Experimental witnessing for entangled states with limited local measurements

Gaoyan Zhu; Chengjie Zhang; Kunkun Wang; Lei Xiao; Peng Xue

doi:10.1364/PRJ.462212

Abstract

We experimentally demonstrate a method for detection of entanglement via construction of entanglement witnesses from a limited fixed set of local measurements (

M

). Such a method does not require a priori knowledge about the form of the entanglement witnesses. It is suitable for a scenario where a full state tomography is not available, but the only resource is a limited set of

M

. We demonstrate the method on pure two-qubit entangled states and mixed two-qubit entangled states, which emerge from photonic implementation of controllable quantum noisy channels. The states we select are motivated by realistic experimental conditions, and we confirm it works well for both cases. Furthermore, possible generalizations to higher-dimensional bipartite systems have been considered, which can potentially detect both decomposable and indecomposable entanglement witnesses. Our experimental results show perfect validity of the method, which indicates that even a limited set of local measurements can be used for quick entanglement detection and further provide a practical test bed for experiments with entanglement witnesses.

1. INTRODUCTION

Quantum entanglement, which lies at the heart of the foundations of quantum mechanics, is a crucial resource for quantum information science [1,2]. For a given state, how to determine whether it is entangled or not is a fundamental question in quantum entanglement theory. In the past two decades since the definition of entanglement was clarified in Ref. [3], tons of research have been reported related to this subject, such as the well-known positive partial transpose criterion [4], the computable cross norm criterion (or matrix realignment criterion) [5,6], the permutation separability criterion [7,8], and entanglement witnesses [9 –17]. Entanglement witnesses accomplish this task without requiring full state tomography. Thus, several types of entanglement witnesses have been defined and studied theoretically [18 –32] and have been demonstrated in various physical systems [33 –39].

Unlike the other criteria in which it is assumed that the state density matrix is preknown, entanglement witnesses are Hermitian operators and designed directly for detection of entanglement for states. Rather, an operator $W$ that acts on a bipartite system is an entanglement witness if and only if $Tr (ρ_{s} W) \geq 0$ for all separable states $ρ_{s}$ and $Tr (ρ_{e} W) < 0$ for, at least, one entangled state $ρ_{e}$ . Entanglement witnesses completely characterize separable states and allow to detect entanglement experimentally [40]. Moreover, a witness $W$ is optimal if for any $R \geq 0$ an operator $W - R$ is no longer a witness [41]. Optimal entanglement witnesses are the best entanglement detectors, that is, $W$ is optimal if and only if there is no other witness that detects more quantum entangled states than $W$ does. A witness, which is not optimal, may be optimized via a suitable optimization procedure [41]. It is, therefore, clear that knowing all optimal entanglement witnesses one is able to detect all entangled states. In a recent work, Riccardi et al. [42] proposed a method to construct entanglement witnesses from a limited fixed set of local measurements, which performed on two-qubit systems. This method fits within a scenario that a full state tomography is not available, whereas only a smaller resource as a measurements set $M$ is available. Such a method completely characterizes the class of entanglement witnesses of the form $W = P^{Γ}$ where the superscript $Γ$ denotes a partial transposition, that can be derived from $M$ . Designing a witness for an arbitrary entangled state, especially for mixed states is not easy since the problem of separability of mixed states appears to be extremely complex; whereas due to the decoherence phenomenon, in laboratories we unavoidably deal with mixed states rather than pure ones.

In this paper, we experimentally demonstrated the method in Ref. [42] for detecting entanglement of both pure and mixed two-qubit states. The mixed states we select are motivated by realistic experimental conditions, which emerge from four typical single-qubit entanglement breaking channels, i.e., Pauli channels, dephasing channels, depolarizing channels, and amplitude damping channels. The features of requiring low measurement resources and no a priori knowledge about the witness forms show the widespread application and experimental feasibility of the method, thus, providing a practical test bed for quick entanglement detection. Furthermore, we experimentally explore the possible extensions to higher-dimensional bipartite systems, which can be used to detect decomposable and indecomposable entanglement witnesses.

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2. THEORETICAL SCENARIOS

A two-qubit entanglement witness can be decomposed as $W = \sum_{i, j} T_{i j} σ_{i} \otimes σ_{j},$ (1)with $σ_{i} \in {σ_{0}, σ_{1}, σ_{2}, σ_{3}}$ , where $σ_{0}$ represents a two-dimensional identity matrix and $σ_{1}$ , $σ_{2}$ , and $σ_{3}$ are Pauli matrices. Given a limited set of measurements $M = {σ_{1} \otimes σ_{1}, σ_{2} \otimes σ_{2}, σ_{3} \otimes σ_{3}}$ , witnesses of the below form can be constructed: $W = α σ_{0} \otimes σ_{0} + \sum_{i = 1,2,3} (a_{i} σ_{0} \otimes σ_{i} + b_{i} σ_{i} \otimes σ_{0}) + \sum_{i = 1,2,3} c_{i} σ_{i} \otimes σ_{i},$ (2)where $α, a_{i}, b_{i}, c_{i}$ are real parameters. We focus only on characterization of witnesses within the family of Eq. (2) that are of the form $W = P^{Γ} .$ (3)Therefore, $P$ consists of only terms from the limited source $M$ .

