1. Introduction

Quantum measurement plays an important role in quantum theory, and is essential for quantum information technologies. In 1932, von Neumann proposed a strong measurement protocol [1], which describes the interaction of the system and the measurement apparatus (or measurement meter) in a dynamical way. After a strong measurement, the quantum state of the system collapses randomly into an eigenstate of the measurement operator with a corresponding probability. In 1988, Aharonov et al. extended the von Neumann measurement to the weak coupling regime, and presented a weak measurement protocol [2] that reveals some information about the system without collapsing its state into an eigenstate. The weak measurement and weak value have become into a public spotlight [3–7] due to their various potential applications, such as small signal amplification [8–12], quantum state and geometric phase determination [13–17], and the demonstration of nonclassical features of quantum mechanics [18–21].

Nonlocal observables of spacelike separated quantum systems were a contentious topic due to the shortage of direct interaction between them [22–24]. In 1931, Landau and Peierls [25] claimed that nonlocal variables cannot be measured instantly without violating relativistic causality. Subsequently, this conjecture was disapproved and nonlocal measurements of some types of observables became possible, given infinite entanglement, under the restriction of relativistic causality [26–28].The nonlocal properties of some simple physical systems can, in principle, be measured at defined times without violating relativistic causality [26–28]. Furthermore, nonlocal observables can be even measured instantaneously with finite entanglement consumption using verification measurements [29–31].

Nonlocal product observables are a specific type of nonlocal observables and are essential for performing many quantum information tasks [32–36], ranging from quantum nonlocal tests [37] and semicausal measurements [38] to stabilizer codes [39] and quantum thermal transistors [40–42]. Nonlocal von Neumann measurements of these observables usually require a maximally entangled channel [28]. In 2016, Brodutch and Cohen proposed a protocol for generating von Neumann measurement Hamiltonian for nonlocal observables [43] and used it to perform both weak and strong measurements of nonlocal product observables via quantum erasure [44–46]. In 2019, Li et al., demonstrated a von Neumann measurement of nonlocal product observables using linear optics and quantum erasure [47]. Meanwhile, strong von Neumann measurement of nonlocal variables [48] and direct measurement of nonlocal wave functions [49] have also been experimentally demonstrated using linear optics. Recently, Vadil and Edamatsu presented a generalized measurement protocol for nonlocal bipartite spin products [50]. They could efficiently perform nonlocal generalized measurements by tuning the entanglement of a meter in a partially entangled pure state to the desired measurement strength [50].

In this article, we propose a generalized protocol for measuring a product observable of a spacelike separated quantum system using a meter with only mixed entanglement. We also provide a specific scheme to perform a nonlocal measurement of a product Pauli observable $\sigma _{z}\otimes \sigma _{z}$ with linear optics. We use the multiple degrees of freedom (DoFs) of single photons [51–55], and denote the polarization and spatial modes as the system and auxiliary (meter) qubits, respectively. The inevitable transmission noise can lead to the desired mixed entangled states in the spatial mode [56], while the noise, in contrast, is detrimental for previous nonlocal measurement protocols via quantum erasure [43,47] and for quantum communication protocols [51]. Moreover, our protocol can be used to perform nonlocal measurements with an arbitrary strength by tuning the meter entanglement [56]. Therefore, our protocol can be useful for various applications involving nonlocal product observables and nonlocal weak values, and for tests of quantum foundations in nonlocal scenarios.

