Articles https://doi.org/10.1038/s41567-019-0482-z

Emergence of the geometric phase from quantum measurement back-action

Young-Wook Cho 1,4*, Yosep Kim 2,4, Yeon-Ho Choi1,3, Yong-Su Kim1,3, Sang-Wook Han1, Sang-Yun Lee 1, Sung Moon1,3 and Yoon-Ho Kim 2*

The state vector representing a quantum system acquires a phase factor following an adiabatic evolution along a closed trajec- tory in phase space. This is the traditional example of a geometric phase, or Pancharatnam–Berry phase, a concept that has now been generalized beyond cyclic adiabatic evolutions to include generalized quantum measurements, and that has been experimentally measured in a variety of physical systems. However, a clear description of the relationship between the emer- gence of a geometric phase and the effects of a series of generalized quantum measurements on a quantum system has not yet been provided. Here we report that a sequence of weak measurements with continuously variable measurement strengths in a quantum optics experiment conclusively reveals that the quantum measurement back-action is the source of the geometric phase—that is, the stronger a quantum measurement, the larger the accumulated geometric phase. We furthermore find that in the limit of strong (projective) measurement there is a direct connection between the geometric phase and the sequential weak value, ordinarily associated with a series of weak quantum measurements.

hen a quantum system undergoes adiabatic state evo- phase. Since the measurement interaction in our work is based on lution along a closed trajectory due to a slow-varying a two-qubit entangling operation26, we clearly demonstrate that the Hamiltonian, it gains a phase factor, now known as quantum measurement back-action results from the sequential W 1 the Berry phase . Together with Pancharatnam’s phase for polar- weak quantum measurements. At the strong-measurement limit ized light2,3, the Berry phase is referred to as the geometric phase, (that is, projection measurement), this generalized geometric phase because the accumulated phase factor was originally related to the is then simplified to the standard geometric phase for well-defined closed geometric path of adiabatic state evolution in an Hilbert state evolution. We furthermore find that, in the limit of strong space. Over the years, the concept of the geometric phase has been projection measurement, there is a direct connection between the significantly generalized, including non-adiabatic4, non-cyclic5,6 geometric phase and the sequential weak value27–29, ordinarily asso- and non-unitary state evolutions7,8, as well as to include evolution of ciated with a series of weak quantum measurements30–34. a mixed quantum state9. Recently, the geometric phase concept has been found to be applicable in many disciplines, including chemis- Schematic and theory try10, materials science11 and fault-tolerant quantum computing12,13. We start by considering the geometric aspect of the quantum state In addition to the theoretical progress, the geometric phase has change due to sequential quantum measurements. For simplicity, let been observed in a wide variety of physical systems, including pho- us first examine the case of sequential quantum measurements for tons14–17, graphene18, superconducting systems19, exciton-polariton projective observables. The initial quantum state |ψ〉 is subject to a condensates20, diamond colour-centres21, among others. ϕ series of projectors Â, B ̂and Π̂ =∣ϕϕ⟩⟨ ∣. The projection postulate The generalized geometric phase for non-adiabatic, non-cyclic ϕ and non-unitary state evolution can be viewed as the geometric phase states that the sequential observable Π̂BÂ̂ causes the initial quan- induced by a series of generalized quantum measurements. As quan- tum state |ψ〉 to evolve according to the trajectory ψ → a → b → ϕ tum measurement inevitably involves state disturbance due to quan- on the Bloch sphere, as shown in Fig. 1a. Note that {ψ, a, b, ϕ} are 3 tum measurement back-action22–24, it has been suggested that quantum real vectors in R . The closed trajectory is formed by taking ϕ = ψ. measurement back-action is at the heart of the quantum nature of the The geometric phase factor exp(iΦG) emerges from the cyclic quan- geometric phase25. However, a complete description of the geometric tum state evolution due to the sequential measurement. To measure phase for a series of generalized quantum measurements has not yet the geometric phase ΦG, a phase reference is required and a Mach– been reported, either theoretically or experimentally. Zehnder interferometer provides such a reference (Fig. 1b). The In this work, we report an experimental demonstration of the detector exhibits interference according to the dynamical phase dif- emergence of the geometric phase that results from the quantum ference Δ between the two probability amplitudes. The qubit state in measurement back-action generated by a series of generalized the upper path is subject to the sequential quantum measurement quantum measurements. By making use of a sequence of weak ̂ϕ ̂ quantum measurements with continuously variable measurement Π BÂ, gaining the geometric phase ΦG, whereas the qubit in the strengths, we have conclusively identified the quantum measure- lower path is projected onto the state |ϕ〉 without incurring any geo- ment back-action as the origin of the geometric phase—that is, the metric phase factor (Fig. 1c). This causes the shift of the interference stronger the measurements, the larger the accumulated geometric fringe by the amount ΦG, which is measurable.