It is proven in Ref. [42] that, there are six families of rank-1 projectors $| ψ ⟩ ⟨ ψ |$ within the class of Eq. (2), $| φ_{1} ⟩ = a | Φ^{+} ⟩ + b | Φ^{-} ⟩; | φ_{2} ⟩ = a | Ψ^{+} ⟩ + b | Ψ^{-} ⟩; | φ_{3} ⟩ = a | Φ^{+} ⟩ + b | Ψ^{+} ⟩; | φ_{4} ⟩ = a | Φ^{-} ⟩ + b | Ψ^{-} ⟩; | φ_{5} ⟩ = a | Φ^{+} ⟩ + i b | Ψ^{-} ⟩; | φ_{6} ⟩ = a | Φ^{-} ⟩ + i b | Ψ^{+} ⟩,$ (4)where $| Φ^{\pm} ⟩ = (| 00 ⟩ \pm | 11 ⟩) / \sqrt{2}$ and $| Ψ^{\pm} ⟩ = (| 01 ⟩ \pm | 10 ⟩) / \sqrt{2}$ are the Bell states; $a, b \in R$ and satisfy $a^{2} + b^{2} = 1$ .

In return, six families of entanglement witness that are of the form as Eq. (3) can be derived: $W_{k} = | ϕ_{k} ⟩ {⟨ ϕ_{k} |}^{Γ},$ (5)where $k = 1, 2, \dots,6$ . As an example, $W_{1} = | φ_{1} ⟩ ⟨ φ_{1} |$ decomposed as the form of Eq. (2) is given by $W_{1} = \frac{1}{4} [σ_{0} \otimes σ_{0} + σ_{3} \otimes σ_{3} + (a^{2} - b^{2}) σ_{1} \otimes σ_{1} + (a^{2} - b^{2}) σ_{2} \otimes σ_{2} + 2 a b (σ_{3} \otimes σ_{0} + σ_{0} \otimes σ_{3})],$ (6)which consists of only terms of measurement $M$ .

Then, for a given two-qubit state $ρ$ , to identify whether it is entangled or not, we perform all the local measurements of $M$ after which $Tr (W_{k} ρ)$ is evaluated for each $k$ . To find a value of $Tr (W_{k} ρ)$ below zero, we minimize the value $Tr (W_{k} ρ)$ over $a$ and $b$ for each $k$ . If a negative value is obtained, then, the state is identified as entangled, and at the same time, a corresponding optimal witness for this state is defined. Such a postprocess implies that the method requires no a priori knowledge about the form of the entanglement witnesses.

3. EXPERIMENTAL DEMONSTRATION

In the experiment, we test the performance of the procedure with pure entangled two-qubit states of the form of Eq. (7) and multiple families of mixed entangled two-qubit states, which are generated by sending one party of the maximally entangled state $| Φ^{+} ⟩$ through four types of single-qubit entanglement breaking channels, i.e., Pauli channels, dephasing channels, depolarizing channels, and amplitude damping channels, acting on one of the qubits. The experimental setup is illustrated in Fig. 1(a).

Figure 1.Experimental setup. (a) Optical structure for the experiments. The entangled photon pairs are produced via the type-I spontaneous parametric down-conversion (SPDC) process by pumping two adjacent nonlinear crystals of BBO with a 405-nm laser diode. Two $α$ -BBO crystals are inserted after the BBOs to compensate the walk-off effect. Mixed states emerge from the single-qubit noisy channel, such as the amplitude damping channel, the Pauli channel, the dephasing channel, and the depolarizing channel, acting on one of the qubits. Local measurements in $M$ are carried out via a sequence of quarter-wave plate (QWP)-HWP-PBS and single-photon detection. Coincidence measurements are then performed via avalanche photodiodes (APDs). Total coincidence counts are about 120,000 over a collection time of 12 s within a 3-ns time window. (b) Schematic of the Pauli channel realized with two liquid crystals (LCs). Voltage sequences applied on the spatial path of photons. $t_{i}$ indicates the time interval of voltage to realize the gate $σ_{i}$ . $T$ is the collection time. $V_{H}$ is the corresponding applied voltage when an LC acts as an HWP, and $V_{I}$ is for an identity operator.