2. Protocol for nonlocal generalized measurement of the product observable $\sigma _{\rm z}\otimes \sigma _{\rm z}$ with mixed entanglement

Consider a bipartite system consisting of two spacelike separated qubits, where Alice and Bob each have local access to one qubit. The mixed state of the system is arbitrary and can be described as

The meter used to measure the target system is initialized to a mixed entangled state of two orthogonal components and can be described as

The coupling to the degenerate subspaces of a product observable can be achieved by controlled-NOT (CNOT) gate operations with the system and meter qubits as the control and target, respectively. After the coupling, a proper measurement on each meter qubit can read out the information about the nonlocal system and complete the nonlocal generalized measurement, shown in Fig. 1. Specifically, the nonlocal generalized measurement process can be described as follows:

 

Fig. 1. Schematics for nonlocal generalized quantum measurement of the product Pauli observable $\sigma _{\rm zA}\otimes \sigma _{\rm zB}$ using mixed entanglement. Here $\rho _S$ and $\rho _M$ represent the states of the system and the meter, respectively. Alice and Bob perform a controlled-NOT gate on their qubits before measuring their meters in $\sigma _{\rm zA}$ and $\sigma _{\rm zB}$ bases.

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(1) Alice and Bob perform CNOT gates on their qubits to couple the system to the meter, and evolve the composite system $\rho _S\otimes \rho _M$ into a symmetric state with respect to the meter subsystem, since the parity of the flip-operation number determines the meter state for the specified initial meter state.

(2) Alice and Bob then perform projective measurements on their meter qubits separately in the $\sigma _{\rm z}$ basis. For simplicity, we denote the eigenstates of $\sigma _{\rm z}$ with eigenvalues $+1$ and $-1$ as $|0\rangle$ and $|1\rangle$, respectively. Then, they can get the measurement result $+1$ or $-1$.

(3) Suppose Alice and Bob can share their measurement results. The global outcomes of the generalized quantum measurement of the product Pauli observable $\sigma _{\rm zA}\otimes \sigma _{\rm zB}$ can be calculated from the four different local outcomes, projecting the system into either the even parity or the odd parity subspace with the corresponding global measurement result $+1$ or $-1$, shown in Table 1.

Table 1. Relationship between local measurement results and global results of the nonlocal product measurement.

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If classical communication is not allowed between Alice and Bob, the protocol can success with a probability of $50\%$ by a modification as follows: one of the two parties Alice (Bob) can post-select the local result $+1$ and the other party can then read out the global result by measuring the corresponding local meter.

3. Nonlocal generalized measurement with linear optics and mixed entanglement

Photons interacting weakly with their environments are regarded as the best physical system for long-distance transmission of quantum states. Various nonlocal operations on single photon pairs can be realized at room temperature with only linear optics, such as entanglement purification and concentration [57–64], multiple quantum entanglement preparation [65–67], and quantum nonlocal measurements [48–50,68]. Moreover, photons with multiple DoFs, such as polarization, spatial mode, time-bin, angular momentum, and frequency, enable high-capacity and compact quantum information processing [51–55], one-step quantum secure direct communication (QSDC) [69–71], and efficient quantum secret sharing [72]. By encoding a photon in both polarization and spatial modes, we present a nonlocal generalized protocol to measure the polarization of two spacelike separated photons, the spatial mode of which performs as a meter in a mixed entangled state. In principle, other DoFs of single photons, such as time-bin and frequency, can also be used to implement a nonlocal generalized measurement of product observables.

The physical implementation of our protocol with linear optics is shown in Fig. 2. A spontaneous parametric down-conversion (SPDC) source, consisting of a type-II $\beta$-barium borate (BBO) crystal and a mirror, generates photon pairs that are entangled in both the spatial mode and the polarization DoFs, which can be referred to as a hyperentangled state [51–53], described as follows:

Here $a_{1}$ and $b_{1}$ ($a_{2}$ and $b_{2}$) are the upper (lower) spatial modes of photons $A$ and $B$ sent to Alice and Bob, respectively. $|ij\rangle =|i\rangle _A\otimes |j\rangle _B$ with $i,j\in \{V,H\}$ represents two polarized photons, the first (second) of which is sent to Alice (Bob). For simplicity, we shall use $|ij\rangle$ to describe the polarization here after if there is no confusion.

