Experimental proposal for performing nonlocal measurement of a product observable

Zhao-Qin Wu; Hui Cao; Hao-Liang Zhang; Shan-Jun Ma; Jie-Hui Huang

doi:10.1364/OE.24.027331

1. Introduction

The causality principle, which states that two separate observers cannot communicate with each other with superluminal velocity, imposes some restrictions on the process of quantum measurement [1]. These restrictions make it hard to perform a nonlocal measurement on spacelike separated systems instantaneously. It was only in 1980 that Aharonov and Albert proved that certain nonlocal observables can be measured at a well-defined time without breaking relativistic causality [2,3], and the explicit methods for performing such instantaneous nonlocal measurements are studied several years later [4]. Popescu and Vaidman presented a more general analysis on the consequences of relativistic causality on nonlocal quantum measurement and concluded that instantaneous nonlocal measurements invariably disturb the measured system so that all local information about the quantum state is erased [5]. In the following, the measurability of nonlocal observables with product-state eigenstates is investigated by assuming classical communication and local interactions [6, 7]. If the measuring devices are allowed to contain quantum entanglement, all nonlocal observables can in principle be measured instantaneously, and several schemes have been proposed based on the entanglement consumption [8–10].

A tensor product operator with the form Ω = ℳ_A ⊗ ℳ_B is a simple example of nonlocal observable, where ℳ_A and ℳ_B are Hermitian operators on the two spacelike separated systems A and B, respectively. Product observables play an important role in quantum theory. For example, they are used as CHSH observables to test local hidden-variable theories [11]. Although the mathematical structure of a product operator is composed of two local operators on two subsystems, it does not mean we can implement it by simply performing two local measurements individually, especially when the two subsystems are spacelike separated. The combined action of two local measurements disturbs the whole system more than the ideal nonlocal measurement [12].

Different from many previous schemes to perform nonlocal measurements in a destructive way, an interesting scheme for creating the von Neumann measurement Hamiltonian for a large class of nonlocal observables is proposed by Brodutch and Cohen recently [12]. In their scheme, an ancillary meter initially prepared in an entangled state is required to couple the system and the target meter, where a local projective measurement has to be made, followed by a process of erasing the measuring result via quantum eraser [13–17]. As an example, here we design an experiment in an optical system to perform the nonlocal measurement of $σ_{z}^{A} \otimes σ_{z}^{B}$ , which is a simple and typical example of product observable and has many applications in quantum optics, e.g. describing the interaction between two spins [18]. It is shown that this product observable with two degenerate eigenvalues could be implemented in a deterministic way without violating relativistic causality. We believe the approach proposed in this paper can be generalized to realize nonlocal measurements in higher dimensional or even multipartite systems.

2. Experimental setup

According to the quantum measurement theory, if an observable Ω with discrete eigenvalues is made on a quantum state represented by the density matrix ρ, its wave function collapses to P̂_iρP̂_i after the measurement, with P̂_i the projection operator onto the eigenspace of the ith eigenvalue of Ω [19–21]. However, such a measurement is not easy to implement if the quantum state is shared by two or more spacelike separated parties. In order to perform a nonlocal measurement among spacelike separated parties, a possible way, just as proposed in the recent work [12], is to couple the system under consideration to a meter, where an entangled ancillary system is usually required to encode the information of the initial state to the system-meter coupling system, and the final projective measurement on the meter induce the collapse of the wave function of the system on demand. Following this idea, here an experiment in the simplest 2 × 2 composite system is designed as following to perform a nonlocal measurement of the product observable $σ_{z}^{A} \otimes σ_{z}^{B}$ .

Two observers Alice (A) and Bob (B), who are separated very far compared with the measuring time spent in the experiment, initially share an arbitrary 2 ⊗ 2 state,