1Center for Quantum information, Korea Institute of Science and Technology, Seoul, Korea. 2Department of Physics, Pohang University of Science and Technology, Pohang, Korea. 3Division of Nano and Information Technology, KIST School, Korea University of Science and Technology, Seoul, Korea. 4These authors contributed equally: Young-Wook Cho, Yosep Kim. *e-mail: [email protected]; [email protected]

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a yˆ = ∣R〉 qubit provides the phase reference to observe the geometric phase shift via quantum interference.

 ϕ We have implemented the quantum circuit in Fig. 2a with pho- tonic polarization qubits and path qubits (Fig. 2b). The system a and meter qubits are encoded in the polarization mode of a sin- b xˆ = D ∣ 〉 gle photon (|H〉 ≡ |0〉, |V〉 ≡ |1〉) and the ancilla and the reference zˆ = H ∣ 〉 qubits are encoded in the path mode of a single photon. Initially, the total quantum state for four qubits is prepared in |ψ〉s|0〉a|+〉m|0〉r, b BS where ∣± ≡∣(0 ±∣1)∕ 2. The subscripts {s, a, m, r} refer to –i 2 ∣ψ〉 ∝ ∣1 + e Δ∣ system, ancilla, meter and reference qubits, respectively. After ∣ψ〉 the single photon encoding the meter and reference qubits passes through the polarizing beam displacer (PBD), the total four- e–iΔ Δ ∣ψ〉 Δ qubit quantum state is ∣∣⊗ψ sa0(∣∣10mr+∣01mr∣∕)2. c A von Neumann type measurement interaction BS ̂ ̂ ̂ ̂ ̂ ̂ i –i 2 M(,ÂB)(= BÂ⊗ σx̂+ II−BÂ) ⊗ is applied between the system ∣ψ〉 Aˆ Bˆ ∝ ∣αe ФG + e Δ∣ ˆˆ qubit and the meter qubit with the aid of the ancilla qubit, condi- BA∣ψ〉 ФG tionally if the state of the reference qubit is |0〉r. Note that, for imple- Πˆ ϕ M̂ ̂ e–iΔ menting (,ÂB), we consider successive interactions acting on Δ ∣ψ〉 Δ the system⊗ancilla⊗meter qubits: Û(Ax=⊗ÂÂσ̂ ⊗ IÎ+ −⊗̂ ) IÎ⊗ ̂ and Û1BB̂ 1(̂ )1̂ 10̂ ̂ 0 .̂ The Bx= ⊗∣ ∣⊗σ̂ + II− ⊗∣ ∣⊗ ++II⊗∣ ∣⊗ Fig. 1 | Conceptual representation of the geometric phase due to quantum ancilla qubit also needs to be projected onto ̂ so that + Πa =∣++∣ measurement back-action. a. A pure quantum state ∣⟩ψ is sequentially 35,36 ϕ Π̂ = IÎ⊗Π̂ a ⊗ .̂ Then, M̂(,ÂB̂)2=ΠTra[ ̂ÛÛ]BA (refs. ), where measured by projectors Â, B ̂and Π̂, represented by vectors {a, b, ϕ} on Tr a[·] is a partial trace over the ancilla qubit. See Methods for further the Bloch sphere. The accumulated geometric phase exp(i ) depends ΦG details on the implementation of M̂(,ÂB)̂. As the ancilla qubit has on the trajectory . , To measure , a phase reference is ψ → a → b → ϕ b ΦG been traced out, the three-qubit state is calculated to be required and a Mach–Zehnder interferometer provides such a reference: the detector exhibits interference according to the dynamical phase −iΔ 2 {0BÂ̂∣⊗ψψ∣∣0(+ I−̂BÂ̂)1∣⊗∣∣0 difference Δ between the two probability amplitudes |1 + e | . c. Only the smrsmr (1) qubit state in the upper path gains the geometric phase ΦG induced by the ϕ +∣ψ sm⊗∣01∣∕r}2 sequential quantum measurements Â, B ̂and Π̂. The interference fringe is given by eeiiΦG −Δ 2 and the geometric phase can be extracted ∣α + ∣ ΦG Since the conditional interaction between the system and meter from the fringe shift. The coefficient α represents the relative amplitude qubits defines the quantum measurement, the weak quantum mea- difference between the two probability amplitudes and is due to quantum surement with the measurement strength γ ∈ [0, 1] on the system measurements on the qubit state in the upper path. BS, beam splitter. qubit is defined by the rotation of the meter qubit R(θg), conditioned1 on the state of the reference qubit, and application of the