Experimental Simulation of the Unruh Effect on an NMR Quantum Simulator

SCIENCE CHINA Physics, Mechanics & Astronomy

. Article . March 2016 Vol.59 No. 3: 630302 doi: 10.1007/s11433-016-5779-7

Experimental simulation of the Unruh eﬀect on an NMR quantum simulator† FangZhou Jin1, HongWei Chen2*, Xing Rong1, Hui Zhou1, MingJun Shi1, Qi Zhang1, ChenYong Ju1,YiFuCai1, ShunLong Luo3,XinHuaPeng1, and JiangFeng Du1*

1Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China; 2High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China; 3Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received December 23, 2015; accepted December 25, 2015; published online January 15, 2016

The Unruh effect is one of the most fundamental manifestations of the fact that the particle content of a field theory is observer dependent. However, there has been so far no experimental verification of this effect, as the associated temperatures lie far below any observable threshold. Recently, physical phenomena, which are of great experimental challenge, have been investigated by quantum simulations in various fields. Here we perform a proof-of-principle simulation of the evolution of fermionic modes under the Unruh effect with a nuclear magnetic resonance (NMR) quantum simulator. By the quantum simulator, we experimentally demonstrate the behavior of Unruh temperature with acceleration, and we further investigate the quantum correlations quantified by quantum discord between two fermionic modes as seen by two relatively accelerated observers. It is shown that the quantum correlations can be created by the Unruh effect from the classically correlated states. Our work may provide a promising way to explore the quantum physics of accelerated systems.

quantum simulation, Unruh eﬀect, quantum correlations

PACS number(s): 03.65.Ud, 03.65.Wj, 04.62.+v

Citation: F. Z. Jin, H. W. Chen, X. Rong, H. Zhou, M. J. Shi, Q. Zhang, C. Y. Ju, Y. F. Cai, S. L. Luo, X. H. Peng, and J. F. Du, Experimental simulation of the Unruh eﬀect on an NMR quantum simulator, Sci. China-Phys. Mech. Astron. 59, 630302 (2016), doi: 10.1007/s11433-016-5779-7

1 Introduction thermal emission of particles from black holes [4] and cosmo- logical horizons [5]. However, this phenomenon is too weak Quantum mechanics and relativity theory are two pillars of to be observed with current technique. There have been a lot modern physics. With their amalgamation, many novel phe- of attempts [6-13] in searching for the observational evidence nomena have been identified. For example, the Unruh ef- of the Unruh effect and in general the experimental observa- fect [1,2] is one of the most significant outcomes of the quan- tion is still of great challenge. To address this issue, quantum tum field theory. It shows that a uniformly accelerated ob- simulators [14-16] may provide a promising approach. The server in the flat spacetime sees a thermal bath emergent from aim of the quantum simulation is to simulate the quantum the Minkowski vacuum of a quantum field [3]. This effect system of interest with a controllable laboratory system de- serves as an important tool to investigate phenomena such as scribed by the same mathematical model. It is widely applied for simulating the quantum systems, which can not be effi-

*Corresponding authors (HongWei Chen, email: hwchen@hmﬂ.ac.cn; ciently simulated by classical computers or are not directly JiangFeng Du, email: [email protected]) tractable by the current techniques in the laboratory. This ap- †Contributed by JiangFeng Du (CAS academician)

c Science China Press and Springer-Verlag Berlin Heidelberg 2016 phys.scichina.com link.springer.com F. Z. Jin, et al. Sci. China-Phys. Mech. Astron. March (2016) Vol. 59 No. 3 630302-2 plication has achieved great developments in accessing vari- (a) ous fields including condensed matter physics [17-20], high energy physics [21-23], cosmology [24-26], and quantum chemistry [27, 28] in recent years. Here, we design a quantum simulator to simulate the evolution of fermionic modes under the Unruh effect. Then we perform a proof-of-principle quantum simulation of the behavior of Unruh temperature with acceleration by a nuclear spin quantum simulator. Furthermore, we investigate the cor- (b) R relations of two fermionic modes under the Unruh effect. In- terestingly, we show that the quantum correlations quantified R y by quantum discord can be created by the Unruh effect. This II phenomenon is characterized by a nonunital quantum chan- R I n nel which captures the acceleration effect.