{| ψ_{0} 〉}_{A B} = α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉 + α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉,

with

\sum_{i = 1}^{4} {| α_{i} |}^{2} = 1

. Here the basis states |H_A(B)〉 and |V_A(B)〉 are associated with two paths of the interferometer on the left (A) and right (B) side, respectively, illustrated as the red solid lines in the lower part of Fig. 1. Such a 2 ⊗ 2 quantum state can be generated in experiment through type-II parametric down-conversion (PDC) [22–28], and the two correlated photons, called photon A and B hereafter, which contain controllable polarization entanglement [29], are sent to the two interferometers. Before the two photons A and B enter their own interferometer, a local operation through a half wave plate (HWP) or a controllable electro-optic phase modulator can be made to rotate the polarization of the photon A or B to an arbitrary direction we want [30, 31]. The polarization beam splitter (PBS) at the entrance of each interferometer then distributes the input photons into two paths based on their polarization, the path “H” for the photons with H polarization, and the path “V” for the photons with V polarization. The HWP near the output port of each interferometer, with its optical-axis direction oriented at 22.5°, is used to turn the “H” polarization of the photon to “V” polarization, so that the photon in the two paths can interfere at the output port.

Fig. 1 A schematic setup for performing the nonlocal measurement $σ_{z}^{A} \otimes σ_{z}^{B}$ . Two photons A and B generated from PDC are sent to two interferometers hold by Alice and Bob, who are separated very far. The nonlocal measurement $σ_{z}^{A} \otimes σ_{z}^{B}$ on the two photons can be turned to local measurement on the photon in the meter ℳ, where an ancillary meter 𝒩 initially prepared in a maximally entangled state and three identical Kerr media (shade rectangles) are used to generate the quantum correlation between the AB two-photon system and the meter ℳ. The polarization of the photon can be adjusted by the half wave plate (grey oval).

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Once the quantum state in Eq. (1) is prepared, our goal in current work is to couple this two-photon system with a two-dimensional meter ℳ through an interaction described by the following transformation,

U = {\hat{P}}_{A B}^{(+ 1)} {| + 〉}_{ℳ} 〈 + | - {\hat{P}}_{A B}^{(- 1)} {| - 〉}_{ℳ} 〈 - 1 |,

where

{\hat{P}}_{A B}^{(\pm 1)}

is projection operator onto the eigenspace of the eigenvalue ±1 of the product observable

σ_{z}^{A} \otimes σ_{z}^{B}

in the Hilbert space of the two photons A and B, and |±〉_ℳ are the eigenvetors of the local measurement on the meter ℳ. In this experimental scheme, the meter ℳ is composed of a two-path interferometer and a single photon ℳ, emitted from a single-photon source, illustrated as the black dotted lines in the right middle of Fig. 1. Please note that the meter ℳ is located at Bob’s side, and Bob can make any operation on it locally. Since the two beam splitters (BS) at the input and output ports of the interferometer are assumed to be 50 : 50 BS, the initial state of the meter ℳ can be written as,

{| ψ_{0} 〉}_{ℳ} = \frac{1}{\sqrt{2}} ({| 0 〉}_{ℳ} + {| 1 〉}_{ℳ}),

where |0〉_ℳ and |1〉_ℳ represent the basis states of the single photon ℳ in the path “0” and “1” of the interferometer, respectively.

If we apply the transformation (2) onto the composite system composed of the two photons and the meter ℳ, by directly choosing |0〉_ℳ and |1〉_ℳ as the measuring basis of the meter ℳ, we achieve the target state,

| ψ_{f} 〉 = U ({| ψ_{0} 〉}_{A B} \otimes {| ψ_{0} 〉}_{ℳ}) = \frac{1}{\sqrt{2}} (c_{+} {| ψ 〉}_{A B}^{(+ 1)} {| + 〉}_{ℳ} - c_{-} {| ψ 〉}_{A B}^{(- 1)} {| - 〉}_{ℳ}),

where

{| ψ 〉}_{A B}^{(\pm 1)}

represents the state vector of the two photons in the eigenspace corresponding to the eigenvalue ±1 of the product observable

σ_{z}^{A} \otimes σ_{z}^{B}

, and the coefficient c_± = 〈ψ^(±1)|ψ₀〉_AB determines the success probability of obtaining the state

{| ψ 〉}_{A B}^{(\pm 1)}

. The state in Eq. (4) is an entangled state between the two-photon system AB and the meter system ℳ, which means we can turn the measurement on the two photons to the measurement on the meter ℳ, and the eigenvector of the observable

σ_{z}^{A} \otimes σ_{z}^{B}

on the two photons is achieved after the collapse of the wave function induced by the measurement on the meter ℳ, and the corresponding eigenvalue can be inferred through the final state of the meter ℳ, |−〉_ℳ for eigenvalue −1, and |+〉_ℳ for eigenvalue 1.