projector Π̂r to the reference qubit (Fig. 2a). In experiment, the conditional rotation One of the key features of our work is that we consider a gener- of the meter qubit is accomplished with HWPs, as shown in Fig. 2b. alized quantum measurement scenario in which the measurement If the reference qubit is in the state |0〉r, the meter qubit is rotated with can be weak: γ ∈ [0, 1], where γ is the weak measurement strength. the HWP set with the angle θg such that |0〉m → cos 2θg|0〉m + sin 2θg|1〉m This has been achieved by introducing auxiliary qubits for register- and |1〉m → sin 2θg|0〉m − cos 2θg|1〉m. The case of θg = 0 corresponds to 22 ing the measurement outcomes . The relevant quantum circuit is the projection measurement, whereas the case of θg = π/8 corresponds shown in Fig. 2a. Initially, the quantum state |ψ〉s is encoded in the to the null measurement, as no measurement information is gained system qubit and undergoes the sequential quantum measurement when projecting the meter qubit onto the computational basis37. The ϕ M̂(,ÂB)̂ and is finally projected via the projector Π̂s . Specifically, measurement strength γ is then related to the angle θg by the following an ancilla qubit is used to temporarily register the measurement relation, γ(θg) = 1 − 4/(3 + cot 2θg) ∈ [0, 1]. Note that the measurement outcome for the first observable Â. Conditioned on the ancilla state, outcome for the sequential observable BÂ̂ is read-out by projecting the measurement interaction between system and meter qubits the meter state onto the computational basis {|0〉m, |1〉m} for the ref- is realized for the observable B.̂ Then, erasing the information in erence qubit |0〉r. We consider the case when the meter state is pro- the ancilla by projecting it on a certain basis ensures that only the jected onto |0〉m as it corresponds to the trajectory ψ → a → b → ϕ in meter qubit remains as the register for sequential observables BÂ̂ the limit of γ → 1. The post-selection process is realized by consider- (refs. 35,36). In other words, the system–meter interaction becomes ing a specific output port at the second PBD, as shown in Fig. 2b. M̂(,ÂB)(̂= BÂ̂⊗ σx̂+ II−̂BÂ̂) ⊗ .̂ The meter qubit is flipped when See Supplementary Note 2 for further details. If the reference qubit BÂ̂ is applied, otherwise it is unchanged. In our scheme, the four pos- is in the state |1〉r, the HWP oriented at π/4 implements the opera- sible measurement outcomes {00, 01, 10, 11} are essentially reduced tion |0 |1 |1 |1 to pass the second PBD without photon loss. 〉m 〉r → 〉m 〉r ϕ to two outcomes. That is, the outcome ‘11’ corresponds to the case Finally, the projector ̂ is applied to the system qubit, thus making Πs ϕ in which the meter qubit is flipped, and the other three outcomes the system–meter qubit state as Π̂ ss∣⊗ψ ∣M m, where correspond to the case in which the meter qubit is unchanged. Since we are particularly interested in the case where BÂ̂ is applied, we 52γγ2 ++1 consider only this case by projecting the meter qubit at the read-out iΦG ∣∝M mmαe0∣+ ∣1 m (2) process. Note that the measurement strength is determined accord- 22 ing to the choice of projection basis for the meter state at the read- out. Using a single meter qubit makes not only the experimental and set-up simpler, but also extracting the geometric phase easier. The (1 − γ) geometric phase ΦG induced by the quantum measurement for vari- ̂ BÂ̂ ϕγ∣ 4 I + ∣ψ ous measurement strengths is extracted by analysing the projected αeiΦG ≡ (3) meter qubit state with the aid of the reference qubit. The reference ϕψ∣

