Emergence of Classical Objectivity of Quantum Darwinism in a Photonic Quantum Simulator

Ming-Cheng Chen1,2, Han-Sen Zhong1,2, Yuan Li1,2, Dian Wu1,2, Xi-Lin Wang1,2, Li Li1,2, Nai-Le Liu1,2, Chao-Yang Lu1,2,∗ and Jian-Wei Pan1,2 1 Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China and 2 CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China. (Dated: August 6, 2019) Quantum-to-classical transition is a fundamental open question in physics frontier. Quantum decoherence theory points out that the inevitable interaction with environment is a sink carrying away quantum coherence, which is responsible for the suppression of quantum superposition in open quantum system. Recently, quantum Darwinism theory further extends the role of environment, serving as communication channel, to explain the classical objectivity emerging in quantum measurement process. Here, we used a six-photon quantum simulator to investigate classical and quantum information proliferation in quantum Darwinism process. In the simulation, many environmental photons are scattered from an observed quantum system and they are collected and used to infer the system’s state. We observed redundancy of system’s classical information and suppression of quantum correlation in the fragments of environmental photons. Our results experimentally show that the classical objectivity of quantum system can be established through quantum Darwinism mechanism.

Key Words: Quantum Measurement, Quantum Darwin- (a) ism, Hovelo Bound, Quantum Discord Single Photons.

Photons Observers

INTRODUCTION

Quantum system Quantum mechanics is a spectacularly successful predic- (b) tive theory, but there is still an unresolved problem about its 0 R rS interpretation in quantum measurement problem [1]. The or- 0 q thodox Copenhagen interpretation separates the world into 1 0 q quantum domain and classical domain, which is bridged by 2 0 rE observation-induced collapse [2]. How the wave function col- q3 0 lapses and classical objectivity emerges from a quantum sub- q4 strate? A detailed mechanism of this quantum-to-classical 0 q5 transition is of fundamentally importance for developing a unified view of our physical world. Figure 1: Quantum Darwinism process. (a) Multiple observers use Quantum decoherence theory identifies that the uncon- independent fragments of the scattered environment photons to re- trolled interactions with the environment can destroy the co- veal the state of observed quantum system. They can determine the herence of a quantum system into a mixed state. In the theory, pointer states of the observed quantum system without perturbing it the environment is traced out and thus the system’s classical and thus agree on the observed outcome. As a result, the quantum behavior is explained in the level of ensemble average [3,4]. system becomes classical and objective in this process. (b) A quan- arXiv:1808.07388v2 [quant-ph] 5 Aug 2019 tum simulator to simulate the system-environment interaction and How the quantum system’s classical objectivity arises in a sin- produce quantum Darwinism states. In the simulator, the fist qubit is gle measurement event is still unresolved. Classical objectiv- quantum system and the environment particles (other qubits) inter- ity is a property that many observers can independently ob- act with the system of arbitrary interaction strengths {θi} in parallel serve and establish a consensus view of the state of a quan- and have no interaction between themselves. Measurements are per- tum system without perturbing it [5]. In a general observation formed on these particles to infer the information of quantum system. process, observers don’t directly touch and interact with the quantum system. They perceive the system by collecting information from its surrounding environment. the system can reach observers. The environment selects sys- Recently, quantum Darwinism explains the emergence of tem’s classical information to broadcast and proliferate, and classical objectivity of a single quantum system through clas- observers use the redundant classical information in local en- sical information broadcasting and proliferating in its envi- vironment fragments to perceive the state of system. In this ronment [5,6]. The key idea is that the environment acts as process, many observers can independently and simultane- communication channel and only classical information about ously query separate fragments of the environment and reach a 2

2 3 THEORY

1 The basic process of quantum Darwinism is shown in Fig. 1(a). A central quantum system (single photon) is monitored EPR EPR EPR by particles (photons) in the environment [16, 17]. The particles are scattered from the system and caught by observers.

