Experimental realization of sequential weak measurements of non-commuting Pauli observables

Jiang-Shan Chen; Meng-Jun Hu; Xiao-Min Hu; Bi-Heng Liu; Yun-Feng Huang; Chuan-Feng Li; Can-Guang Guo; Yong-Sheng Zhang

doi:10.1364/OE.27.006089

1. Introduction

Limited by the Heisenberg uncertainty principle [1], conventional quantum measurement theory forbids joint information extraction for non-commuting observables because such measurement inevitably collapses the system into one of the eigenstates of the observable. This awkward situation can be partially relaxed, however, by weak measurements in which less information is obtained but with smaller disturbance to the system. Weak measurement, which was proposed by Aharonov, Albert and Vaidman in their 1988 original paper [2], has attracted wide interest in recent years. When the interaction between the quantum system and probe is sufficiently weak and a proper post-selected state is given to the quantum system after interaction, Aharonov, Albert and Vaidman showed that the measured observable takes a strange value, which they called the weak value (WV). Specifically, the WV of an observable $\hat{A}$ is defined as ${〈 \hat{A} 〉}_{w} = 〈 ψ_{f} | \hat{A} | ψ_{i} 〉 / 〈 ψ_{f} | ψ_{i} 〉$ with |ψ_i〉 and |ψ_f〉 respectively representing the pre-selected and post-selected states of the quantum system. The WV has a strange nature even from the point view of standard quantum theory. It is in general a complex value and can be beyond the eigenvalue spectrum of an observable. Despite long-standing controversy over the physical meaning of the WV [3–9], the strange nature of the WV itself shows that it has powerful applications in solving quantum paradoxes [10–14], quantum state reconstruction [15–17] and signal amplification [18–20].

The fact that one hardly disturbs the system when making weak measurements implies that non-commuting observables can be measured in succession and joint properties can be extracted via so-called sequential weak values (SWVs) [13, 21]. The measurement of SWVs experimentally is thus of great interest from both fundamental and applicative points of view. Most experiments reported to date have made only single-observable weak measurements or joint weak measurements on commuting observables [22–26] and the first experimental measurement of the SWVs of two incompatible observables in a photonic system was reported only recently [21,27], in which case the use of the spatial distribution of photons as the pointer restricts the ability to make further sequential measurements. It is still difficult to measure SWVs, either the real part or imaginary part, of non-commuting observables.

In this article, we propose the realization of the measurement of SWVs of arbitrary non-commuting observables and experimentally demonstrate the first measurement of SWVs of three non-commuting Pauli observables in a photonic system with genuine single photons. Central to our proposal is the realization of sequential weak measurements (SWMs) of non-commuting Pauli observables.

2. SWMs and SWVs

We begin with a brief review of SWMs and SWVs. In a typical weak measurement, the system with a pre-selected state is first weakly coupled to a pointer and then post-selected into a specific state. The real part and imaginary part of the WV of a measured observable can be obtained when we measure conjugate observables of a pointer, e.g., position and momentum. In most theoretical discussions and experiments, pointers are chosen to have a Gaussian continuous distribution with a von Neumann-type interaction Hamiltonian $\hat{H} = g \hat{A} \otimes \hat{P}$ [2,17,21]. We here adopt a discrete pointer (e.g., the polarization degree of freedom (DOF) of photons) and the interaction Hamiltonian has the form $\hat{H} = γ \hat{A} \otimes {\hat{σ}}_{y}$ , in which different eigenstates of observable $\hat{A}$ result in different rotations of the pointer [15,28]. The number of SWMs is limited in the case of a continuous pointer. This is because there are three independent spatial DOFs of photons. Besides the spatial distribution along the direction of propagation, only two independent spatial distributions can be used as sequential pointers and the number of sequential measurements is thus limited. Our proposal adopts the discrete pointer, which is commonly used [15,28–32] and appropriate for the purpose here. In our realization of a weak measurement, the polarization information of photons is first encoded into the path DOF and the polarization DOF is then used as the pointer. After the weak interaction and measurement of the pointer, the information of the system encoded in the path DOF is then transformed back into the polarization DOF. The measurement results are now encoded into the path of photons. Because the number of path DOFs is not limited, the number of sequential measurements is also not limited in our case. The state of the pointer, after the system is post-selected into state |ψ_f〉, becomes (unnormalized)