R -y II 2 The fermionic modes under the Unruh effect Figure 1 (Color online) Sketch of the world lines for two observers Al- ice (A) and Rob (R), and quantum circuit for quantum simulating the Unruh We first briefly introduce the fermionic modes under the Un- effect. (a) Alice is an inertial observer and Rob is a uniformly accelerated ruh effect. Consider an inertial observer Alice (A) and a observer. The sets (t, z)and(τ, ζ) denote the Minkowski and Rindler coor- uniformly accelerated observer Rob (R) with a constant ac- dinates, respectively. The right and left Rindler wedges are the regions with celeration a. As shown in Figure 1(a), Alice moves in the |t| < z and |t| < −z, respectively. They are separated by the Rindler horizon Minkowski plane with coordinate (t, z). The setting of the so that they are causally disconnected from each other. (b) The quantum circuit to simulate the evolution of fermionic modes via Unruh effect. The uniform acceleration can be conveniently described by the four qubits are used to model the Unruh effect on qubit 1. R (ϕ) = e−iϕnˆ·σ/ˆ 2, τ, ζ nˆ Rindler coordinate ( ) with two disconnected regions I and wheren ˆ = (nx, ny, nz) is a real unit vector in three dimensions andσ ˆ denotes II. One can describe the uniformly accelerated Rob to travel the three component vector of Pauli matrices. Ry(2θ) is the rotation of angle on a hyperbola constrained to region I, as shown in Figure 2θ around the y axis. 1(a). Rob has no access to field modes in the causally disconnected region II. Therefore, he must trace over the inaccessible region II, which unavoidably leads to the detection of 3 The quantum simulator and the Unruh tem- a mixed state. The corresponding thermal equilibrium state perature relates to the Unruh temperature T = a/2πck ,where is B Here we propose a nuclear spin quantum simulator to simu- the reduced Planck constant, c is the speed of light and k is B late the evolution of fermionic modes in noninertial frames. the Boltzmann constant. For T = 1 K, one requires a ≈ 1021 In Figure 1(b) the quantum circuit for simulating the quan- m/s2, which is experimentally inaccessible now. tum states under the Unruh effect is depicted. The qubit 1 With the emergence of quantum information theory [29], represents the particle mode observed by Rob with initial increasing interests are devoted to relativistic effects in this zero acceleration. The qubits 2, 3, 4, which are initialized field [30-46]. Here we first consider the Unruh effect in quan- in the state |0, represent the antiparticle mode in region II, tum information beyond the single-mode approximation [39]. | | the antiparticle mode in region I and the particle mode in Let 0Ω U and 1Ω U be Unruh vacuum state and one-particle ϕ − ϕ = region II, respectively. By keeping cos 2 inz sin 2 qR, state respectively, then, for the fermion case, they can be de- − + ϕ = ( inx ny)sin 2 qL (Note that qR and qL can be complex scribed by [39-42] 2 2 numbers with |qR| + |qL| = 1.) and θ = r, the transfor- 2 mation of the state in the quantum circuit is equivalent with |0Ω = cos r |0000Ω − sin r cos r|0011Ω U the one under the Unruh effect in eq. (1) (see Appendix A1 2 + sin r cos r|1100Ω − sin r |1111Ω , (1) for details). Thus it enables us to explore novel phenomena | + = | − | 1Ω U qR(cos r 1000 Ω sin r 1011 Ω) under the Unruh effect, such as the behavior of the Unruh + qL(sin r|1101Ω + cos r|0001Ω) , temperature and the dynamics of quantum correlations with relativistic effects via quantum simulation. 2 2 where qR and qL are complex numbers with |qR| + |qL| = 1. For a proof-of-principle quantum simulation of the quan- The dimensionless acceleration parameter r is defined by tum states under the Unruh effect, we turn to the case of = −2πΩc + −1/2 ∈ , π/ , ∈ , ∞ = cos r [exp( a ) 1] with r [0 4] a [0 ), qR 1, which corresponds to the single-mode approxima- and Ω is the frequency of Unruh mode. Here we use the nota- tion and has been greatly discussed [32,34-37,43-45]. In this | = | +| −| −| + ϕ = tion pqmn Ω pΩ I qΩ II mΩ I nΩ II, where the subscripts case, the quantum circuit corresponds the case of 0in ± indicate particle and antiparticle and the subscripts I and II Figure 1(b), and the quantum circuit reduces to the upper two refer to the Rindler regions I and II, respectively. qubits (see Appendix A2 for details). F. Z. Jin, et al. Sci. China-Phys. Mech. Astron. March (2016) Vol. 59 No. 3 630302-3

