Entanglement plays a key role in quantum information processing, from quantum computation^[1] to quantum teleportation.^[2] In particular, entanglement can provide more security for quantum cryptography.^[3] In such applications, we need to quantify the entanglement of a quantum system. There have been several well-known entanglement measures, such as concurrence,^[4] entanglement of formation,^[5] negativity^[6–8] and random robustness.^[9] All of these require estimating a large number of density matrix elements, which is difficult work for bipartite and multipartite quantum states. If we only need to verify if a state has entanglement or not, the entanglement witness (EW) is a good choice. By using a proper EW operator, one can quickly receive a “yes” or “no” result, which is much easier to achieve.

However, several studies show that conventional EW (CEW) is measurement-device-dependent, because errors and misalignments of the measurement devices can lead to incorrect estimations of the quantities, incurring erroneous conclusions.^[10–12] Moreover, another problem of CEW is that^[13] one should have some information about the tested states first, then the proper EW operator to detect the state can be found or designed, otherwise one has to try the EW operators randomly.

The recent work by Shahandeh et al.^[14] has introduced a measurement-device-independent (MDI) and universal entanglement witness (UEW) theory based on semiquantum game framework,^[15] which can be used to deal with the drawbacks of CEW. A schematic diagram of the game framework is shown in Fig. 1. Alice and Bob (players) share state . Charlie (referee) prepares quantum question states and and sends to Alice and Bob respectively during each round. Alice and Bob, respectively, measure the question state and their own qubit, and then send their measurement outcomes x, y to Charlie. Local operations and classical communication (LOCC) are allowed during the game. Charlie then calculates the reward function from their classical responses. If the outcome of the reward function is larger than 0, Alice and Bob win the game and get the payoff (equal to the outcome) from Charlie. Thus the aim of Alice and Bob is to make the payoff as large as they can. In this game, whether the payoff will be larger than 0 is connected to whether there is entanglement in two players’ shared state. And how large the payoff can be is connected to how much entanglement in their shared state. As referee, Charlie should ensure that two players cannot win the game by cheating, i.e., they will never win the game if their shared state does not have entanglement.

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. The scheme of extremal semiquantum nonlocal game. Players (Alice and Bob) share state , and then, they receive quantum questions , from referee (Charlie). After performimg local POVM on their own part of shared state and the quantum question state, they return classical answers x and y to Charlie. LOCC is allowed in this game. If the shared state is entangled, then Alice and Bob can always win this game (achieve payoff value larger than 0) as long as they perform suitable measurements.

To prevent two players from cheating, there are two requirements for Charlie’s quantum question states and . First, the question states should be nonorthogonal to each other. Second, they should be able to form an extremal decomposable entanglement witness operator .^[16] Then, only if Alice and Bob share negative-partial-transpose (NPT)^[17] entangled states, will they possibly be able to achieve payoff larger than 0. Otherwise, no matter what measurements they perform, they will never win, i.e., there will never be positive outcomes. This is the MDI characteristic, which prevents Alice and Bob from cheating. In other words, two players cannot get positive rewards by cheating (e.g., performing some special measurements) if they do not share a pair of entangled states.

To achieve the maximum positive outcome, two players should also try to find the optimal positive operator-valued measure (POVM). It should be stressed here that, the maximum payoff that the players possibly achieve is proportional to the amount of NPT entanglement in the shared state. Therefore, if we replace the shared state with unknown states that need to be witnessed, upon this semiquantum game framework, we are able to design an MDI and universal entanglement witness (MDI-UEW).

In this paper, we present and experimentally demonstrate an MDI-UEW that has the ability to detect and quantify entanglement at the same time. We first show that NPT entangled states should be classified into different collections based on their optimal EW operators. Without loss of generality, we choose entangled states from four representative collections (including both pure and mixed states) as examples to explain the MDI-UEW in detail. Our results demonstrate the universality of MDI-UEW in witnessing NPT entanglement of a low-dimensional bipartite system and verify the ability of its outcome to quantify entanglement, which is comparable to concurrence. Furthermore, we perform a theoretical analysis and a numerical simulation to reveal that the outcome of MDI-UEW for an entangled state is less than or equal to its concurrence.