In our experiment, polarizations of the photon pairs are generated in a pure entangled state, $| ϕ ⟩ = α | H H ⟩ + \sqrt{1 - α^{2}} | V V ⟩,$ (7)by setting the angle of the half-wave plate (HWP) in front of the $β$ -barium borate (BBO) to $0.5 \arcsin α$ with a fidelity higher than 97%. Here $| H ⟩ = | 0 ⟩$ and $| V ⟩ = | 1 ⟩$ represent the horizontal and vertical polarizations of photons, respectively. The fidelity is defined [43] as $F (ρ_{th}, ρ_{\exp}) = {(Tr \sqrt{\sqrt{ρ_{th}} ρ_{\exp} \sqrt{ρ_{th}}})}^{2}$ , which can be evaluated from the reconstruction of the states via a full state tomography [44].

The mixed two-qubit entangled states emerge from four typical types of quantum noisy channels, i.e., Pauli channels, dephasing channels, depolarizing channels, and amplitude damping channels, acting on one of the qubits. These noisy channels are selected to show the validity of the method when applying it in a realistic experimental environment. The initial state is prepared in $(| H H ⟩ + | V V ⟩) / \sqrt{2}$ . Quantum noisy channels are completely positive trace-preserving maps $N$ acting on a quantum state in a $d$ -dimensional Hilbert space $H^{d}$ , which can be mathematically described as $ρ \mapsto \sum_{i} M_{i} ρ M_{i}^{†},$ (8)where $M_{i}$ are the Kraus operators, satisfying $\sum_{i} M_{i}^{†} ρ M_{i} = I$ .

Taking a single-qubit Pauli channel as an example (see Appendix A for details on realizations of other noisy channels), the Kraus operators are given by $M_{1} = \sqrt{p_{0}} σ_{0}$ , $M_{2} = \sqrt{p_{1}} σ_{1}$ , $M_{3} = \sqrt{p_{3}} σ_{2}$ , and $M_{4} = \sqrt{p_{3}} σ_{3}$ with $\sum_{i} p_{i} = 1$ . The transformation of a single-qubit state under the Pauli channel is $ρ \mapsto p_{0} ρ + p_{1} σ_{1} ρ σ_{1} + p_{2} σ_{2} ρ σ_{2} + p_{3} σ_{3} ρ σ_{3} .$ (9)The experimental realization of the single-qubit Pauli channel is illustrated in Fig. 1(a), consisting of a sequence of LCs on the path of the second photon, one having its fast axis horizontal and the other oriented at 45° with respect to the horizontal [45,46]. Depending on the applied voltage $V$ , it is possible to change the retardation between ordinary and extraordinary polarized radiations. That is, by applying either $V_{I}$ or $V_{H}$ to an LC in Fig. 1(b), it can be made to act as either a full wave or an HWP, respectively. Thus, by varying independently the voltages applied to LCs for different time intervals, we then apply the four Pauli operators to the second photon with different values of the weight $p$ . More precisely, the optical axes of ${LC}_{1}$ and ${LC}_{2}$ are fixed to 45° and 0° corresponding to the horizontal direction, respectively. Then, ${LC}_{1}$ and ${LC}_{2}$ act as an HWP at 45° ( $σ_{1}$ ) and an HWP at 0° ( $σ_{3}$ ), respectively, when voltage $V_{H}$ is applied, whereas both ${LC}_{1}$ and ${LC}_{2}$ act as an identity operation ( $σ_{0}$ ) when voltage $V_{I}$ is applied. When both of the LCs are applied with $V_{H}$ , the unit realizes a $σ_{2}$ operation. The weighted parameter $p_{i}$ ( $i \in {0,1,2,3}$ ) can be varied by changing the time-interval $t_{i}$ , which corresponds to each $σ_{i}$ of the voltage. By fixing the relative value of the probability parameters to ${p_{0}, p_{1}, p_{2}, p_{3}} = {1 - p / 6, p / 12, p / 18, p / 36}$ , the number of parameters is reduced from 4 to 1. By setting the time intervals along with the relation $t_{0} : t_{1} : t_{2} : t_{3} = (1 - p / 6) : p / 12 : p / 18 : p / 36$ , the Pauli channels with different values of parameter $p$ are implemented. Here, $\sum_{i} t_{i} = T$ , and $T$ is the collection time.

To verify the validity of the method for experimental witnessing for entangled states with limited local measurements, we compare the witness values and the logarithmic negativity of the states [47,48]. The logarithmic negativity $E_{N} (ρ) = \log_{2} {∥ ρ^{Γ_{B}} ∥}_{1}$ is a notion of entanglement measure. Here, ${∥ \dots ∥}_{1}$ denotes the trace norm, and $Γ_{B}$ denotes the partial transpose. It varies between 0 and 1, indicating the separated and maximally entangled states, respectively, and can be evaluated with the states reconstructed via a full state tomography.