 

Fig. 2. Schematics of nonlocal generalized measurement on bipartite spin products with mixed entanglement in a linear optical system. The photon pairs are generated by an SPDC source. The photon polarization is used as the system, and the spatial mode is used as the meter. The beam splitter (BS) implements a Hadamard operation on the spatial mode. Polarization beam splitter (PBS) transmits $H$-polarized photons and reflects $V$-polarized ones, and thus can be referred to as a CNOT gate on the polarization (control) and spatial mode (target) of the same photon. HWP and QWP are, respectively, half-wave plate and quarter-wave plate, which are combined to implement polarization rotations. The upper (lower) modes are denoted as $a_{1}$ and $b_{1}$ ($a_{2}$ and $b_{2}$) for photons sent to Alice and Bob, respectively.

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The photon $A$ ($B$) is transmitted to Alice (Bob) through a multicore fiber [56]. The channel noise environment will inevitably disturb the photon-pair state. For a multicore fiber channel, the dominant noise for the spatial modes is dephasing, since the crosstalk between the cores is about -$45$ dB$/100$ km [73], and the dominant noise for the polarization is random rotation due to thermal fluctuation, vibration, and birefringence effects of the fiber channel. Therefore, the state of the photon pair after transmission can be evolved into a mixed state [56]

The spatial mode of the photon pair is in a mixed entangled state $\rho _{s1}$ composed of $|{\Phi ^{+}}\rangle _{s}$ and $|{\Phi ^{-}}\rangle _{s}$. The spatial-mode entangled states of a photon pair can be described as

Here the parameter $F_{1}$ ($F_2$) represents the probability that the spatial mode of the photon pair is in state $|{\Phi ^{+}} \rangle _{s}$ ($|{\Phi ^{-}}\rangle _{s}$). Therefore, $F_1$ can also be referred to as the spatial mode fidelity [74] of the transmission with respect to the initial state $|{\Phi ^{+}} \rangle _{s}$, and $F_2$ can be referred to as the probability with a phase-flip error. The polarization of the photon pair evolves into a mixed state $\rho _{p1}$, in which the channel noise introduces bit flip, phase flip, and phase-bit flip for the polarization DoF with probabilities $a_2$, $a_3$, and $a_4$, respectively. The polarization Bell states can be described as

The mixed spatial-mode entanglement can not be directly used to perform nonlocal measurement on the product operator $\sigma ^p_{\rm zA}\otimes \sigma ^p_{\rm zB}$ with $\sigma ^p_{\rm zA}=|H\rangle _A\langle H|-|V\rangle _A\langle V|$ and $\sigma ^p_{\rm zB}=|H\rangle _B\langle H|-|V\rangle _B\langle V|$, since the polarization parities of $|{\Phi ^{+}}\rangle _{s}$ and $|{\Phi ^{-}}\rangle _{s}$ are both even and can not be distinguished by local measurements in the $\sigma ^s_{\rm zA}\otimes \sigma ^s_{\rm zB}$ basis with $\sigma ^s_{\rm zA}=|a_1\rangle \langle a_1|-|a_2\rangle \langle a_2|$ and $\sigma ^s_{\rm zB}=|b_1\rangle \langle b_1|-|b_2\rangle \langle b_2|$. Fortunately, a pair of polarization-independent beam splitters (BSs) interfering the lower and upper modes of each photon can transform the phase-flip state into bit-flip state and complete the following transformations of the spatial modes: $|{\Phi ^{+}}\rangle _{s}\stackrel {\rm BS}\longrightarrow |{\Phi ^{+}}\rangle _{s}$, $|{\Phi ^{-}}\rangle _{s} \stackrel {\rm BS}\longrightarrow |{\Psi ^{+}}\rangle _{s}$, when the BSs apply a Hadamard operation on the spatial mode DoF with the transformation matrix $\frac {1}{\sqrt {2}}\!\left ( \begin {smallmatrix} 1 & 1 \\ 1 & -1 \end {smallmatrix} \right )$. Subsequently, a set of half-wave plate (HWP) and quarter-wave plate (QWP) placed at each output port of two BS can evolve the photon pair polarization into another state

