −y 2 x 2 y coupling between 1H and 13C nuclei at J ≈ 215.1 Hz. indices states for nucleus where the pulse is applied and The fourth and fifth terms represent the radio frequency U 3 represents a free evolution under J coupling. In 2J (rf) field that may be applied to 1H and 13C , respectively. this case, a pulse sequence that correctly implement a The π/2 pulse has a time setup of 7.4 µs for 1H and 9.6 true CNOT gate, i.e., with the correct phases, are indeed µs for 13C at transmitter channel; or a time setup of 7.9 necessary to produce the correct output for the witness. µs for 1H and 10.2 µs for 13C at decoupler channel. The protocol described in Refs. [4, 5] was used for quan- 1 tum state tomography (QST). As described above, the Spin-lattice relaxation times (T1)of2.5sand7sfor H and 13C, respectively, were measured using the inversion experimental procedure depicted in Fig. 1 of the main text is ran three times for each initial state in order to recovery pulse sequence. The recycle delay was set to 40 H measure the magnetization hσ1 iξi in the states ξi that s in all experiments. The effective transversal relaxation H C times, measured as the inverse of the spectrum line width leads to the two-point correlation functions hσi ⊗ σi iρ. ∗ So, the witness given in equation (2) of the main text is at halt maximum, were found to be T2 = 0.31 s and ∗ 1 13 directly measured. T2 =0.12 s for H and C nuclei, respectively. In order to experimentally demonstrate the witnessing protocol, two initial states were prepared by mapping Measures of correlations them into the deviation density matrix using the general pulse sequence scheme shown in Fig. 1 of the main text. The first state corresponds to a quantum correlated Bell- The correlations in the high-mixed state find in the ab Iab −5 diagonal state, which is obtained from the thermal equi- NMR context, ρ = /4+ ε∆ρ (ε ≈ 10 ), are com- librium by applying the pulse sequence for producing the puted by expanding the symmetric version of quantum pseudo-pure state |11i, followed by the pulses that imple- discord [6, 7] ment a pseudo EPR gate [3]. It is worth mentioning that, Q(ρab)= I(ρab) − max I(χab), (4) despite a true EPR gate would produce the same state, a a {Πi ,Πj } there is no need of using it in our propose, since the same kind of deviation density matrix can be produced with a where the quantum mutual information (expanded in the pseudo EPR gate, with the advantage of using a smaller leading order in ε), is given by number of pulses. A classically correlated Bell-diagonal state, which is obtained by applying a z-gradient pulse ε2 I(ρab) ≈ 2tr[(∆ρab)2] − tr (∆ρa)2 − tr (∆ρb)2 , after the aforementioned pulse sequence, was also pre- ln 2 6 and the measurement-induced mutual information is fiers can be computed directly from the experimentally reconstructed deviation matrix and they were employed ε2 to observe the quantumness of a room temperature NMR I(χab) ≈ 2tr (∆χab)2 − tr (∆χa)2 − tr (∆χb)2 , ln 2 quadrupolar system [7]. with χab = Iab/4+ ε∆χab as the state obtained from ρab through a complete projective measurement map ab a b ab a b a(b) (∆χ = i,j Πi ⊗ Πj (∆ρ )Πi ⊗ Πj ). ∆ρ = ∗ P [email protected] tr {∆ρab} is the reduced deviation matrix while ∆χa(b) b(a) [1] I. S. Oliveira et al., NMR Quantum Information Process- stands for the reduced measured deviation matrix in the ing (Elsevier, Amsterdam, 2007). subspace a(b). The classical counterpart of Eq. 4 is [2] E. M. Fortunato et al., J. Chem. Phys. 116, 7599 (2002). AB AB C(ρ ) = max a b I(χ ). It is worth mentioning [3] I. L. Chuanget al., Proc. R. Soc. A 454, 447 (1998). {Πi ,Πj } [4] G. L. Long, H. Y. Yan, and Y. Sun, J. Opt. B: Quantum that Q(ρab) = 0 if and only if ρab can be cast in terms Semiclass. Opt. 3, 376 (2001). of local orthogonal basis. In other words, the symmetric [5] J. Teles, et al., J. Chem. Phys. 126, 154506 (2007). ab quantum discord is zero if and only if ρ has only clas- [6] J. Maziero, L. C. Céleri, and R. M. Serra, arXiv:1004.2082. sical correlations or no correlations at all (in this case [7] D. O. Soares-Pinto et al., Phys. Rev. A 81, 062118 (2010). the correlations present in ρab can be locally broadcast [8] M. Piani, P. Horodecki, and R. Horodecki, Phys. Rev. [8]). The aforementioned symmetric correlation quanti- Lett. 100, 090502 (2008). This figure "fig1.png" is available in "png" format from:

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