| {\tilde{ϕ}}_{p} 〉 = 〈 ψ_{f} | e^{- i γ \hat{A} \otimes {\hat{σ}}_{y}} | ψ_{i} 〉 | 0_{p} 〉,

where γ reflects the strength of the system-pointer coupling, |0_p〉 is the initial state of the pointer and the natural unit is used such that ħ ≡ 1. The pointer belongs to a qubit space spanned by the orthogonal states {|0_p〉, |1_p〉}. When the coupling is sufficiently weak (i.e., γ << 1), the pointer state |φ̃_p〉 can be approximately rewritten as

| {\tilde{φ}}_{p} 〉 \approx 〈 ψ_{f} | (1 - i γ \hat{A} \otimes {\hat{σ}}_{y}) | ψ_{i} 〉 | 0_{p} 〉 \approx 〈 ψ_{f} | ψ_{i} 〉 e^{- i γ {〈 \hat{A} 〉}_{w} \otimes {\hat{σ}}_{y}} | 0_{p} 〉,

where

{〈 \hat{A} 〉}_{w} = 〈 ψ_{f} | \hat{A} | ψ_{i} 〉 / 〈 ψ_{f} | ψ_{i} 〉

is the WV. The post-selection of the system causes

θ {〈 \hat{A} 〉}_{w}

rotation of the pointer in the weak coupling case and

{〈 \hat{A} 〉}_{w}

can be extracted by measuring conjugate observables of the pointer. The real part and imaginary part of

{〈 \hat{A} 〉}_{w}

are respectively obtained from the expectation values of pointer observables σ̂₊ and σ̂_R. To get the relationship between the WV and the expectation values of pointer observables, we use the definitions that 〈σ̂_+(R)〉_p = 〈φ̃_p|σ̂_+(R)|φ_p〉/〈φ̃_p|φ̃_p〉 and iσ̂_y = |0_p〉〈1_p| − |1_p〉〈0_p|, and we finally deduce that

{〈 {\hat{σ}}_{+} 〉}_{p} = 2 γ Re {〈 \hat{A} 〉}_{w},

{〈 {\hat{σ}}_{R} 〉}_{p} = 2 γ Im {〈 \hat{A} 〉}_{w},

where σ̂₊ ≡ |+〉〈+| − |−〉〈−|, σ̂_R ≡ |R〉〈R| − |L〉〈L| and

| \pm 〉 = (| 0 〉 \pm | 1 〉) / \sqrt{2}

,

| R 〉 = (| 0 〉 + i | 1 〉) / \sqrt{2}

,

| L 〉 = (| 0 〉 - i | 1 〉) / \sqrt{2}

.

We now consider SWMs of N arbitrary observables ${\hat{A}}_{1}, {\hat{A}}_{2}, \dots, {\hat{A}}_{N}$ , in which N ancilla detectors are used. After interactions of the system and for N pointers, the state of the composite system, assuming that the weak coupling strength θ is the same for each weak measurement without loss of generality, becomes

{| Ψ 〉}_{s p_{1} \dots p_{N}} = e^{- i γ_{N} {\hat{A}}_{N} \otimes {\hat{σ}}_{y}} \dots e^{- i γ_{1} {\hat{A}}_{1} \otimes {\hat{σ}}_{y}} | ψ_{i} 〉 | 0_{p_{1}} 〉 \dots | 0_{p_{N}} 〉 .

After SWMs, post-selecting the system on |ψ_f〉 gives the state of pointers as

{| Φ 〉}_{p_{1} \dots p_{N}} = {〈 ψ_{f} | Ψ 〉}_{{sp}_{1} \dots p_{N}} .