So far, the state of the polarization (system) and the spatial mode (meter) of two spacelike separated photons has been generated, and it can be described as

We can complete the nonlocal generalized measurement of the product Pauli observable $\sigma ^p_{\rm zA}\otimes \sigma ^p_{\rm zB}$ by performing project measurement on the spatial modes of photons A and B, which can be achieved by either nondestructive measurements or post-selecting measurements. For a proof-of-principle demonstration, post-selecting measurements can be used, in a manner similar to what has been suggested for quantum entanglement distillation [59]. Finally, a full tomography of the two-photon polarization state can be achieved by a combination of a QWP, a HWP, a PBS and two single-photon detectors in each spatial mode [47].

The results of the nonlocal generalized measurement are shown in Table 2. The local result of the projective measurement of spatial modes for Alice and Bob, heralded by the click of one single-photon detector $D_a$ ($a=1,2,3,4$) and by the click of one detector $D_b$ ($b=5,6,7,8$), can be either the upper mode denoted by $+1$ or the lower mode denoted by $-1$. The global result of nonlocal generalized measurement of the product observable $\sigma ^p_{\rm zA}\otimes \sigma ^p_{\rm zB}$ will be $+1$ when two local measurement results of the meter qubits are identical, otherwise the global result will be $-1$.

Table 2. Relationship between local measurement results and the global results of the nonlocal generalized product measurement of $\sigma ^p_{\rm zA}\otimes \sigma ^p_{\rm zB}$ on two polarized photons.

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After the projective measurement of spatial modes (meter qubits), the final state of the polarization (system qubits) can be described by applying the following four measurement operators on the polarization DoF of the photon pair:

These measurement operators can be rewritten in a more compact form by applying superoperators, $I_{\pm }(\rho _p)=M_{\pm }\rho _pM^{\dagger }_{\pm }$, on the system. The positive operator-valued measure (POVM) effects for the whole process, corresponding to the projective measurement on meter qubits, can be described as

The corresponding state of the system conditioned on a global measurement result $i\in \{+,-\}$ can be described as

4. Measurement strength and concurrence

The measurement strength is closely related to the entanglement of the meter and thus the probability of state $|\Phi ^+\rangle _s$ in the mixed meter state. The entanglement of a mixed state of a bipartite system can be described by the entanglement of formation [75], an explicit formula of which can be described as [76,77]

In practice, ${\mathcal {E}}(C)$ monotonically increases with the concurrence $C$, and we can take the concurrence as a measure of entanglement with

Here $\lambda _{i}$ with ${i}\in \left \{1,2,3,4\right \}$ are eigenvalues of the Hermitian matrix $R=\sqrt {\sqrt {\rho }\widetilde {\rho }\sqrt {\rho }}$ and thus can also be referred to as the square roots of the eigenvalues of the matrix $\rho \widetilde {\rho }$. The $\lambda _{i}$ is in a descending order with $\lambda _{1}\geq \lambda _{2}\geq \lambda _{3}\geq \lambda _{4}$. The spin-flip state $\widetilde {\rho }$ of the bipartite system is [77]

In our nonlocal generalized measurement with linear optics, the meter qubits are encoded by the spatial modes of a photon pair, and their state just before the measurement is $\rho _{s2}=F_{1}|{\Phi ^{+}}\rangle _{s}\langle {\Phi ^{+}}|+F_{2}|{\Psi ^{+}} \rangle _{s}\langle {\Psi ^{+}}|$, which is a conjugate matrix with $\rho _{s2}=\rho _{s2}^{*}$. The spin-flip state of the meter, according to Eq. (20), can be described as

The corresponding matrix $N$ for calculating the concurrence of the meter state can be described as follows:

The four eigenvalues of matrix $N$ in the descending order for $F_1\geq F_2$ are $\lambda _{1}=F_{1}^{2}$, $\lambda _{2}= F_{2}^{2}$, $\lambda _{3}=0$, and $\lambda _{4}= 0$, respectively. Therefore, the concurrence of the meter state is