Our experimental setup for MDI-UEW is shown in Fig. 2, where four photons are used. In the source part, two 2-mm β-barium-borate (BBO) crystals produce the original entangled photon pairs by type-I spontaneous parametric down conversion (SPDC). The states of the photon pairs have the form of .

Fig. 2.

Figure Option ViewDownloadNew Window

Fig. 2. Experimental setup for MDI-UEW. The original entangled photon pairs are generated by type-I SPDC in two 2-mm BBO crystals with the form of . The reflected parts of non-polarizing beam splitters NPBS1 and NPBS2 are used as question states. The transmitted parts of NPBS1 and NPBS2 are used to prepare . By selecting a suitable phase compensator PC1, |Φ+〉 or |Φ−〉 can be prepared. When half wave-plate HWP2 is set at an angle of 45°, and HWP1 is set at an angle of 0°, |Ψ+〉, |Ψ−〉 can be prepared. The bypath of NPBS3 is used to prepare the second Bell state component in the Bell diagonal state. HWP4 and HWP5 are used to change the ratio between H-polarization and V-polarization in |Φ+〉, then a redundant amount of H- or V-polarization will be discarded when passing through beam displacer BD2. Then states cos θ|HH〉 + sin θ|VV 〉 are prepared. Wave-plate combinations in path 1 and path 4 are used to rotate the polarizations to encode photons 1 and 4 to the desired question states, and . The measurement part consists of four QWP-HWP-PBS combinations, where QWP and PBS denoting quarter wave-plate and polarizing beam splitter, respectively. All the four photons are separately detected by single-photon avalanche diodes, but the four-photon events are coincidently counted together and analyzed by a picosecond time analyzer with well-tuned delays.

Then, with some probability, two photon pairs are respectively distributed into paths 2 and 3 and paths 1 and 4 with two non-polarizing beam splitters, NPBS1 and NPBS2. The photon pairs going through the NPBSes simultaneously to paths 2 and 3 are used as target bipartite states . We name them photon pairs 2 and 3. While photon pairs are used to prepare the question states and if they are reflected by the NPBSes simultaneously, we name them photon pairs 1 and 4. Two polarizers make them separable because only one component of the original state |HH〉 can pass through the polarizers.

In this experiment, the tested (input) states are chosen from four representative collections of entanglement, including both pure states and mixed states. Using phase compensators (PCs), time delay plates (TDPs), and half-wave plates (HWPs) in path 2 and path 3, we prepare all Bell states and several Bell diagonal states (mixture of any two Bell states in a different proportion).^[24]

By carefully selecting the phase of PC1, |Φ⁺〉 or |Φ⁻〉 are prepared. When HWP2 is set at an angle of 45° and HWP1 is set at an angle of 0°, we can prepare state |Ψ⁺〉 and |Ψ⁻〉 as well.

NPBS3 and NPBS4 are used to prepare the second Bell state component of a Bell diagonal state. HWP3 is used to tune the polarization of photon 3 while PC2 is used to tune the relative phase between different polarizations. TDP is used to make the transmitting part and reflecting part of NPBS3 decoherent.

States cos θ |HH 〉 + sin θ |VV 〉 are prepared from |Φ⁺〉 with the help of beam displacers BD1 and BD2. We change the ratio between H-polarization and V-polarization of photon 3 by adjusting the angle of HWP4 and HWP5. Then redundant amount of H- or V-polarization will be discarded when passing through BD2 and finally we obtain the desired states with θ ∈ [0, π/2].

The HWP-QWP (quarter-wave plate) combinations in path 1 and path 4 are used to prepare the question states in the collection of |± x〉, |± y〉, |± z〉 (eigenstates of Pauli matrices). There are a total of 12 pairs of (see Appendix A).

A complete set of POVM , used in our experiment, can be expanded into combinations of , , , and I^{⊗ 4} (the Pauli matrix and identity matrix). Detailed information is provided in Appendix B. Four QWP-HWP-PBS (polarizing beam splitter) combinations in the measurement part are sufficient to perform these measurements. By carefully adjusting the time delay between photon pairs 2 and 3 and pairs 1 and 4, we collect four-photon events and calculate the rewards for Alice and Bob according to Eq. (1).

The experimental results are shown in Fig. 3 and Table 1. The error bars represent one standard deviation, which is derived by using the Monte Carlo method. There are two comparisons in Fig. 3. One is the entanglement-detection ability between CEW and MDI-UEW, and the other is the entanglement-quantification ability between entanglement-witness measure and concurrence. Here, concurrence is used as a reference and to make a uniform standard to judge whether a state is entangled or not.