The SWV is here defined as

{〈 {\hat{A}}_{1} \dots {\hat{A}}_{N} 〉}_{w} = 〈 ψ_{f} | {\hat{A}}_{N} \dots {\hat{A}}_{1} | ψ_{i} 〉 / 〈 ψ_{f} | ψ_{i} 〉

, which can be obtained by measuring with N pointers and using N ancilla detectors. In the case of N = 3, by setting γ₁=γ₂=γ₃=γ, we obtain

\begin{array}{l} {〈 {\hat{σ}}_{+} {\hat{σ}}_{+} {\hat{σ}}_{+} 〉}_{p_{1} p_{2} p_{3}} & = 2 γ^{3} Re [{〈 {\hat{A}}_{1} {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w} + {〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} {〈 {\hat{A}}_{3} 〉}_{w}^{†} \\ + {〈 {\hat{A}}_{1} {\hat{A}}_{3} 〉}_{w} {〈 {\hat{A}}_{2} 〉}_{w}^{†} + {〈 {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w} {〈 {\hat{A}}_{1} 〉}_{w}^{†}] . \end{array}

Similarly, the imaginary part of the SWV is determined as

\begin{array}{l} {〈 {\hat{σ}}_{R} {\hat{σ}}_{R} {\hat{σ}}_{R} 〉}_{p_{1} p_{2} p_{3}} & = 2 γ^{3} Im [- {〈 {\hat{A}}_{1} {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w} + {〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} {〈 {\hat{A}}_{3} 〉}_{w}^{†} \\ + {〈 {\hat{A}}_{1} {\hat{A}}_{3} 〉}_{w} {〈 {\hat{A}}_{2} 〉}_{w}^{†} + {〈 {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w} {〈 {\hat{A}}_{1} 〉}_{w}^{†}] . \end{array}

In inferring

{〈 {\hat{A}}_{1} {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w}

experimentally, we must perform separate trials to measure the SWV

{〈 {\hat{A}}_{i} {\hat{A}}_{j} 〉}_{w} (i < j)

and WV

{〈 {\hat{A}}_{k} 〉}_{w}

. This implies that more resources are needed to infer multiple-observable SWVs. It is noted that the real part and imaginary part of SWVs depend on the number of σ̂_R in the measured expectation value [13]. The real part of the SWV is obtained if there is an even number of σ̂_R, and the imaginary part of the SWV is obtained otherwise. For example, the measurement of expectation values 〈σ̂_Rσ̂_Rσ̂₊〉_p₁p₂p₃ and 〈σ̂₊σ̂_Rσ̂₊〉_p₁p₂p₃ determines the real part and imaginary part of SWV respectively.

We should distinguish the SWVs and joint weak values (JWVs) discussed in [33] and related works [11,12,32,34,35]. In order to obtain JWVs of N observables ${\hat{A}}_{1}, {\hat{A}}_{2}, \dots, {\hat{A}}_{N}$ , N different couplings to N distinct pointer observables ${\hat{σ}}_{y}^{i}$ simultaneously are required before the post-selection of the state and the interaction Hamiltonian takes form as $\hat{H} = \sum_{i} γ_{i} {\hat{A}}_{i} \otimes {\hat{σ}}_{y}^{i}$ . In the case of SWMs, however, N observables of one system is measured sequentially by using N pointers, e.g., the first interaction is described by unitary evolution ${\hat{U}}_{1} = e^{- i γ_{1} {\hat{A}}_{1} \otimes σ_{y}}$ , the second interaction is described by ${\hat{U}}_{2} = e^{- i γ_{2} {\hat{A}}_{2} \otimes σ_{y}}$ , and the Nth interaction is described by ${\hat{U}}_{N} = e^{- i γ_{N} {\hat{A}}_{N} \otimes σ_{y}}$ . The SWVs and the JWVs, though may have similar formula to obtain them, they are defined in different measurement framework.