For different measurement strengths, the mixed entanglement of the spatial mode required for measuring the polarization of two spacelike separated photons should be changed accordingly. For $C_1$ approaches unity, a strong nonlocal measurement can be achieved, while for $C_1$ approaches zero, only a weak measurement can be achieved. Furthermore, for a general $C_1$, a nonlocal measurement at any intermediate strength $S=C_1$ can be achieved with the protocol presented in Fig. 1. Generally, the concurrence $C_1$ of the meter is assumed to be no less than the measurement strength, otherwise entanglement purification will be required to increase the concurrence of the meter to the desired level [78] . For the specific measurement protocol implemented with linear optics, we use the polarization and the spatial modes of a photon pair as the system and the meter, respectively. The distribution of spatial mode entanglement can be faithful, and efficient distribution of high-dimensional spatial mode entanglement through a $11$ km multicore fiber has been experimentally demonstrated with an average fidelity of $F=0.921$ [73]. Therefore, the linear protocol can be implemented directly for nonlocal measurements with a weak and/or mediate measurement strength. Note that the aforementioned results can also be obtained for other systems, when the meter state is initialized to a similar mixed state and the corresponding two-qubit CNOT gate is available [79–81].

5. Discussion and summary

We show that a generalized quantum measurement of a product observable can be implemented with mixed entanglement, which in practice can be more resource-efficient compared with those based on quantum erasure [43] or partially entangled pure states [50]. We can measure a product observable $\sigma _{\rm z}\otimes \sigma _{\rm z}$ on an arbitrary state of a spacelike separated two-qubit system, when the meter is prepared in a mixed entangled state composed of $|\Phi ^+\rangle _{M}$ and $|\Psi ^+\rangle _{M}$. This protocol is thus particularly useful for the case with a bit-flip quantum channel, a phase-flip quantum channel or a bit-phase flip channel [82], which evolves a maximally entangled state into a mixed one consisting of two components $|\Phi ^+\rangle _{M}$ and $|\Psi ^+\rangle _{M}$, up to single-qubit rotations of meter qubits. For a bit-phase flip channel, the single-qubit rotations can be implemented by $\sigma ^M_{\rm A}\otimes \sigma ^M_{\rm B}$ with $\sigma ^M_{\rm A}=|0\rangle \langle 0|-i|1\rangle \langle 1|$ and $\sigma ^M_{\rm B}=|0\rangle \langle 0|+i|1\rangle \langle 1|$ rotations of meter qubits A and B, respectively.

The mixed entangled meter in principle can be referred to as a binary with even and odd parities, and thus leads to distinct results after its interaction with the system, which is the essential for our nonlocal generalized measurement using mixed entanglement. We can perform the nonlocal generalized measurement directly when the required measurement strength equals the concurrence of the meter, rather than performing entanglement purification [57–63] to prepare maximally entangled states for the measurement with quantum erasure. Furthermore, this protocol can be generalized to measure other product observables [43,50] by introducing additional single-qubit rotations. For instance, to measure the nonlocal product observable $\sigma _{\rm x}\otimes \sigma _{\rm x}$, a Hadamard operation should be applied on each system qubit before and after the nonlocal generalized quantum measurement described in Fig. 1.

In summary, we have shown that it is possible to perform a nonlocal generalized quantum measurement of a product observable with mixed entanglement. The measurement strength equals the concurrence of the meter, and thus an arbitrary-strength nonlocal measurement can be achieved by properly modifying the entanglement of the meter. Furthermore, we present a feasible implementation protocol with linear optics. The post-measurement states and statistics after the nonlocal measurement are the same as those based on quantum erasure with maximal entanglement. All these distinguishing features make our protocol efficient for various applications involving nonlocal product observables in combination with tests of quantum foundations in nonlocal scenarios.