However, the transformation described in Eq. (2) can not be directly realized in current situation, because we have assumed that the two observers Alice and Bob are separated very far compared with the total interacting time. That is why we need to employ an ancilla, called meter 𝒩 in [12], to assist the generation of the coupling between the two photons AB and the meter ℳ. Similar to the AB two-photon system, the ancilla here is also composed of two single photons, called 𝒩_A and 𝒩_B in the following, which are generated through type-II parametric down-conversion, and enter the two two-path interferometers located at Alice and Bob’s site, respectively. Please see the blue dashed lines for this ancillary meter 𝒩 in the upper part of Fig. 1. We assume the ancillary meter 𝒩 is initially prepared in a maximally entangled state,

{| ψ_{0} 〉}_{𝒩} = \frac{1}{2} [{| 1 〉}_{𝒩_{A}} {(| 1 〉 + | 0 〉)}_{𝒩_{B}} + {| 0 〉}_{𝒩_{A}} {(| 1 〉 - | 0 〉)}_{𝒩_{B}}],

where |1〉_S (|0〉_S) represents the state with one photon in the path “1” (“0”) of the two top interferometer in Fig. 1, with S = 𝒩_A, 𝒩_B. In order to achieve the above state, a HWP is placed before the PBS at the entrance of the two interferometers to adjust the polarization of the photon 𝒩_A and 𝒩_B.

In the following, three identical Kerr media should be prepared. As is well known, variant Kerr media, such as a three-level atomic system through four-wave mixing [32], a microstructured optical fiber [33], a four-level atomic system [34], or even a microcavity at room temperature [35], could be used to couple two optical modes and generate nonlinear effects, which has many potential applications, such as quantum nondemolition measurement of photon number [36–39], and generation of the controlled-PHASE gate [40]. Although the actual Kerr interaction is usually implemented in experiment at high light intensities, which is favor of the observation of large Kerr nonlinearity, much progress has been made on theoretical and experimental investigation on large Kerr nonlinearity of weak fields [41]. Here we use Kerr media to couple the AB two-photon system, the ancillary meter 𝒩 and the target meter ℳ. It is assumed that once two single photons meet in the Kerr medium, a cross-Kerr interaction described by the Hamiltonian [42],

H = g a_{1}^{†} a_{1} a_{2}^{†} a_{2},

takes place between the two photons, where

a_{k}^{†}

(a_k) (k = 1, 2) are the creation (annihilation) operators of the two photons, and g is the coupling strength. An experimental scheme for realizing this kind of cross-Kerr interaction between two weak fields is proposed in a four-level atomic system [43,44]. In fact, each optical mode includes at most one excitation, and no photon is absorbed or created during the interaction in the present scheme, so the Kerr nonlinearity is not the key point we care about. The only effect caused by the interaction is a phase change to the interacting terms. More details are shown in the next section.

With the AB two-photon system prepared in the state (1), the meter ℳ in the state (3), the ancillary meter 𝒩 in the state (5) initially, and three Kerr media described above at hand, we need to follow four sequential steps to accomplish the experiment.

Alice uses the Kerr medium “0” to couple the photon A in path “H” and the photon 𝒩_A in path “0” locally, and a quantum correlation between the AB two-photon state and the ancillary meter 𝒩_A is then established after the interaction. The shade rectangles in Fig. 1 stand for the Kerr media we used in experiment.
A projective measurement on the photon 𝒩_A, i.e. the A part of the ancillary meter 𝒩, is made by Alice, which destroys the correlation between the photon 𝒩_A and the rest photons, no matter what result is achieved. After this projective measurement, the photon 𝒩_A can be traced out in the theoretical analysis of the experiment, but the AB two-photon system is still entangled with the photon 𝒩_B.
Bob uses the Kerr medium “1” to couple the photon 𝒩_B in path “1” and the photon ℳ in path “1”, and the Kerr medium “2” to couple the photon B in path “V” and the photon ℳ in path “0” simultaneously. Through the interaction induced by the two Kerr media “1” and “2”, the meter ℳ is entangled to AB two-photon system and the photon 𝒩_B.
Bob makes a projective measurement on the photon 𝒩_B to erase the previous result of the projective measurement on the photon 𝒩_A made by Alice, which at the same time destroys the correlation between the photon 𝒩_B and the rest photons. Once the entangled state between the AB two-photon system and the meter ℳ, described in Eq. (4), is derived after tracing out the photon 𝒩_B, the nonlocal measurement of the product observable $σ_{z}^{A} \otimes σ_{z}^{B}$ on the AB two-photon system turns to the local measurement on the meter ℳ.