In the case that the measured observable is of Pauli type (i.e., $\hat{A} \equiv {\hat{σ}}_{A} = \vec{σ} \cdot {\vec{n}}_{A}$ ), we can obtain the Pauli observable WV 〈σ̂_A〉_w without approximation because e^{−iγσ̂_A⊗σ̂_y} = cosγ −isinγ(σ̂_A⊗σ̂_y). By analogy to the deduction of Eq. (3), we obtain the exact expressions

{〈 {\hat{σ}}_{+} 〉}_{p} = \frac{sin (2 γ) Re ({〈 {\hat{σ}}_{A} 〉}_{w})}{{cos}^{2} (γ) + {sin}^{2} (γ) {| {〈 {\hat{σ}}_{A} 〉}_{w} |}^{2}},

{〈 {\hat{σ}}_{R} 〉}_{p} = \frac{sin (2 γ) Im ({〈 {\hat{σ}}_{A} 〉}_{w})}{{cos}^{2} (γ) + {sin}^{2} (γ) {| {〈 {\hat{σ}}_{A} 〉}_{w} |}^{2}},

which reduce to Eqs. (3) and (4) naturally in the first-order approximation. These expressions are exact for any strength of measurement coupling, while Eqs. (3) and (4) are shown in a good approximation only for very weak coupling. Exact expressions that determine the multiple Pauli observable SWVs are also obtainable. For the case that N = 2,

{〈 {\hat{σ}}_{+} {\hat{σ}}_{+} 〉}_{p_{1} p_{2}} = \frac{\frac{1}{2} Re ({〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} + {〈 {\hat{A}}_{2} 〉}_{w} {〈 {\hat{A}}_{1} 〉}_{w}^{*}) {sin}^{2} 2 γ)}{{cos}^{4} γ + ({| {〈 {\hat{A}}_{1} 〉}_{w} |}^{2} + {| {〈 {\hat{A}}_{2} 〉}_{w} |}^{2}) {cos}^{2} γ {sin}^{2} γ + {| {〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} |}^{2} {sin}^{4} γ}

{〈 {\hat{σ}}_{R} {\hat{σ}}_{+} 〉}_{p_{1} p_{2}} = \frac{\frac{1}{2} Im ({〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} - {〈 {\hat{A}}_{1} 〉}_{w} {〈 {\hat{A}}_{2} 〉}_{w}^{*}) {sin}^{2} 2 γ)}{{cos}^{4} γ + ({| {〈 {\hat{A}}_{1} 〉}_{w} |}^{2} + {| {〈 {\hat{A}}_{2} 〉}_{w} |}^{2}) {cos}^{2} γ {sin}^{2} γ + {| {〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} |}^{2} {sin}^{4} γ} .

For the case that N = 3,

\begin{array}{l} {〈 {\hat{σ}}_{+} {\hat{σ}}_{+} {\hat{σ}}_{+} 〉}_{p_{1} p_{2} p_{3}} = \\ \frac{2 Re ({〈 {\hat{A}}_{1} {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w} + {〈 {\hat{A}}_{3} 〉}_{w}^{*} {〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} + {〈 {\hat{A}}_{2} 〉}_{w}^{*} {〈 {\hat{A}}_{1} {\hat{A}}_{3} 〉}_{w} + {〈 {\hat{A}}_{1} 〉}_{w}^{*} {〈 {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w}) {tan}^{3} γ}{K} \end{array}

\begin{array}{l} {〈 {\hat{σ}}_{R} {\hat{σ}}_{R} {\hat{σ}}_{R} 〉}_{p_{1} p_{2} p_{3}} = \\ \frac{2 Im (- {〈 {\hat{A}}_{1} {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w} + {〈 {\hat{A}}_{3} 〉}_{w}^{*} {〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} + {〈 {\hat{A}}_{2} 〉}_{w}^{*} {〈 {\hat{A}}_{1} {\hat{A}}_{3} 〉}_{w} + {〈 {\hat{A}}_{1} 〉}_{w}^{*} {〈 {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w}) {tan}^{3} γ}{K}, \end{array}