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a c (A, B ) 22.5° A B 0° D1 ∣+〉m∣0〉r –1 –1 ϕ System ∣ψ〉s R(θA) R (θA)RR(θB) (θB) Πs Meter and + reference Πa + ∣0〉r Ancilla ∣0〉a Πa 22.5° ϕ ∣1〉r Πs θB Meter + R( ) ∆ ∣ 〉m θg m 0°

∣0〉a 0 1 Reference ∣ 〉r Πr θB θA 1 b (A, B ) ∣ 〉a θg θA 1 ∣ψ〉 ∣0〉 22.5° Πr ∣ψ〉s A B ϕ s a Πs D System 1 System and ∣0〉a + Πa ancilla –22.5° PBD Ancilla HWP Δ ∣+〉m∣0〉r 4 θ Meter and g QWP PBS θm reference D2 PBS PPBS PBD ( 0 + 0 1 )/ 2 π π Δ π θ 1〉m∣ 〉r ∣ 〉m∣ 〉r 4 4 4 4 m D HWP HWP QWP QWP 2 Measurement strength γ(θg) = 1 – 4/ (3 + cot 2θg) (fixed) (fixed)

Fig. 2 | Schematic of the experiment. a. Quantum circuit for measuring the geometric phase due to quantum measurement back-action. Measurements

 and B,̂ respectively, are set by R(θA) and R(θB). The sequential measurement M̂̂(,ÂB) between the system qubit and the meter qubit is mediated by the ϕ ancilla qubit and the measurement strength γ is set by R(θg). The system qubit is then projected by Π̂s and the information on the ancilla and the reference + 1 qubits is erased by applying projections Π̂a and Π̂r, respectively. The geometric phase ΦG accumulated on the system qubit due to quantum measurement back-action is read out by measuring the meter qubit (see text for more details). b, Conceptual experimental set-up for the quantum circuit shown in a. The system and meter qubits are encoded in the polarization mode of a single photon and the ancilla and reference qubits are encoded in the path mode of a single photon. The meter and reference qubits are entangled by a CNOT operation: (∣∣10mr+∕∣∣01mr)2. The state |1〉m|0〉r carries the information on the system qubit measured by observables  and B ̂sequentially. To extract the geometric phase ΦG on the system qubit, the meter qubit is scanned by rotating the HWP angle Δ with the measurement basis fixed at θm = 22.5°. c, Schematic of the experimental set-up. The wave plates without denoted angles are fixed at 45°. The quantum measurement M̂̂(,ÂB) is implemented with two-photon quantum interference, thus allowing us to explore the quantum nature of the geometric phase (that is, the geometric phase due to quantum measurement back-action). PBS, polarizing beam splitter; PPBS, partially polarizing beam splitter; PBD, polarizing beam displacer; HWP, half-wave plate; QWP, quarter-wave plate.

Here, 0 and is the geometric phase due to quantum α≥ ΦG With geometric phase 3,000 measurement. 1,200 Phase reference ФG = 0.705 ± 0.020 In order to extract the geometric phase ΦG, the final meter state in equation (3) is analysed with a set of a quarter-wave plate (QWP 2,000 900 set at π/4), a half-wave plate (HWP set at Δ/4), a quarter wave plate (QWP set at /4), a half-wave plate (HWP set at ), and a polariz- π θm 1,000 ing beam splitter (PBS), as shown in Fig. 2b. The coincidence count 600

N between the detectors D1 and D2 then exhibits an interference Coincidence counts 300 fringe as a function of Δ. In particular, with the projection angle set 0 at θm = π/8, varying the HWP angle Δ in Fig. 2b has the same effect as scanning the path length difference in Fig. 1c. From equations (2) –2π –π 0 π 2π and (3), the coincidence count N is calculated to be Δ (rad)