Fig. 3.

Figure Option ViewDownloadNew Window

Fig. 3. Comparison between the MDI-UEW and CEW on entanglement detection ability and comparison between the EWM and concurrence on entanglement quantification ability. CEW is calculated by . EWM represents , which is the witness outcome of MDI-UEW. (a) Experimental results for the Bell states. The related collections of Bell states are labelled in the parentheses. The orange bars represent the experimental results of the EWM, the green bars represent the CEW, and the blue bars represent the estimated concurrence. (b) Experimental results of cos θ|HH〉 + sin θ|VV〉 (θ ∈ [0,π/2]). The theoretical values of the EWM (EWMth, red dashed line) and concurrence (Con.th, black solid line) are both sin 2θ, while the theoretical value of CEW (blue solid line) is –1. Red circular points represent the experimental results of EWM (EWMexp), whereas the black square points represent the estimated concurrence (Con.exp). Blue triangle points represent the results of CEW without any ability for entanglement detection or quantification for the target states.

In Fig. 3(a), four Bell states are tested by both CEW and MDI-UEW. MDI-UEW successfully witnesses the entanglement in these states (orange bars) because their values are all larger than 0, while CEW only witnesses the entanglement in state |Ψ⁻〉 (green bars) because only one bar is larger than 0.

The EW operator in our experiment is W₀ = – 2|Φ⁺ 〉 〈Φ⁺|^T_B, which is only designed for state |Ψ⁻〉 and its related collections C_|Φ⁺〉 in CEW. According to the analysis in Subsection 2.1, four Bell states come from four different collections, thus one EW operator is not sufficient to correctly detect entanglement in them based on CEW theory. Therefore, the other three Bell states are inevitably recognized as seperable states, where their outcomes are less than 0. The design of an MDI-UEW is based on a specific EW operator, but it does not depend on it. Any decomposable extremal EW operator is available. Figure 3(a) shows that entangled states in different collections can all be detected by an MDI-UEW. This demonstrates the universality of MDI-UEW.

In Fig. 3(b), the tested states are cosθ |HH〉 + sinθ |VV〉, where θ ∈ [0, π/2]. They are entangled states and in collection C_|Ψ⁻〉 when θ ≠ 0 or π/2. The EW operator W₀ is unable to detect their entanglement in CEW theory. The amount of entanglement in this series of states varies according to the value of θ. The black square points are the experimental values of their concurrence and the black solid line is the theoretical value. Concurrence is used here as reference. The blue triangle points are the experimental results of CEW. We can find that, no matter how much entanglement is in the states, it always gives out –1; in other words, these states are all recognized as separable states by CEW, which is incorrect. The blue solid line is the theoretical value. The red circular points are experimental results of MDI-UEW and the red solid line shows their theoretical values. When θ equals 0 or π/2, the corresponding state is |HH〉 or |VV〉. They are separable states, thus the values of them are 0, which is a correct detection. With the increase of θ, the states become entangled and the MDI-UEW gives out values larger than 0 for them. Moreover, these values are associated with the amount of entanglement in the states (referred to concurrence), which indicates that the outcome of MDI-UEW is possible to be used as a kind of entanglement measure.

Figure 3(b) also shows an interesting fact that, once the EW operator is mismatched to the states in CEW, the expect value will always be –1, no matter whether the states are maximally entangled, partially entangled, or even separable. In other words, negative value of CEW provides no information.

MDI-UEW also works for mixed states. Table 1 shows the experimental results for the Bell diagonal states. The EW operator W₀ only witnesses entangled states in collections C_|Φ⁺〉 according to CEW theory. State 1 is in this collection, and thus it can be correctly detected. However, state 2 and state 3 are in the collections C_|Ψ⁻〉 and C_|Ψ⁺〉, respectively. Therefore, it is not surprising that the outcomes of CEW for these two are negative and they are judged as separable. Meanwhile, MDI-UEW successfully overcomes this problem, because its outcomes for these three states are positive.