4 LO 5 LO 6 These environment particles serve as individual memory cells which are imprinted of system’s pointer-state information. QWP When there are random interactions among the environment’s HWP Filter particles, the stored information will be inevitably scrambled PBS (q) out. Hence, only non-interaction environment is good mem- BD ory for redundant records of system’s state. BBO Coupler EPR LO BS PA We design a quantum Darwinism simulator shown in Fig.1(b) to simulate non-interaction environment, such as Figure 2: Experimental setup. A six-photon interferometer is used daily photonic environment. In the simulator, a central to produce the system-environment composite quantum Darwinism qubit interacts with the environment qubits through two-qubit states. Photon 1 is the central quantum system and photons 2 ∼ 6 controlled-rotation gates U(θ) = |0i h0| ⊗ I + |1i h1| ⊗ Ry(θ) are the environment. Infrared laser pulses (775 nm wavelength, 120 with random angles to mimic the random scattering process, fs pulse duration, 80 MHz repetition rate) pass through three beta- barium borate (BBO) nonlinear crystals sequentially to produce three where Ry(θ) rotates a qubit by angle θ along the y axis of pairs of Einstein-Podolsky-Rosen(EPR) polarization-entangled pho- Bloch sphere. When the system qubit is initialized in super- tons. The two components of EPR state are independently produced position state α|0iS + β|1iS, the simulator will produce Dar- and coherently combined by beam displacers (BDs). Three single winist states with branch structure photons, one from each pair (photons 1, 2 and 3), interfere on two po- θ θ larization beam splitters (PBSs) to generate an entangled six photons α|0i ⊗N |0i + β|1i ⊗N (cos i |0i + sin i |1i ), in Greenberger-Horne-Zeilinger (GHZ) state. Two single photons (5 S i=1 i S i=1 2 i 2 i and 6) pass through polarization-dependent Mach-Zehnder (MZ) in- (1) terferometers to realize local non-unitary operations (LOs). In the MZ interferometers, two polarization components of input photons where |α|2 + |β|2 = 1 and N is the number of environment are separated by polarization beam splitters and recombined on bal- qubits. anced beam splitters (BSs). The internal half-wave plates (HWP) are The interaction with environment selects preferred pointer set at angle θ5/4 and θ6/4, respectively. All the photons are sent to polarization analysis (PA) setups, each consisting of a quarter- states of the observed system, which are the states left un- wave plate (QWP), a HWP, a spectrum filter, and a PBS. The photons changed under the interactions and thus multiple records of are finally detected by fiber-coupled single-photon detectors and six- the state can be faithfully copied into environment. For inter- fold coincidence counting are registered. Note that our experimental actions generated from a Hamiltonian form of Hi = giA⊗Bi, setup is not a faithful simulator of the scattering process. Instead, the eigenstates of monitored observable A are the pointer we mainly want to simulate the process where scattered photons are states, where A and Bi are two observables on system and en- used to infer the state of quantum system. vironment particle i, respectively [5]. In our simulator setting, A = (σI − σZ )/2, Bi = σX and gi∆t = θi/2, therefore, the −iHi∆t pointer states from interaction U(θi) = e are |0i and |1i, respectively. consensus about the system’s classical state. Specifically, the The pointer states can be quantified by the disappearance of quantum Darwinism theory singles out a branch structure of quantum coherence system-environment(observers) composite quantum states [7– 9] from measurement-like interaction to explain the appear- C(ρS) = Hcl(ρS) − H(ρS) (2) ance of classical pointer states. The classical pointer states are the eigenstates of the measurement observable. where ρS is the reduced density matrix of system, H(ρS) = −tr(ρSlog2ρS) is quantum von-Neumann entropy and In this work, we report a test of quantum Darwinism princi- Hcl(ρS) = −tr(pslog2ps) is classical Shannon entropy, ps ple on a photonic quantum simulator [10, 11] in view of infor- is the diagonal elements of density matrix ρS in pointer-state mation theory. We measured the information correlations be- bases [18, 19]. The reality of pointer states will emerge when tween system and environment , where the system is a single the quantum system is completely decohered by the environ- photon and the environment is another five photons. Quan- ment. In this case, the classical entropy will equal to the quantum mutual information, Holevo bound, and quantum discord tum entropy. The efficiency of decoherence depends on the [12–15] are used to account for the total correlation, classical initial states of environment [17, 20]. Impure or misaligned correlation, and pure quantum correlation, respectively. We (close to the eigenstate of observable Bi) environment will re- used these correlations to investigate information broadcast- duce the decoherence efficiency. In our simulation, |0i states ing and proliferating. are used as initial environment states with optimal efficiency. 3

(a) (b)

rij

11 11 1 000 1 11 00 → 1 000 11 0 → 0 00 1 11 00 0 → → 111 00 11 00 1 0 111 00 (c) (d) 1 00

1.6 1.6

1.2 1.2

0.8 0.8

0.4 0.4 Mutual information

0.0 0.0 2 3 4 5 6 2 3 4 5 6 23 24 25 26 34 35 36 45 46 56 23 24 25 26 34 35 36 45 46 56 234 235 236 245 246 256 345 346 356 456 234 235 236 245 246 256 345 346 356 456 2345 2346 2356 2456 3456 2345 2346 2356 2456 3456 23456 Scattered photons Scattered photons 23456

Figure 3: Experimental results. (a)(b) The absolute value of restructured density matrix elements of Darwinism states for parameter θ~A and θ~B , respectively. The states are estimated by quantum state tomography with 729 measure settings. (c)(d) The mutual information between the system and the 31 different pieces of environment fragments for parameter θ~A and θ~B , respectively. The red line indicates the classical entropy of system and the dotted line marks 70% of the classical entropy to guide the eye. The short arrows label four special environment fragments for further analysis in the main text.