Fig. 3 | Extraction of the geometric phase. The measurement-induced  22  2 52γγ++1  52γγ++1 quantum geometric phase ΦG is extracted by comparing the polarization N ∝+α  + α cos(ΔΦ++G0θ ) (4)  8  2 interferogram as a function of Δ in comparison to the reference interferogram. The experimental dataset for the measurement setting, ψ = ϕ = {0, 1, 0}, a = {0, 0, 1} and b = −{sin(7π/13), 0, cos(7π/13)}, is where θ0 is the initial phase factor. Hence, the geometric phase ΦG due to quantum measurement corresponds to the phase shift from shown as red solid circles. For setting the reference phase, the condition for a reference interference fringe (that is, without the sequential mea- null measurement back-action is used by setting ψ = a = b = ϕ = {0, 0, 1} surement). (see the blue solid circles). From the sinusoidal curve fitting (solid lines), the geometric phase is measured to be 0.705 0.020 rad. The result For the closed trajectory case, that is, ϕ = ψ, we obtain ΦG = ± is in a good agreement with the theoretical value of 0.725 rad. The error bar represents 1 s.d. due to Poissonian counting statistics. −1 γψ ⋅()ba× ΦG = tan (5) 1(++γ aa⋅⋅ψψbb+ ⋅ ) measurement strength γ, which is related to the amount of quan- The above relation clearly demonstrates the fact that the geomet- tum measurement back-action. Note that equation (5) reduces ric phase depends not only on the geometric vectors but also the to the standard geometric phase derived from Pancharatnam’s

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a b c /2 (rad) π ФG γ = 1 π/4 γ = 0.57 – /2 – /4 0 /4 /2 = 0 y = ∣R〉 π π π π γ

(rad) 0 ФG (rad) γ = 0 G Ф  ϕ –π/4 π/2 0 –π/2 1.0 –π/2 a b γ = 1 (rad) 0.8 x = ∣D〉 –π θg ν –π/2 γ = 0.57 π/8 0.6 z = ∣H 〉 γ = 0 4θB + π 4θ 0 0.4 B (rad) π/16 Visibility π/2 ) 0.2 (θ g 0 γ π 0.0 γ = 1 Measurement strength –π –π/2 0 π/2 π

4θB (rad)

Fig. 4 | Emergence of the geometric phase from quantum measurement back-action. a, The quantum state trajectory based on the geodesic hypothesis ϕ due to sequential projective quantum measurements. The initial state |ψ〉, the final projection Π̂, and the first measurement  are fixed, respectively, at ψ = ϕ = {0, 1, 0} and a = {0, 0, 1}. The second measurement B ̂is varied with b = −{sin 4θB, 0, cos 4θB} such that the geometric phase ΦG is accumulated by increasing the angle θB. For weak measurements, it is in general not possible to define the corresponding vectors a and b. b, The measurement- induced geometric phase ΦG as a function of the measurement direction θB and the measurement strength γ(θg). There is good agreement between the experimental data (black dots) and the theoretical result given by equation (6) (surface plot). c, The data show that the geometric phase is indeed due to quantum measurement back-action; that is, the stronger the measurements, the larger the accumulated geometric phase. When the measurement strength γ goes to zero, no geometric phase is observed because there are no changes to the quantum state due to the measurements (that is, no measurement back-action). Note that, at γ = 1, a sudden phase jump is observed. This singular behaviour can be understood by the geodesic hypothesis or by the fact that no phase can be defined as the visibility V = 0 at this point. The error bars are obtained by performing 500 Monte Carlo simulation runs by taking into account the Poisson statistics in measured coincidence counts.