The above four steps should be performed sequently in the same reference frame. Since the measuring result of each step does not affect the operation in the next step (but does affect its measuring result), we can start the operation in the next step once the operation in the previous one is finished, and do not need to wait for the result of the previous step. In other words, we can in principle implement the four sequential operations in a very short time, compared with the distance between Alice and Bob.

3. Evolution of the system and discussions

In this section, we regard the three systems, AB two-photon system, the ancillary meter 𝒩 and the target meter ℳ, as a whole, and present the details about the evolution of the whole system during the experiment, accompanied by some discussions. In the first step of the experiment, a Kerr medium “0” is used to couple the photon A in path “H” and the photon 𝒩_A in path “0” through the interaction described by the Hamiltonian in Eq. (6), which turns the initial state of the whole system |ψ₀〉 = |ψ₀〉_AB ⊗ |ψ₀〉_𝒩 ⊗ |ψ₀〉_ℳ to,

\begin{matrix} | ψ 〉 = & \frac{1}{2} [(α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉 + α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 1 〉}_{𝒩_{A}} {(| 1 〉 + | 0 〉)}_{𝒩_{B}} \\ + e^{i ϕ} (α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉) {| 0 〉}_{𝒩_{A}} {(| 1 〉 - | 0 〉)}_{𝒩_{B}} \\ + (α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 0 〉}_{𝒩_{A}} {(| 1 〉 - | 0 〉)}_{𝒩_{B}}] \otimes {| ψ_{0} 〉}_{ℳ}, \end{matrix}

where the phase change ϕ = gt, induced by the Kerr medium, is a constant with a fixed coupling strength g and interaction duration t. In the following, we assume the two parameters g and t are appropriately set so that the phase change is equal to π, and this assumption is also valid for the other two Kerr media. Here we see the two systems, AB two-photon system and the ancillary meter 𝒩, are entangled to each other after the interaction in the Kerr medium “0”, with entanglement of von Neumann entropy [19] E(|ψ₁〉) = −η₁ ln η₁ − η₂ ln η₂, with η₁ = |α₁|² + |α₂|² and η₂ = |α₃|² + |α₄|².

By combining the two paths “0” and “1” through a 50 : 50 BS at the output of the interferometer, the projective measurement on the photon 𝒩_A made by Alice turns the whole system in the above state (7) to,

| ψ_{2} 〉 = [(α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉) {| 1 〉}_{𝒩_{B}} + (α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 0 〉}_{𝒩_{B}}] \otimes {| ψ_{0} 〉}_{ℳ} \otimes {| - 〉}_{𝒩_{A}},

or

| ψ_{2}^{'} 〉 = [(α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉) {| 0 〉}_{𝒩_{B}} + (α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 1 〉}_{𝒩_{B}}] \otimes {| ψ_{0} 〉}_{ℳ} \otimes {| - 〉}_{𝒩_{A}},

with equal probability, where

{| \pm 〉}_{𝒩_{A}} = \frac{1}{\sqrt{2}} {(| 1 〉 \pm | 0 〉)}_{𝒩_{A}}

are eigenvectors of this projective measurement. Although the photon 𝒩_A is now separable to other photons in the system, the entanglement between the AB two-photon system and the ancillary meter 𝒩 (now represented by photon 𝒩_B alone) is preserved, with entanglement of von Neumann entropy E(|ψ₂〉) = E(|ψ′₂〉) = E(|ψ₁〉), no matter what result is obtained for the photon 𝒩_A, |−〉_{𝒩_A} or |+〉_{𝒩_A}.