where

\begin{array}{l} K = & 1 + ({| {〈 {\hat{A}}_{1} 〉}_{w} |}^{2} + {| {〈 {\hat{A}}_{2} 〉}_{w} |}^{2} + {| {〈 {\hat{A}}_{3} 〉}_{w} |}^{2}) {tan}^{2} γ + ({| {〈 {\hat{A}}_{1} {\hat{A}}_{2} 〉}_{w} |}^{2} \\ + {| {〈 {\hat{A}}_{1} {\hat{A}}_{3} 〉}_{w} |}^{2} + {| {〈 {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w} |}^{2}) {tan}^{4} γ + {| {〈 {\hat{A}}_{1} {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w} |}^{2} {tan}^{6} γ . \end{array}

The strength of measurement γ is assumed the same for each measurement. Corresponding to our experiment, we measure the “one-operator” weak values, the “two-operator” weak values and “three-operator” weak values at different experimental runs.

3. Experimental realization of sequential weak measurements of photons

Figure 1 shows the experimental setup for making the SWMs of polarization observables of single photons. The polarization observable of photons is a Pauli-type observable, and the WV and SWV are thus determined using Eqs. (9)–(15). The setup consists of five parts, i.e., the heralded single-photons source, initial state preparation, SWMs, post-selection and detection.

Fig. 1 Experimental setup for realizing sequential weak measurements of three non-commuting polarization observables of photons. The single photons are produced by generating a pair of photons via spontaneous parametric down conversion (SPDC) with idler photons used as triggers. The signal photons, after initial state preparation, are sent into weak measurement modules a, b, and c sequentially. After sequential weak measurements, post-selection and coincidence counting are performed. Modules a, b and c respectively realize weak measurements of polarization observables σ̂_y, σ̂_z and σ̂_φ. Q1 is rotated at 0, Q3 is rotated at π/2, H1, H3 and H10 are rotated at π/8, H4, H8, H9, H11, H15, H16, H18, H22 and H23 are rotated at π/4, H17 and H24 are rotated at φ/2 (in our experiment, we take φ = π/3). H16, H12 and H20 are rotated at γ/2, while H15, H13 and H19 are rotated at −γ/2 (in our experiment, we measure SWV with coupling parameter γ = 25° and γ = 30°, respectively.) Q2, H7, Q4, H14, Q5, and H21 combining PBSs after them in the corresponding modules are used to make projective measurements on the corresponding pointers, and these wave plates are rotated at some angles according to which observables of these pointers would be measured.

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The single-photons source is realized by heralding the coincidence of a pair of photons. The pair of photons centred at 808 nm is generated via SPDC by pumping a 2-mm-thick type-I beta barium borate (BBO) crystal with a 808-nm mode-lock Ti:sapphire laser (repetition rate : 76 MHz) that is frequency-doubled to 404 nm. The coincidence counting rate is about 270 per second, and we count for 30 seconds to record the coincidences in our measurement. After coupling to a single-mode fibre (SMF), the idler photons are directly sent to a silicon single-photon avalanche detector (SPD) while the signal photons are connected to a launcher and then emitted along the free-space path. The signal photons are prepared into the initial state $| ψ_{i} 〉 = (| H 〉 + | V 〉) / \sqrt{2}$ using a polarizing beam splitter (PBS) and a half-wave plate (HWP) rotated at π/8 after the PBS, where |H〉 and |V〉 respectively represent the horizontal and vertical polarization states. The SWMs of polarization observables of the signal photons are made via the weak measurement modules a, b and c as shown in Fig. 1. After passing through module c, the signal photons are projected into the final state |ψ_f〉 = cosθ|H〉 + sinθ|V〉 via an HWP rotated at θ/2 and a PBS. The signal photons, after the PBS, are then coupled to an SMF and sent to an SPD for coincidence detection.