2,3 connection when γ → 1 (that is equation (5) is mathematically the the geometric phase with the increase of θB, but there is a sudden same with the half-solid-angle subjected by the geodesic lines con- phase jump at 4θB = 0, more clearly shown in Fig. 4c. This singu- necting the three vertices {ψ, a, b} when γ → 1). See Supplementary lar behaviour is not observed in the geometric phase obtained via Note 1 for detailed derivation. conventional unitary evolutions and it can be understood with the geodesic hypothesis of the state collapse due to quantum measure- Experimental geometric phase ment. The geodesic hypothesis states that, when a qubit state is col- In experiment, the initial state and the final projection are set at lapsed to an eigenstate due to a projective quantum measurement, ψ = ϕ = {0, 1, 0}. The measurement  is fixed at a = {0, 0, 1} and the the quantum state follows the geodesic line on the Bloch sphere 8,38 measurement B ̂can be varied, b = −{sin 4θB, 0, cos 4θB}. From equa- during its collapse into the eigenstate . The sudden phase jump tion (5), we then have observed in Fig. 4b,c for γ = 1 occurs when the vertices a and b are antipodal on the surface of the Bloch sphere so that no geodesic  ()sin4  lines can be defined39. Therefore, our experimental results at γ = 1 −1 γθg θB  ΦθGg(,B θ )t= an   (6) can be regarded as a clear manifestation of the geodesic hypoth-  1(−γθg)cos 4θB  esis8,38. Although the geodesic hypothesis is an essential prerequisite to reveal the measurement-induced geometric phase, it is a philo- where γ(θg) is the measurement strength. Figure 3 shows a par- sophical notion, as the state collapse is in fact discontinuous and ticular example of the interference fringe observed for 4θB = 7π/13 instantaneous. It is also interesting that the continuous nonlinear with γ = 1. The reference interference fringe is obtained with null stochastic model for the quantum measurement process also con- measurement back-action by setting ψ = a = b = ϕ (that is by set- forms to the collapse of a quantum state only when the geodesic ting all observables to be commuting and the initial state to be trajectory is assumed40. The phase singularity41 can also be seen by their eigenstate). The interference fringe with geometric phase looking at the interference visibility. As depicted in Fig. 4c, when the is shifted from the reference interference fringe by the amount phase jump occurs, the visibility becomes zero. ΦG = 0.705±0.020 rad. The experimentally extracted geometric As the measurement becomes weaker (γ < 1), the measurement- phase from the phase shift measurement is in good agreement with induced geometric phase becomes smaller (Fig. 4b,c). Note also that the theoretically predicted value of 0.725 rad. the phase singularity no longer exists if γ ≠ 1. Moreover, in experi- For the initial state and sequential projectors described above, ment, we find that there is no measurement-induced geometric the quantum state trajectory based on the geodesic hypothesis due phase when γ = 0. This observation is consistent with the theoretical to sequential projective quantum measurements (that is, γ = 1) is results in equations (5) and (6) shown in Fig. 4a. Note that the B ̂measurement direction is varied The expression for the geometric phase in equation (5) clearly by choosing the angle θB. It is clear that, as the angle θB increases, indicates that the geometric phase is in fact due to quantum mea- the accumulated geometric phase ΦG should get bigger as well. surement back-action. The quantum measurement back-action The experimental geometric phase due to quantum measurement refers to the fact that the quantum state of the measured system 22–24 as a function of the measurement strength γ(θg) ∈ [0, 1] and the gets altered non-unitarily due to the measurement . The numer- −1 measurement direction θB is shown in Fig. 4b. For the projection ator in the argument of tan in equation (5), γψ·(b × a), is in measurement (γ = 1), we observe the expected linear increase of fact identical to −2TiBr(γψ[,̂ Â])∣∣ψ , which quantifies the