(a) (a)

(b)

Figure 2 (Color online) Experimental sample and experimental pulse ff sequence for quantum simulating the Unruh e ect. (a) The three-qubit Figure 3 (Color online) Experimental results of the quantum simulation. 13 NMR quantum simulator consists of a sample of C-labeled Diethyl- (a) The experimental populations of the states |0 (red squares) and |1 2 H H fluoromalonate dissolved in H-labeled chloroform. The parameters of the (green circles) with different values of acceleration parameter r. The red line NMR couplings of this molecule are listed in the table. (b) The green dashed and green line are the corresponding theoretical values. (b) Experimental block includes the experimental pulse sequence for simulating the Unruh ef- results (blue triangles) of simulating the Unruh temperature with the accel- 1 19 fect. JHF is the scalar coupling strength between Hand F. The rotation eration parameter. The blue line is the theoretical prediction. The horizontal θ angle is related to the simulated acceleration parameter r. axis is the acceleration parameter r and the vertical axis is the Unruh temperature in unit of ω/kB for simplicity. We now experimentally simulate the behavior of the quan- 19 tum states under the Unruh effect for the case of qR = 1on quantum state seen by the accelerated observer and F nu- a nuclear spin quantum simulator, which consists of a 13C, a clear spin as the ancilla. First we prepare the pseudo-pure 1Handa19F nuclear spin. Our experiments were performed state (PPS), by spatial averaging. Then the spin 1H under- on a Bruker Avance III 400 MHz spectrometer equipped with goes the simulated Unruh effect and its final state becomes a QXI probe at room temperature. We used a sample of 13C- a thermal equilibrium state with the simulated Unruh tem- labeled Diethyl-fluoromalonate dissolved in d-chloroform as perature T (see Appendix A3 for details). Figure 3(a) shows a three-qubit quantum simulator, where the nuclear spins of the experimental populations of the ground states |0H and 13 1 1 Cand H were used to simulate the states observed by Alice the excited state |1H for spin H with different values of the and Rob, and 19F was taken as an ancilla qubit. The coupling acceleration parameter r.Forsmallr,spin1H has high pop- and relaxation time for all three spins are shown in Figure ulation in the ground state, while for large r, the spin has 2(a). In this three nuclear-spin-qubits system, the thermal almost equal populations in both ground and excited states. equilibrium state is given by ρ = (1 − ε)1/8 + ερd in the Figure 3(b) shows that the experimentally simulated Unruh ε ∼ −5 high temperature approximation, where 10 . Deviation temperature (blue triangles, in units of ω/kB for simplicity) density matrix ρd can be prepared to the form of our interest increases along with r, which agrees nicely with the theoreti- in the experiments [47]. Our calculation of the correlations cal prediction (the blue line). is based on the experimental ρd, which is obtained by quantum tomography. The whole duration of the quantum sim- 4 The quantum correlations under the Unruh ulation is about 20 ms and hence, the decay caused by the effect relaxation is negligible. The relevant sources of experimental errors may come from the inhomogeneities of RF fields, Now we investigate quantum correlations under the Unruh ef- static magnetic fields, and imperfect calibration of rotations. fect in the single-mode approximation. Quantum correlations The quantum circuit to simulate the Unruh effect on the quan- can be characterized by quantum discord [48, 49] which has tum states can be realized by the pulse sequence depicted in been proven to be powerful in various tasks in quantum in- Figure 2(b). The first block labeled by “state preparation” formation processing [50]. It has been shown [36,37,46] that is to initialize the system. The second block with the green quantum discord is observer-dependent for both bosonic and dashed line includes the pulse sequence to simulate the Un- fermionic systems via the Unruh effect. Since a uniformly ruh effect. The last block labeled with “tomography” stands accelerated observer is unable to access information about for the procedure of reconstructing the final quantum states. the whole spacetime, from the perspective of the observer a By the nuclear spin quantum simulator, we demonstrate communication horizon appears, resulting in the loss of in- the behavior of the Unruh temperature with the acceleration, formation and accordingly the degradation of correlations. that is depicted in Figure 3. We use 1H nuclear spin as the However, we show that the quantum correlations quantified F. Z. Jin, et al. Sci. China-Phys. Mech. Astron. March (2016) Vol. 59 No. 3 630302-4