The important component of our setup is the weak measurement module that makes the weak measurement of the arbitrary polarization observable. We here take module c, which makes the weak measurement of observable σ̂_φ ≡ |φ〉〈φ| − |φ^⊥〉〈φ^⊥| with |φ〉 = cosφ|H〉 + sinφ|V〉 and |φ^⊥〉 = sinφ|H〉 −cosφ|V〉, as an example to explain how the process works. The basic idea is that we first transform the measurement basis {|φ〉, |φ^⊥〉} into {|H〉, |V〉} via H17 and the following beam displacer (BD), then make the weak measurement of observable σ̂_z ≡ |H〉〈H | − |V〉〈V| using optical elements between H18 and H23 [36]. We complete the weak measurement of σ̂_φ by transforming the system back to the measurement basis {|φ〉, |φ^⊥〉} via H24. Both H17 and H24 are rotated at φ/2. Suppose that photons prepared in the polarization state α|φ〉 + β|φ^⊥〉 are sent to module c. After photons pass through H17, their state transforms to α|H〉 + β|V〉. To realize weak interaction e^{−iγσ̂_z⊗σ̂_y}, we encode the information of the system (i.e., polarization of photons) into the DOF of the optical path and use the DOF of polarization as the pointer. This is done using a BD, and three HWPs (i.e., H18, H19 and H20). The rotation angles of H18, H19 and H20 are respectively π/4, −γ/2 and γ/2 radians. The parameter γ, which reflects the strength of the measurement, can be continuously adjusted in our case. The composite state of photons, after weak interaction, becomes α|0〉 ⊗ (cosγ|H〉 − sinγ|V〉) + β|1〉 ⊗ (cosγ|H〉 + sinγ|V〉), where the path states |0〉, |1〉 respectively represent photons flying along the down arm and up arm. We then make projective measurements on the pointer via Q5, H21 and a PBS. Here we only use the transmission arm of PBS, but we can also make standard measurement of observables of the pointer, i.e., there is no post-selection involved in the measurement of the pointer. This is done through setting two rotations of Q5 and H21 according to measurement basis of the pointer. We measure the pointer observables σ̂₊ and σ̂_R to get WVs and SWVs, and the basis for these projective measurements on the pointer are respectively {|+〉, |−〉} and {|R〉, |L〉}. The information of the system that is encoded in the path basis {|0〉, |1〉} needs to be recoded back into the polarization basis for subsequent measurement, which is done by recombining the light of the two arms via H22 and a BD, in which H22 is rotated at π/4 and placed in down arm. H23 rotated at π/4 after the BD is used to exchange the polarization states |H〉 and |V〉. The system is transformed back to the measurement basis {|φ〉, |φ^⊥〉} via H24 and this finishes the weak measurement of observable σ̂_φ.

The measurement of the pointer for each weak measurement, as described above, is made via a polarization analyser placed between BDs in the module. The outcome of the measurement is encoded in the path of outgoing photons. By making a projective measurement of the basis state of an observable of the pointer separately, the expectation value of the observable is calculated by combining outcomes of projective measurements. The expectation values of multiple observables, which are required to obtain SWVs, can be obtained similarly. For example, the expectation value 〈σ̂₊σ̂₊σ̂₊〉_p₁p₂p₃, which is needed to evaluate ${〈 {\hat{A}}_{1} {\hat{A}}_{2} {\hat{A}}_{3} 〉}_{w}$ , can be measured as follows. Notice that the measurement basis of each pointer is {|+〉, |−〉}, and σ̂₊σ̂₊σ̂₊ has eight eigenstates (projective measurement basis of observable σ̂₊σ̂₊σ̂₊) but two eigenvalues ±1 for degeneracy. We rotate Q2, H7, Q4, H14, Q5 and H21 at some angles according to projective measurement base and record the coincidences. We perform projective measurements on σ̂₊σ̂₊σ̂₊ and collect the coincidences N_±1 corresponding to eigenvalues ±1. Finally, we calculate the value (N₊₁ − N₋₁)/(N₊₁ + N₋₁), and that value is just 〈σ̂₊σ̂₊σ̂₊〉_p₁p₂p₃.