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Table 1 | Experimental results Our experiment has conclusively identified the quantum mea- surement back-action as the origin of the geometric phase in the ϕ Experimental results measurement process, demonstrating the genuine quantum nature of the geometric phase. We hope our work triggers further stud- {0, 1, 0} ΦG =−0.5190±=.016,0V .850±0.011 ϕ ies to generalize our results to include higher dimensions and more BÂ̂ (0.484 0.012) (0.276 0.010)i w =±−± observables. Finally, our investigation on the measurement-induced ϕ ̂ geometric phase has led us to connect the geometric phase to the B w =−0.4330.250i [theory]  1 1 1  weak value: in the limit of strong (projective) measurement, there  ,,−  ΦG =−0.0140±=.013,0V .945±0.010  3 3 3    ϕ is a direct connection between the geometric phase and the weak ̂ B w =±(0.712 0.023)−±(0.010 0.009)i value. We believe that accessing the sequential weak value and the ϕ BÂ̂ = 0.683 [theory] ability to perform direct quantum state/process characterization via w only projection measurements will greatly improve the applicability The experimental geometric phase ΦG, the interference visibility V, and the extracted ϕ of the weak value in experimental quantum information research. ̂ sequential weak value B w are shown for two different projector settings ϕ. The initial state is ψ = {1 ∕∕∕3,13,1 3}. The observables  and B ̂are projectors set to be a = {0, 0, 1} and b = {1, 0, 0}, respectively. The results are in good agreement with the theoretical predictions. Online content The errors are obtained by performing 500 Monte Carlo simulation runs by taking into account Any methods, additional references, Nature Research reporting the Poisson statistics in measured coincidence counts. summaries, source data, extended data, supplementary informa- tion, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and incompatibility of two observables  and B ̂as well as the mea- code availability are available at https://doi.org/10.1038/s41567- surement strength γ. As both [BÂ,]̂ and γ determine the degree of 019-0482-z. quantum measurement back-action, our result clearly identifies the quantum measurement back-action as the origin of the geometric Received: 5 November 2018; Accepted: 22 February 2019; phase—that is, the stronger the measurements, the larger the accu- Published online: 15 April 2019 mulated geometric phase25. Another interesting aspect of our work is that we able to experi- References mentally demonstrate the connection between the geometric phase 1. Berry, M. V. Quantum phase factors accompanying adiabatic changes. and the weak value, suggested theoretically in recent years27–29. Proc. R. Soc. Lond. A 392, 45–57 (1984). An important and generally accepted notion on the weak value is 2. Pancharatnam, S. Generalized theory of interference, and its applications. Part I. Coherent pencils. Proc. Indian Acad. Sci. Sect. A 44, 247–262 (1956). that a weak measurement interaction (that is, γ≪1) is essential to 3. Berry, M. V. Te adiabatic phase and Pancharatnam’s phase for polarized extract the weak value. As such a weak measurement interaction light. J. Mod. Opt. 34, 1401–1407 (1987). would introduce almost no quantum measurement back-action, it is 4. Aharonov, Y. & Anandan, J. Phase change during a cyclic quantum natural to ask why there would be a mathematical link between the evolution. Phys. 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A 50, 305302 (2017). 30. Aharonov, Y., Albert, D. Z. & Vaidman, L. How the result of a measurement Acknowledgements of a component of the spin of a spin-1/2 particle can turn out to be 100. This work was supported by the KIST institutional programmes (project nos Phys. Rev. Lett. 60, 1351–1354 (1988). 2E29580 and 2E27800-18-P044), by the ICT R&D programme of MSIP/IITP (grant 31. Cho, Y.-W., Lim, H.-T., Ra, Y.-S. & Kim, Y.-H. Weak value measurement with no. B0101-16-1355) and by the National Research Foundation of Korea (grant nos an incoherent measuring device. New. J. Phys. 12, 023036 (2010). 2016R1A2A1A05005202 and 2016R1A4A1008978). Y.K. acknowledges support from 32. Dressel, J. et al. Colloquium: understanding quantum weak values: basics and the Global PhD Fellowship by the National Research Foundation of Korea (grant no. applications. Rev. Mod. Phys. 86, 307–316 (2014). 2015H1A2A1033028). 33. Jordan, A. N. & Korotkov, A. N. 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J. et al. Measurement of quantum weak values of photon Competing interests polarization. Phys. Rev. Lett. 94, 239904 (2005). The authors declare no competing interests. 38. Facchi, P., Klein, A. G., Pascazio, S. & Schulman, L. S. Berry phase from a quantum Zeno efect. Phys. Lett. A 257, 232–240 (1999). Additional information 39. Wong, H. M., Cheng, K. M. & Chu, M.-C. Quantum geometric phase Supplementary information is available for this paper at https://doi.org/10.1038/ between orthogonal states. Phys. Rev. Lett. 94, 070406 (2005). s41567-019-0482-z. 40. Patel, A. & Kumar, P. Weak measurements, quantum-state collapse, and the Born rule. Phys. Rev. A 96, 022108 (2017). Reprints and permissions information is available at www.nature.com/reprints. 41. Camacho, R. M. et al. Realization of an all-optical zero to π cross-phase Correspondence and requests for materials should be addressed to Y.-W.C. or Y.-H.K. modulation jump. Phys. Rev. Lett. 102, 013902 (2009). Journal peer review information: Nature Physics thanks Andrew Jordan and the other 42. Curic, D. et al. Experimental investigation of measurement-induced anonymous reviewer(s) for their contribution to the peer review of this work. disturbance and time symmetry in quantum physics. Phys. Rev. A 97, 042128 (2018). Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in 43. Bednorz, A., Franke, K. & Belzig, W. Noninvasiveness and time symmetry of published maps and institutional affiliations. weak measurements. New J. Phys. 15, 023043 (2013). © The Author(s), under exclusive licence to Springer Nature Limited 2019