ff (a) by quantum discord can be created by the Unruh e ect from (b) the classically correlated states. For the state detected by Alice and Rob (denoted as c 3 c3 ρAR), the total correlation is quantified by mutual information ρ = ρ + ρ − ρ ρ = − ρ ρ MI( AR) S ( A) S ( R) S ( AR), where S ( ) Tr[ log2 ] is the von Neumann entropy and ρA and ρR are the reduced- density matrices of ρAR. When a positive-operator-value {ΠR} measure k is applied on particle R, the maximal acces- c r 1 r sible information about particle A gives the classical correla- k tion CCR(ρAR) = max{ΠR}[S (ρA) − pkS (ρ )], where pk = k k A 1⊗ΠR ρ 1⊗ΠR † ρk = 1⊗ΠR ρ 1⊗ΠR † / Tr[( k ) AR( k ) ], A TrR[( k ) AR( k ) ] pk. (c) The quantum part of the total correlation is defined as quantum discord QDR(ρAR) = MI(ρAR)−CCR(ρAR). In the following, we use QD (CC) to refer to QDR (CCR) for convenience. We consider the Bell-diagonal state

3 1 ρ = 1 + c σA ⊗ σR (2) AR 4 i i i i=1

σA(R) in an inertial frame, where i are the Pauli matrices. From eq. (2), we expand the Minkowski particle states into Rindler Figure 4 (Color online) The behavior of the quantum correlations under regions I and II particle and antiparticle states. Since Rob is Unruh eﬀect. (a) The positive (P) and negative (N) regions of the partial causally disconnected from region II, we trace over the mode derivative of QD with respect to r. The axes denote the acceleration parame- in this region II. Then the quantum state shared by Alice and ter r, the Bell-diagonal-state parameters c1 and c3, respectively. In the green Rob can be expressed as: (red) region N (P), QD is degraded (enhanced) with increased acceleration. (b) The sectional plot of (a) with c1 = 1. (c) Experimental results of simulating the behavior of quantum correlations via the Unruh eﬀect with nuclear ρ =1 1 + σA ⊗ σR + σA ⊗ σR ARI c1 cos r x x c2 cos r y y spin quantum simulator. The experimental values of MI (black squares), QD 4 (red dots) and CC (olive diamonds) are derived from the experimentally re- + 2 σA ⊗ σR − 2 1A ⊗ σR . c3 cos r z z sin r z (3) constructed quantum states. The curves are the corresponding theoretical predications. Insets: the zoom-in plot of the sectional region. This is an X-type state which has an analytical formula of

QD [51, 52]. which is related to a Bell-diagonal state with c1 = 1, c2 = For clarity, we focus on a class of Bell-diagonal states with c3 = 0. This state is a classically correlated state, in which 0 c1 1, 0 c3 = −c2 1, where −c1 + 2c3 1isfor there exists zero quantum discord. After the quantum sim- the valid states. We evaluate the derivative ∂QD/∂r (see Ap- ulator, the final states have been reconstructed by quantum pendix A4 for details) and find that its positivity condition di- tomography. The related values of QD, CC and MI can be vides the parameter space of the acceleration r and the initial obtained with these reconstructed density matrices (see Ap- state coefficients c1 and c3 into two parts, which is shown in pendix A6 for details). We depict the behaviours of CC and Figure 4(a). The red region P has a positive ∂QD/∂r in which MI in Figure 4(c), and show that they both decay along with QD increases along with the acceleration, and the green re- r. Strikingly, it is explicitly shown that QD can be created gion N is the opposite. from this classically correlated state via the Unruh effect. Our In order to demonstrate this phenomenon more clearly, we result implies that, although the total correlations decay, the set c1 = 1. We depict ∂QD/∂r with r and c3 as shown in non-classical correlations can be generated under the Unruh Figure 4(b). These results show that the behaviour of QD effect. Note that the quantum entanglement in this case is in noninertial frames is not only observer-dependent but also always zero. initial-state-dependent. From Figures 4(a) and 4(b), one can explicitly see that it is possible to enhance QD via the Un- 5 Conclusion and discussion ruh effect. It is worth remarking that even with a classically correlated two-qubit state (c1 = 1, c2 = c3 = 0), QD can be To summarize, we have performed a proof-of-principle quan- created by the Unruh effect. This remarkable phenomenon tum simulations of the evolution of fermionic modes under can be ascribed to the nonunital characteristics of the Unruh the Unruh effect, whose observation is current technically in- channel. (see Appendix A5 for details) accessible. Furthermore, we have discovered that quantum The experimental results for simulating the behavior of correlations quantified by quantum discord can be created by quantum correlations under the Unruh effect are shown in the Unruh effect. This indicates that the Unruh effect can Figure 4(c) . We first prepare the state of spin 13Cand1H, affect the behavior of quantum correlations (decreasing or in- F. Z. Jin, et al. Sci. China-Phys. Mech. Astron. March (2016) Vol. 59 No. 3 630302-5 creasing) in several aspects, depending on the observer’s ac- 26 J. Steinhauer, Nat Phys. 10, 864 (2014). celeration as well as the initial modes shared by observers. 27 A. Spuru-Guzik, A. D. Dutoi, P. J. Love, and M. 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The qubits 2, 3, 4, which are initialized in the state |0, represent the antiparticle mode in region R II y II, the antiparticle mode in region I and the particle mode in region II respectively. When the first qubit is in ground state Figure a1 (Color online) The quantum circuit for simulating the Unruh ff = |01, after the operations U(ϕ, nˆ,θ) of the quantum circuit the e ect in the case of qR 1. The qubit 1 is related to the states detected the final state becomes accelerated observer Rob, and the qubit 2 is used to model the Unruh effect on qubit 1.