4. Results

In our experiment, modules a, b and c respectively make the weak measurements of polarization observables σ̂_y, σ̂_z and σ̂_φ. Here σ̂_y ≡ |R〉〈R| − |L〉〈L|, σ̂_z ≡ |H〉〈H| − |V〉〈V|. We make the measurement in the case of σ̂_φ=π/3 such that the SWVs of three non-commuting observables, (i.e., 〈σ̂_yσ̂_zσ̂_φ=π/3〉_w) can be obtained as shown in Fig. 2, where boxes represent experimental data and dashed lines represent theoretical predictions. The measurement strength of all three modules, which is determined by parameter γ, is taken to be the same value in our experiment. Two different cases with γ = 25° and γ = 30° are considered in our experiment as shown in Fig. 2(a) in comparison with Fig. 2(b), to verify that SWVs are independent of the measurement strength when Pauli-type observables are measured. Each weak measurement module is, of course, allowed to have a different measurement strength but this will not affect the measured SWVs. Because γ can be continuously tuned via rotating an HWP, the modules with γ = 0 perform no measurement at all, allowing us to make a weak measurement of one Pauli observable and SWMs of two non-commuting Pauli observables. Figure 2 shows the measured SWVs of two and three non-commuting Pauli observables in the case of different post-selected states |ψ_f〉 = cosθ|H〉 + sinθ|V〉. The WV of the Pauli observable and SWVs of non-commuting Pauli observables σ̂_x, σ̂_y, σ̂_z are also measured. The error bars in Fig. 2 are evaluated by Poissonian counting statistics. Considering the statistical errors and possible imperfections of optical elements, our results fit well with theoretical predictions. The results of three sequential measurements are worse than the results of two sequential measurements because system errors accumulate when more measurement modules are added.

Fig. 2 (a) and (b) show the results of our experiment to measure SWV with coupling parameter γ = 25° and γ = 30°, respectively. The θ corresponds to the parameter of the post-selected state |ψ_f〉 = cosθ|H〉 + sinθ|V〉. Red and blue parts respectively represent the real and imaginary parts of SWVs that we measured. The boxes represent experimental data and dashed lines represent theoretical predictions rather than fitted curves. Error bars are evaluated on the basis of Poissonian counting statistical.

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5. Discussion and conclusion

Although we only demonstrated the SWMs of three non-commuting observables, our proposal allows the measurement of non-commuting Pauli observables using a suitable number of weak measurement modules. The controllable measurement strength and modular design of a weak measurement make our proposal suitable for various weak measurement tasks with a photonic system.

As two examples, our scheme may find direct applications in testing Leggett-Garg inequalities [26,37,38] and realizing unbounded randomness certification [39,40]. In both cases, multiple measurements made sequentially on one single system are required, which can be realized by applying multiple strength-controllable weak-measurement modules shown here without post-selection in the last stage. Additionally, our weak measurement module can be directly used in areas such as counterfactual computation [13,41,42], the direct measurement of matrix density [16,17], and direct process tomography [27,30,43–49].

In conclusion, we proposed how to realize SWMs of non-commuting Pauli observables and experimentally demonstrated for the first time the measurement of SWVs of three non-commuting observables with heralded single photons, which is impossible with previously reported methods. The results presented here may not only improve our understanding of the mysterious quantum world but also find important applications in quantum information processing.

Funding

National Key Research and Development Program of China (2017YFA0304100, 2016YFA0301300, 2016YFA0301700); National Natural Science Foundation of China (NSFC) (11774335, 11874345, 11504253, 11674306, 11821404, 61590932); Key Research Program of Frontier Sciences (CAS) (QYZDY-SSW-SLH003); Fundamental Research Funds for the Central Universities; Anhui Initiative in Quantum Information Technologies (AHY020100, AHY060300).

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Experimental realization of sequential weak measurements of non-commuting Pauli observables

Abstract

1. Introduction

2. SWMs and SWVs

3. Experimental realization of sequential weak measurements of photons

4. Results

5. Discussion and conclusion

Funding

References

Cited By

Figures (2)

Equations (15)

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