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Methods Extracting the weak value. The sequential weak value can be extracted from the Experimental details. Te correlated photon pairs are generated via ultrafast interference fringe in equation (4) with γ = 1. The modulus and the argument 2 pumped spontaneous parametric down-conversion at a 1 mm type-II beamlike of the weak value, respectively, are determined by the visibility V =∕2(αα+ 1) ϕ ϕ beta barium borate (BBO) crystal. Te photon pairs are then collected into and the phase shift (that is, BÂ̂ and arg BÂ̂ ). Note that, from ΦG ∣ w∣=α w = ΦG single-mode fbres and delivered to the experimental set-up shown in Fig. 2c. the visibility V, there exist two mathematical solutions for α, the modulus of the Interference flters of 3 nm bandwidth centred at 780 nm are used for the weak value. One of the solutions is less than 1 and the other solution is larger high-visibility two-photon interference. than 1. Additional information is necessary to determine the correct value of α. Note that, from equation (4), if N(Δ, θm = 0) < N(Δ, θm = π/4), then α < 1, and if Implementation of ̂̂(,ÂB). To implement the measurement interaction N(Δ, θm = 0) > N(Δ, θm = π/4), then α > 1. This relation is sufficient to determine M̂(,ÂB)̂ between system and meter qubits, we introduce an ancilla qubit. the correct value of α. Note that N(Δ, θm = 0) and N(Δ, θm = π/4) are independent The system–ancilla–meter joint quantum state is initially prepared in of Δ. Alternatively, one can use the following formulae to determine the real and |ψ〉s|0〉a|1〉m. In experiment, we exploit the path mode to encode the ancilla imaginary values of the weak value: qubit as depicted in Fig. 2c. We consider successive interactions acting on ϕ ̂ ̂ ̂ ̂ Re BÂ̂ [(NN0, 8) (4,8)] 2(N ,4) the system⊗ancilla⊗meter qubits: ÛAx=⊗ÂÂσ̂⊗II+ ()−⊗II⊗ and w =π∕− π∕ π∕ ∕πΔ ∕ ÛBx= BB̂⊗∣11∣⊗σ̂+()II−̂ ̂ ⊗∣11∣⊗ ̂+ IÎ⊗∣00∣⊗ .̂ The ancilla qubit + + also needs to be projected onto Π̂ a =∣++∣ so that Π̂ = IÎ⊗Π̂ a ⊗ .̂ ϕ Then, M̂(,ÂB)2̂ Tr [ ̂ÛÛ] (refs.35,36), where Tr [·] is a partial trace Im BÂ̂ [(NN38,8)(8, 8)]2N(, 4) =Πa BA a w =π∕π∕− π∕ π∕ ∕πΔ ∕ over the ancilla qubit. Note that ÛA is a CNOT-type gate between the system and the ancilla qubits, which is realized with two HWPs (θA) and a PBD. ÛB is a Toffoli-type gate operation, which is realized via a two-photon Data availability quantum-interference-based CNOT gate with PPBSs; Π̂ is a simple projector The data that support the plots within this paper and other findings of this study for the ancilla qubit. are available from the corresponding authors upon reasonable request.

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