√ In quantum Darwinism process, the information about the the superposition state (|0i + |1i)/ 2. Two sets of rotation- ~ ◦ ◦ ◦ ◦ ◦ quantum system is broadcast to the environment. Local ob- angle parameter, θA = (180 , 180 , 180 , 180 , 180 ) and ~ ◦ ◦ ◦ ◦ ◦ servers can only access small fragments of the whole environ- θB = (180 , 180 , 180 , 72 , 100 ), are used in the experiment.The quantum mutual information ments from the following considerations. Note that here the ~ phases in θB are chosen merely to represent nonorthogonal I(S: E ) = H(ρ ) + H(ρ ) − H(ρ ) i S Ei SEi (3) case to simulate the small environment fragment without spe- cific optimization. can be used to quantify how much information an observer Ei (accessing the environment fragment Ei) knows about the A real environment fragment can contain many elemen- quantum system [5]. When I(S: Ei) ≈ H(ρS) for all ob- tary subsystems. If the fragment is large, its quantum states servers {Ei}, the system’s state can be determined by all the can be simplified and expressed as orthogonal logical states observers and thus the quantum system becomes objective. n n θi θi |0Li = ⊗i=1|0ii and |1Li = ⊗i=1(cos 2 |0ii + sin 2 |1ii), Qn θi due to h0L | 1Li = i=1 cos 2 → 0 for sufficiently large number n of subsystems in a fragment. Furthermore, if the EXPERIMENT fragment is small, its quantum states can be expressed as θ θ nonorthogonal logical states |0Li and cos 2 |0Li + sin 2 |1Li. We use a photonic simulator [10, 11] (shown in Fig.2) In our simulation, the photonic qubits represent the quantum to produce quantum Darwinism states consisted of a central state of environment fragments logically and an observer can system qubit (photon 1) and five environment qubits (pho- access one or more fragments to infer the system’s state. The ~ tons 2 ∼ 6 ) . The quantum state of system is observed at parameter θA is used to simulate 5 large environment frag- 4 √ 1.6 r |VVVVVV i)/ 2 [24]. The success probability is 0.25. (a) Total q Classical A (3) Two photons from the GHZ state are further operated 1.2 Quantum by polarization-dependent Mach-Zehnder (MZ) interferome- 0.8 ter. In the interferometer, the H and V components are sep- 0.4 arated by polarization beam splitters and then recombined

0.8 The experiment runs with repetition rate 80 MHz and the single photons are measured with collecting and detecting 0.4 efﬁciency of 0.65. Thus, we obtain six-photon coincidence

Information obtained 2,3,4,5,6 0.0 r counting rates about 5 counts/second. The quantum states are 1.6 (c) qB measured through quantum state tomography. There are 729 1.2 measurement settings and about 700 counts in each setting. 0.8 Fig.3(a) and (b) show the measured density matrices. For set- ~ ting θA, the quantum state ﬁdelity is 0.859±0.002 and the pu- 0.4 ~ rity is 0.777±0.004. For setting θB, the quantum state ﬁdelity

Information obtained 5,6,2,3,4 0.0 r is 0.703±0.004 and the purity is 0.692±0.006. The standard 1.6 (d) qB deviation is estimated from Monte Carlo method with 100 tri- 1.2 als. 0.8 The quantum coherence C(ρS) of system qubit in equation ~ ~ 0.4 (2) are 0.001(4) and 0.020(6) in process θA and θB, respec-