U(ϕ, nˆ,θ)|01020304 2 = cos θ |01020304−sin θ cos θ|01021314 tion of the quantum states under the Unruh effect, nuclear 2 1 19 + sin θ cos θ|11120304−sin θ |11121314 , (a1) spins H (Rob) and F (ancilla) were used. where the the subscripts 1, 2, 3 and 4 refer to the qubit 1, 2, 3 and 4 of Figure 1(b), respectively. When the first qubit is in Appendix A3 Experimental quantum simulation of the ff excited state |11, after the operations U(ϕ, nˆ,θ)thefinal state quantum states under Unruh e ect becomes The experimental quantum simulation of the quantum states ϕ, ,θ | under the Unruh effect is performed on a nuclear spin quan- U ( nˆ ) 11020304 ϕ ϕ tum simulator. The nuclear spin quantum simulator con- = cos − in sin (cos θ|1 0 0 0 −sin θ|1 0 1 1 ) 13 1 19 2 z 2 1 2 3 4 1 2 3 4 sists of a C, a Handa F nuclear spin. Nuclear spins ϕ 13Cand1H are used to simulated the quantum states seen + (−in + n )sin (sin θ|1 1 0 1 + cos θ|0 0 0 1 ). x y 2 1 2 3 4 1 2 3 4 by Alice and Rob, respectively. The 19F nuclear spin is (a2) taken as the ancilla. Starting from the equilibrium state, we Comparing these two equations with eq. (1), we see that by first initialized the system into the pseudopure state (PPS) ϕ ϕ ϕ ρ000 = 1− 1 + |000 000| by spatial averaging [53], with the keeping cos − inz sin = qR,(−inx + ny)sin = qL and 8 2 2 2 ≈ −5 1 × θ = r, the forms of them are equivalent. Thus it enables us to polarization 10 and is a 8 8 unit matrix. The exper- simulate the evolution of fermionic modes under the Unruh imental result of the PPS is shown in Figure a2(a). We per- effect as well as to explore novel phenomena under the Unruh formed a full state tomography [54], which involves the ap- effect. plication of 16 readout pulses and recording of the spectra of all three spins to obtain the coefficients for the 64 operators, which comprise a complete operator basis of the three-spin Appendix A2 The transformation and quantum circuit system. The information of the 1Hand19F spins was obtained in the case of q =1 R through the 13C spectra transferred by a SWAP gate [55]. Fig- For a proof-of-principle experimental quantum simulation of ures a2(b) and a2(c) show the tomographically reconstructed ff ρ000 the quantum states under the Unruh e ect, we turn to the case density matrix exp with the state fidelity 0.99. of qR=1 in eq. (1), the Minkowski vacuum state |0M and one- particle state |1M can be expressed as [35, 37]:

|0M = cos r|0I|0II + sin r|1I|1II , (a3) |1M = |1I|0II . Correspondingly, the effective quantum circuit reduces to the upper two qubits, as Figure a1 shown. With the qubit 2 initialized in the state |0,wefirst perform a controlled-NOT gate on qubit 1 (conditional on the control bit being set to |0), and then apply a controlled-rotation on the qubit 2 with the rotation angle 2θ around the y axis, and eventually perform a controlled-NOT gate on qubit 1 (conditional on the control bit being set to |0). The transformation Uˆ (θ)ofthe quantum circuit is the same with the two-mode squeezing op- ˆ = † † − erator U(r) exp[r(bI bII bIbII)]. In our experiment, we used a sample of 13C-labeled Figure a2 (Color online) Results of pseudo-pure state tomography. (a) 13C experimental spectra after a [π/2] readout pulse applied to the carbon Diethyl-fluoromalonate dissolved in d-chloroform as a three- y for the initial pseudo-pure state ρ000. (b) and (c) Real and imaginary parts 13 qubit quantum simulator, where the nuclear spins of Cand of the experimental density matrices for the initial state |000 respectively. 1H were used to simulate the states observed by Alice and The rows and columns represent the standard computational basis in binary Rob, and 19F was taken as an ancilla qubit. For the simula- order, from |000 to |111. F. Z. Jin, et al. Sci. China-Phys. Mech. Astron. March (2016) Vol. 59 No. 3 630302-7

Then the second nuclear spin 1H (in ground state |0) un- (a) dergoes the simulated Unruh effect with the quantum circuit and the experimental pulse sequence showing in Figure 2(b). The populations of 1H’sfinal state were used to obtain the simulated Unruh temperature. After a [π/2]y readout pulse appliedonthe13C spin, the integrated intensities of the two peaks in 13C NMR spectrum (as shown in Figure a3), given by a and b, is related to the population of 1H in states |0 and |1 [56]. During this procedure, we decoupled the an- r cilla spin 19F to get rid of its influence. Then the population 1 (b) of H in state |0, |1 can be calculated as P0 = a/(a + b) and P1 = b/(a + b) respectively. We can obtain the stim- ω ulated Unruh temperature by T = (ln P0 )−1 according to kB P1 the Fermi-Dirac statistics [35, 37], here ω is the frequency of Minkowski mode. In the main text, the temperature is shown in units of ω for simplicity. kB

Appendix A4 The calculation of quantum correlations for X-type states r

The states of eq. (3) is an X-type state which has an ana- (c) lytical formula of QD [51]. The optimal measurements are σx or σz, while occasionally there may be very small deviation [52]. Note that the deviation is so small that it doesn’t affect the conclusions in our main text. We confirm this by comparing quantum discord derived from the analytical formula and the numerically computed values (see Figure a4). The method to numerically obtain the values of the quantum discord according the definition is described in the next sec- r tion. ρ The MI of ARI is given by Figure a4 (Color online) Calculation of the quantum discord in the case = ρ + ρ − ρ of c1 = 1, 0 c3 = −c2 1. (a) Quantum discord obtained according to MI S ( A) S ( RI ) S ( ARI ) eqs. (a4) and (a5); (b) quantum discord numerically obtained according their 2 4 definitions; (c) the deviation between the (a) and (b). The maximal deviation = 1 − q log q + χ log χ , (a4) − i 2 i i 2 i is less than 3 × 10 6. i=1 i=1 2 with q1,2 = (1 ± sin r)/2, and 1 2 4 χ , = 1 − c cos r ± (c + c )2 cos2 r + sin r , 1 2 4 3 1 2 1 2 4 χ , = 1 + c cos r ± (c − c )2 cos2 r + sin r . 3 4 4 3 1 2 ρ The CC of ARI can be expressed as: = ρ − ρk CC max[S ( A) pkS ( A)] {ΠRI } k k 2 4 = − Λ Λ + λ λ , 1 min ilog2 i ilog2 i (a5) {α=0,π/2} i=1 i=1

2 with Λ1,2 = (1 ± cos α sin r)/2, and r 1 2 λ1,2,3,4 = 1 ± cos α sin r 13 ﬀ 4 Figure a3 (Color online) Experimental C spectrum for di erent stimu- lated acceleration parameters r. The integrated intensities of the two peaks ± { 2, 2} 2 2 α + 2 4 2 α . give the relative population of ground state |0 and excited state |1 of 1H. max c1 c2 cos r sin c3 cos r cos F. Z. Jin, et al. Sci. China-Phys. Mech. Astron. March (2016) Vol. 59 No. 3 630302-8