In the third step, two Kerr media are used to couple the mater ℳ to the AB two-photon system and the ancillary meter 𝒩. As assumed in the previous case, a fixed phase change π is induced by the interaction in each Kerr medium, and the above quantum state turns to,

\begin{array}{l} | ψ_{3} 〉 & = & \frac{1}{\sqrt{2}} {[[(- α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉) {| 1 〉}_{𝒩_{B}} + (- α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 0 〉}_{𝒩_{B}}] {| 0 〉}_{ℳ} \\ + & [- (α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉) {| 1 〉}_{𝒩_{B}} + (- α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 0 〉}_{𝒩_{B}}] {| 1 〉}_{ℳ}} \otimes {| - 〉}_{𝒩_{A}}, \end{array}

or

\begin{array}{l} | ψ_{3}^{'} 〉 & = & \frac{1}{\sqrt{2}} {[(- α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉) {| 0 〉}_{𝒩_{B}} + (- α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 1 〉}_{𝒩_{B}}] {| 0 〉}_{ℳ} \\ + & [(α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉) {| 0 〉}_{𝒩_{B}} - (α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 1 〉}_{𝒩_{B}}] {| 1 〉}_{ℳ}} \otimes {| - 〉}_{𝒩_{A}} . \end{array}

The entanglement contained in the state of the whole system is enhanced through the interaction in the third step. If we regard the whole system as a bipartite system by choosing the photon 𝒩_B as one part and the photons A, B and ℳ as the other, the corresponding entanglement of von Neumann entropy is

E (| ψ_{3} 〉) = E (| ψ_{3}^{'} 〉) = - \sum_{i = 1}^{4} {| α_{i} |}^{2} ln {| α_{i} |}^{2}

, which is larger than the previous one in the second step. The same result is obtained if the bipartite entanglement between the photon ℳ and the composite system composed of the photons A, B and 𝒩 is considered.

The projective measurement on the photon 𝒩_B with eigenvectors ${| \pm 〉}_{𝒩_{B}} = \frac{1}{\sqrt{2}} {(| 1 〉 \pm | 0 〉)}_{𝒩_{B}}$ causes the collapse of the wave function of the whole system. Accordingly, the above states turn to,

\begin{array}{l} | ψ_{4}^{(\pm)} 〉 & = & \frac{1}{\sqrt{2}} [(- α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉 - α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 0 〉}_{ℳ} \\ \pm & (α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉 - α_{3} | V_{A} H_{B} 〉 - α_{4} | V_{A} V_{B} 〉) {| 1 〉}_{ℳ}] \otimes {| + 〉}_{𝒩_{B}} \otimes {| \pm 〉}_{𝒩_{A}}, \end{array}

for the measuring result |+〉_{𝒩_B}, and

\begin{array}{l} | ψ_{4}^{(-)} 〉 & = & \frac{1}{\sqrt{2}} [(- α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉 + α_{3} | V_{A} H_{B} 〉 - α_{4} | V_{A} V_{B} 〉) {| 0 〉}_{ℳ} \\ \pm & (α_{1} | H_{A} H_{B} 〉 + α_{2} | H_{A} V_{B} 〉 + α_{3} | V_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) {| 1 〉}_{ℳ}] \otimes {| - 〉}_{𝒩_{B}} \otimes {| \pm 〉}_{𝒩_{A}}, \end{array}

for the measuring result |−〉_{𝒩_B}. In the measuring basis of the photon ℳ,

{{| \pm 〉}_{ℳ} = \frac{1}{\sqrt{2}} {(| 1 〉 \pm | 0 〉)}_{ℳ}}

, the above states can be rewritten as,

| ψ_{4}^{(+)} 〉 = (c_{\mp} {| ψ 〉}_{A B}^{(\mp)} {| + 〉}_{ℳ} - c_{\pm} {| ψ 〉}_{A B}^{(\pm)} {| - 〉}_{ℳ}) \otimes {| + 〉}_{𝒩_{B}} \otimes {| + 〉}_{𝒩_{A}},

and

| ψ_{4}^{(-)} 〉 = (c_{\mp} {| ϕ 〉}_{A B}^{(\mp)} {| + 〉}_{ℳ} - c_{\pm} {| ϕ 〉}_{A B}^{(\pm)} {| - 〉}_{ℳ}) \otimes {| - 〉}_{𝒩_{B}} \otimes {| \pm 〉}_{𝒩_{A}} .