Information obtained 2,5,3,6,4 tively, which indicates that the environment has fully deco- 0.0 0 0.2 0.4 0.6 0.8 1 hered the system. The quantum mutual information I(S: Ei), Fraction of environment the equation (3), between the system and 31 different combi- nations of environment fragments are shown in the Fig.3(c) Figure 4: Accessible information of local observers. (a) Quantum ~ and (d), respectively. The results have two significant features. Darwinism process θA. The total correlation, classical correlation The mutual information quickly approaches the system’s en- (Holevo bound) and pure quantum correlation (quantum discord) between the system and the environment are shown. The fraction of tropy H(ρS) (exceeding 70%, mainly limited by the purity of environment increase in the order of photons 2,3,4,5 and 6. (b)(c)(d) prepared entangled quantum states) when accessing small en- Quantum Darwinism process θ~B . The orders of environment photons vironment fragments, and the mutual information is saturated are 23456, 56234 and 25364, respectively. One standard deviation is when increasing the size of environment fragments. smaller than the size of data marker. Redundancy plateau of clas- The arrows in Fig.3(d) show that the environment frag- sical information correlation in environment fragments is observed ments 5 and 6 are two low-fidelity records of the system’s meanwhile pure quantum correlation is suppressed due to the incom- states, which is expected from the corresponding nonorthog- pleteness of environment. Local observers use only small fraction θ~ of environment to infer the system, so only classical information is onal rotating angles in process B. On the other hand, envi- accessible. ronment fragments 2346 and 2345 have mutual information exceeding the system’s entropy. This indicates that the mutual information contains more information than system’s in- ~ formation alone. We further analyze the information compo- ments and parameter θB is used to simulate 3 large environment fragments and 2 small environment fragments. sitions by dividing it into the locally-accessible classical information and the extra information from pure quantum cor- The experimental setup is shown in Fig.2. The qubits are relations. We show the classical correlation and quantum cor- encoded in the horizontal (H) and vertical (V ) polarization of relation between system and environment fragments in Fig. single photons, which are produced by spontaneous paramet- 4. ric down-conversion (SPDC) process [21, 22]. The Darwin- ism states in equation (1) are synthesized in three steps. The first one, Hovelo bound χ(S: Ei), measures the capac- ity of environment acting as communication channel to deliver (1) Preparation of three pairs of polarization-entangled system’s classical information. The Hovelo bound photons√ in Einstein-Podolsky-Rosen (EPR) state (|HHi + |VV i)/ 2 [23] from SPDC process. With a laser pump X X χ(S: Ei) = max{H( psρE |s) − psH(ρE |s)} (4) power of 0.9 W , the generation probability of two twin pho- {M } i i s s s tons is 0.025 and the fidelity of EPR state is above 99%. (2) Three EPR pairs are combined on two polariza- is maximum mutual information of the classical-quantum tion beam splitters, postselecting the entangled subspace state between system and environment fragments with opti- of |HHi hHH| + |VV i hVV |, to produce a six-photon mal measurement {Ms} on the system [14], where ρEi|s is Greenberger-Horne-Zeilinger (GHZ) state (|HHHHHHi + quantum state of environment Ei conditioned on a measured 5 result s on the system with probability ps . The second one, step to test the quantum Darwinism in small-scale controllable quantum discord quantum environment. We expect further works to investigate the quantum Darwinism with more complex (e.g., larger-scale D(S: Ei) = I(S: Ei) − χ(S: Ei) (5) and mixed) quantum environment [17, 20] along with the con- siderable progress of current experimental quantum simula- measures the loss of information due to the observers can tion technology [11, 29–31] and high-efficient quantum state only locally access the environment, which quantifies the pure characterization technology [32, 33]. quantum correlation between environment and system [14]. Acknowledgement: This work was supported by the Na- In Fig.4, the classical correlations and quantum correla- tional Natural Science Foundation of China, the Chinese tions display very different features. The classical correla- Academy of Sciences, the National Fundamental Research tions have initial rise and saturate at classical plateau, which Program, the Anhui Initiative in Quantum Information Tech- indicate the environment has recorded redundant copies of nologies and the Postdoctoral Innovation Talents Support Pro- system’s classical information for independent observers. In gram. sharp contrast, the quantum correlations raise when the nearly Note: After completing our work, we became aware of two whole environment is accessed, manifesting quantum correla- related experiments, one using two photons generating a clus- tions cannot be shared between the observers [25]. The Fig. ter state [34] and the other one using single NV center [35] to 4 (b) and (c) also demonstrate the effects of low-fidelity en- ~ demonstrate quantum Darwinisim. vironment fragments (photons 5 and 6 in process θB ), which will lead to early raise of quantum correlation (Fig.4 (b)) or Author Contributions: M.-C.C., C.-Y.L., and J.-W.P. con- delay the raise of classical correlation (Fig.4 (c)). vinced and designed the work.M.-C.C., H.-S.Z., Y.L., D.W., X.-L.W., L.L., and N.-L.L. performed the experiment. M.- C.C., analyzed the data and wrote the manuscript with input DISCUSSION from all authors. C.-Y.L. and J.-W.P. supervised the project.

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