QD can be determined by QD = MI − CC. Then we can We observe that the form of the Unruh channel is simi- obtain ∂QD/∂r , which is shown in Figure 4(a). lar to that of the amplitude damping channel [29]. Accord- In the case of c1 = 1, c2 = −c3, the classical correlation ingly, the Unruh eﬀect in the single-mode approximation can can be expressed as: be characterized by operating the quantum Unruh channel lo- ⎛ ⎞ cally on mode R. While the quantum entanglement does not ⎜ 2 ⎟ ⎜ ⎟ increase through any local quantum channel, QD can be pro- CC = 2 ⎝⎜1 + pilog pi⎠⎟ , (a6) 2 duced by the local nonunital channel [57-59]. Since in our i=1 design the Unruh eﬀect under the single mode approximation = ± / with p1,2 (1 cosr) 4. The quantum mutual information is corresponds to a quantum channel, we ascribe the profound given by quantum discord creation to its nonunital channel character-

2 4 istics. = − + λ λ , MI 1 qilog2qi ilog2 i (a7) i=1 i=1 Appendix A6 Calculating the correlations from experi- 2 mental results with q1,2 = (1 ± sin r)/2, and The values of the correlations in Figure 4(c) were obtained 1 2 2 2 4 λ1,2 = 1 − c3 cos r ± (1 − c3) cos r + sin r , from the experimental quantum states ρCH. We experimen- 4 tally reconstruct the density matrix ρCH by state tomography, 1 2 2 2 4 λ3,4 = 1 + c3 cos r ± (1 + c3) cos r + sin r . where the subscripts C and H represent the qubit of the nu- 4 clear spin of 13Cand1H respectively. Then we calculate the QD can be determined by QD = MI−CC. We then calculated correlations according their definitions. The mutual informa- the values of ∂QD/∂r in this case, which is shown in Figure tion of ρCH is calculated with MI(ρCH) = S (ρC) + S (ρH) − ρ 4(b). S ( CH). The classical correlation and quantum discord are ρ = ρ − ρk obtained with CCH ( CH) max{ΠH }[S ( C) k pkS ( C)] and ρ = ρ − ρk Appendix A5 Unruh channel and the creation of quan- QDH( CH) MI( CH) CCH( CH) by optimizing over all tum discord one-qubit measurements. It has been proven [60] that the optimal measurement is always projective for two-qubit states, In order to better understand the underlying physics, we an- so it is sufficient to maximize over all the following projec- alyze the creation of quantum discord from the perspective tive measurements {1⊗|Θk >< Θk |, k = , ⊥},where| Θ = of quantum channels. A quantum channel Φ is a trace- iφ −iφ cos θ1|0 + e sin θ1|1 and |Θ⊥ = e sin θ1|0−cos θ1|1 preserving, completely positive map which can be expressed † presents an arbitrary basis of H formed by two orthogonal as ρ → Φ(ρ) = E ρE , with E (Kraus operators) satisfy- π i i i i states on the Bloch sphere, with 0 θ1 and 0 φ 2π. † = 1 2 ing i Ei Ei [29]. Consider an arbitrary density matrix Then the correlations can be obtained from the reconstructed ρ of fermionic modes A and R in an inertial frame, AR.When density matrices ρCH. To estimate the error bar of the cor- mode R is observed by uniformly accelerated Rob, by using relations, we define interpolated density matrices ρ ,where eq. (a3) and tracing over the disconnected Rindler region- each element of ρ is near the corresponding element of ρCH II, we get ρ . In terms of quantum channels, we can write ARI within errors. The each element error of experimental density ρ = (1 ⊗ Φ )(ρ ), where Φ is the Unruh channel. Note ARI U AR U matrices comes from the corresponding error propagation by that the Kraus operators are associated with this Unruh chan- the state tomography. Then we compute correlations of thou- ρ ρ nel by sands of and pick out the maximum (minimum) ones max cosr 0 00 ρ ff ρ ρ E0 = , E1 = . (a8) ( min). The di erence of correlations between max ( min)and 01 sinr 0 ρ CH denote the upper (lower) error bar, as shown in Figure 1 † It follows that ρ = = (1 ⊗ E )ρ (1 ⊗ E ). 4(c). ARI j 0 j AR j