The analysis above has confirmed the advantage of MDI-UEW in detecting entanglement over CEW. Next we turn to the comparison between EWM and concurrence in the entanglement-quantification ability. Concurrence is a well-defined entanglement measure, and if EWM can provide similar quantification for entangled states, this can guarantee the ability of EWM to be a kind of entanglement measure.

Concurrences of states in Fig. 3 are achieved by performing full state tomography. The blue bars in Fig. 3(a) are the experimental estimated concurrences of Bell states, which are maximally entangled states with theoretical values as 1. The orange bars are the experimental results of EWM, and all of them have reached the theoretical value 1. In Fig. 3(b), the theoretical values of both EWM and concurrence are sin 2θ, and the experimental results achieve perfect agreement with the theoretical predictions. The states in Table 1 are mixed states. The concurrence of them is |2p – 1| in theory, so does the EWM. Summarizing the experimental results, the ability of EWM as a kind of entanglement measure has been confirmed. However, we should stress that, the evidence here is not sufficient to claim that EWM is the same as concurrence.

Table 1.

Table 1.

Table 1. Experimental results for the Bell diagonal states. . States p EWMth. EWMexp. CEWth. CEWexp. State 1 0.7 0.4 0.393 ± 0.029 0.4 0.393 ± 0.033 State 2 0.55 0.1 0.076 ± 0.031 –1 –0.980 ± 0.020 State 3 0.4 0.2 0.173 ± 0.031 –0.2 –0.211 ± 0.027 State 1 = p|Ψ−〉〈Ψ− | + (1 – p)|Φ+〉〈Φ+ |, State 2 = p|Φ+〉〈Φ+ | + (1 – p)|Φ−〉〈Φ− |, State 3 = p|Ψ−〉〈Φ− | + (1 – p)|Φ−〉〈Φ− |.

Table 1. Experimental results for the Bell diagonal states. .

The values of the concurrence and EWM are equal for the tested states, however, they are actually two different entanglement measures, and EWM of a state is less than or equal to its concurrence. Negativity is the bridge for these two measures, which is also an entanglement measure.

According to Theorem 23 and Lemma 6 in Ref. [23], the definition of negativity of a two-qubit state is as follows:

Comparing Eq. (5) with Eq. (6), it is obvious that the EWM is equivalent to the negativity of the states in a two-qubit system.

Then, it has been proved that the negativity of a state is less than or equal to its concurrence. Therefore, the same is true for the EWM in a two-qubit system. And the negativity/EWM will equal concurrence if the eigenstate with negative eigenvalue of is a Bell state (up to local unitary transformation) (see Theorem 25 in Ref. [23]).

The states we used in our experiment are from the four collections C_|Ψ⁻〉, C_{|Ψ⁺ 〉}, C_{|Ψ⁻〉, and C|Ψ–+〉}, thus their eigenstates with negative eigenvalues are Bell states. Note that, the related eigenstates with negative eigenvalues are labelled at the subscript of each collection. For example, any state in collection C_|Ψ⁻〉 has common eigenstate with negative eigenvalue |Ψ⁻〉. Therefore, the EWM of the input states is theoretically equal to the concurrence as shown in Fig. 3. In other cases, EWM is always less than concurrence.

Figure 4 shows numerical simulation of EWM and concurrence for a series of asymmetric states (which are useful in one-way steering^[25,26]

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Concurrence and EWM for a series of asymmetric states. States are defined in Eq. (7), the two parameters (p,θ) are united as x, which is p = 0.01x and θ = 0.01xπ. EWM of a state is less than or equal to its concurrence.

We introduce MDI-UEW to solve the two drawbacks of CEW, which is measurement-device-dependent and needs prior information about the tested states. The MDI characteristic of MDI-UEW removes the trust from the measurement devices (including measurement bases and measurement agents). And the characteristic of universality enables us to witness NPT entanglement of unknown bipartite states with limited dimensions. Four representative collections of NPT entangled states are used as examples in our experiment, and MDI-UEW successfully witnesses all of them while CEW is only able to detect one collection. Moreover, we find EWM (the outcome of MDI-UEW) can directly quantify the amount of entanglement. This gives MDI-UEW another advantage over CEW. Concurrence of the target states is estimated as a reference in our experiment. We have demonstrated that EWM has sufficient ability to quantify entanglement comparable to concurrence and theoretically prove that EWM is always less than or equal to concurrence, followed with a numerical demonstration for a series of asymmetric states.