Here

{| ψ 〉}_{A B}^{(-)} = (α_{2} | H_{A} V_{B} 〉 - α_{3} | V_{A} H_{B} 〉) / c_{-}

and

{| ϕ 〉}_{A B}^{(-)} = (α_{2} | H_{A} V_{B} 〉 + α_{3} | V_{A} H_{B} 〉) / c_{-}

with

c_{-} = \sqrt{{| α_{2} |}^{2} + {| α_{3} |}^{2}}

are two state vectors in the two-dimensional Hilbert space of the eigenvalue −1 of the product observable

σ_{z}^{A} \otimes σ_{z}^{B}

, and the other two state vectors

{| ψ 〉}_{A B}^{(+)} = - (α_{1} | H_{A} H_{B} 〉 - α_{4} | V_{A} V_{B} 〉) / c_{+}

and

{| ϕ 〉}_{A B}^{(+)} = - (α_{1} | H_{A} H_{B} 〉 + α_{4} | V_{A} V_{B} 〉) / c_{+}

with

c_{+} = \sqrt{{| α_{1} |}^{2} + {| α_{4} |}^{2}}

are associated with the degenerate eigenvalue 1. Here we see the target state in Eq. (4) is already achieved in the above states by tracing out the meter 𝒩. That is to say, the nonlocal measurement of the product observable

σ_{z}^{A} \otimes σ_{z}^{B}

can be realized in this scheme with full success probability, no matter what eigenvector the wave function collapses to after the projective measurement on the photons 𝒩_A and 𝒩_B.

Among the four eigenvectors ${| ψ 〉}_{A B}^{(\pm)}$ and ${| ϕ 〉}_{A B}^{(\pm)}$ , the observers Alice and Bob do not know which one is finally achieved for the AB two-photon system after the nonlocal measurement $σ_{z}^{A} \otimes σ_{z}^{B}$ , unless Alice tells the result of the projective measurement on the photon 𝒩_A to Bob through a classical channel, where a time depending on the distance between Alice and Bob is required, and then Bob can find it out based on the message from Alice and his own measuring results on the photons 𝒩_B and ℳ. So the relativistic causality is not violated here. The potential information encoded in the initial state (1) by Alice through the current scheme can not be decoded by Bob at a speed faster than light. At the same time, since all interactions in the present scheme are performed locally, and there is no need for any photon to travel a long distance, it almost takes no time to accomplish the experiment proposed here. In other words, the measurement proposed in this scheme can in principle be performed instantaneously or in a very short time, compared with the distance between Alice and Bob, which is why we call it “nonlocal measurement”.

4. Conclusions

To summarize, following the idea in [12], we designed an experiment to perform the nonlocal measurement of product observable $σ_{z}^{A} \otimes σ_{z}^{B}$ in an optical system. In order to entangle the two-photon system with a single photon in a two-path interferometer, an ancillary meter initially prepared in a maximally entangled state is required. Three media are used to couple the three systems through a cross-Kerr interaction, which only causes phase change to the interacting terms, and no photon is absorbed or created during the interaction. Under particular experimental settings, it is found that the nonlocal measurement on the two-photon system turns to local measurement on the single photon in a two-path interferometer. Owing to the degeneracy of the nonlocal observable considered here, all outcomes of the measurement are eigenstates of the nonlocal observable, so that the nonlocal measurement could be performed in a deterministic way, no matter what result is obtained in the ancillary system. As a consequence of the relativistic causality, we do not know what result is finally achieved after the nonlocal measurement, unless the two observers located far way from each other are allowed to share the information they obtained. This scheme is valid for an arbitrary two-qubit state, no matter it is entangled or not.

We believe the approach proposed in this scheme could be generalized to realize nonlocal measurements of a higher dimensional or even multipartite system, which play an important role in many subjects of fundamental physics, e.g., quantum measurement, quantum nonlocality, and so on. Since the nonlocal observable could in principle be measured in experiment, as one of its potential applications, it can be used to extend the measurement of weak value [45] from local observables to nonlocal ones, and make a comparison with the scheme introduced in [46], where an interesting method for extracting the weak value of a nonlocal measurement based on local weak measurements is proposed.

Funding

National Natural Science Foundation of China (NSFC) (11364022, 11664018).

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Experimental proposal for performing nonlocal measurement of a product observable

Abstract

1. Introduction

2. Experimental setup

3. Evolution of the system and discussions

4. Conclusions

Funding

References and links

Cited By

Figures (1)

Equations (15)

Optics Express