In the following, we show explicitly the performance of the witnesses on a family of pure states $| ϕ ⟩$ [Eq. (7)] and mixed states [Eq. (8)], which emerge from several single-qubit quantum noisy channels.

In Fig. 2, we show the experimental results of the witness and negativity for both pure and mixed states. For pure states, the experimental results of $E_{N}$ is 0 only when $α = 0, 1$ , which indicates the states are separated. As expected, for $α = 0$ and 1, the experimental results of the entanglement witnesses are $0.0109 \pm 0.0018$ and $0.0093 \pm 0.0017$ , respectively. For the rest states, $Tr (W ρ)$ are always less than 0, which proves the method is effective in detecting entangled two-qubit pure states.

Figure 2.Experimental results for two-qubit systems. (a) Entanglement witness value as a function of state parameter $α$ for the pure states and of parameter $p$ of the quantum noisy channels for the mixed states. (b) Negativities versus state parameter $α$ or noisy parameter $p$ . The solid curves indicate the theoretical predictions, and the symbols are for the experimental results. Error bar indicates the statistical uncertainty which is obtained via the Monte Carlo simulation method.

For the mixed states, which emerge from the Pauli channel, negativities are always larger than 0 (from $0.4812 \pm 0.0049$ to $0.3136 \pm 0.0105$ ) for $0 \leq p \leq 1$ . As expected, the experimental results of the entanglement witnesses are less than 0 (from $- 0.4818 \pm 0.0065$ to $- 0.3194 \pm 0.0059$ ). Entanglement is detected for $0 \leq p \leq 1$ . For the dephasing channel, the experimental results of the negativities decrease from $0.4505 \pm 0.0060$ to $0.0015 \pm 0.0058$ (the theoretical prediction is 0 for $p = 0.5$ ) and, then, increase to $0.4583 \pm 0.0070$ . The experimental results of the entanglement witness increase from $- 0.4691 \pm 0.0064$ to $0.0152 \pm 0.0055$ for $0 \leq p \leq 0.5$ and decrease to $- 0.4748 \pm 0.0064$ for $0.5 < p \leq 1$ . For the depolarizing channel with $0 \leq p < 0.5$ , the experimental results of the negativity and the entanglement witness decrease from $0.4666 \pm 0.0019$ to 0 and increase from $- 0.4958 \pm 0.0066$ to $0.0182 \pm 0.0052$ , respectively. For $p \geq 0.5$ , the negativity is theoretically predicted to be 0, and the experimental results of the witness are larger than 0. This indicates the entanglement is detected for $0 \leq p < 0.5$ , and no entanglement is detected for $p \geq 0.5$ . For the amplitude damping channel, the experimental results of the negativity are always larger than 0 for $0 \leq p \leq 1$ . The experimental results of the entanglement witness are always less than 0. Entanglement is detected for $0 \leq p \leq 1$ . Thus, the method of the construction and optimization of entanglement witnesses is valid for the family of the mixed states emerging from the noisy channels.

4. EXTENSIONS OF ENTANGLEMENT WITNESS TO HIGHER-DIMENSIONAL BIPARTITE SYSTEMS

Finally, we show the demonstration of extensions of entanglement witness to photonic higher-dimensional bipartite states [49 –51], which are constructed with the limited local measurements. A straightforward generalization consists of the following: $W_{i j k} = {| φ_{i} ⟩}_{j k} {⟨ φ_{i} |}_{j k}^{Γ},$ (10)where ${| φ_{1} ⟩}_{j k} = a {| ϕ^{+} ⟩}_{j k} + b {| ϕ^{-} ⟩}_{j k}, {| φ_{2} ⟩}_{j k} = a {| ψ^{+} ⟩}_{j k} + b {| ψ^{-} ⟩}_{j k}, {| φ_{3} ⟩}_{j k} = a {| ϕ^{+} ⟩}_{j k} + b {| ψ^{+} ⟩}_{j k}, {| φ_{4} ⟩}_{j k} = a {| ϕ^{-} ⟩}_{j k} + b {| ψ^{-} ⟩}_{j k}, {| φ_{5} ⟩}_{j k} = a {| ϕ^{+} ⟩}_{j k} + i b {| ψ^{-} ⟩}_{j k}, {| φ_{6} ⟩}_{j k} = a {| ϕ^{-} ⟩}_{j k} + i b {| ψ^{+} ⟩}_{j k},$ and ${| ϕ^{\pm} ⟩}_{j k} = (| j j ⟩ \pm | k k ⟩) / \sqrt{2}$ and ${| ψ^{\pm} ⟩}_{j k} = (| j k ⟩ \pm | k j ⟩) / \sqrt{2}$ are four Bell states, $a, b \in R$ , and satisfy $a^{2} + b^{2} = 1$ . Note that for $d = 3$ , $W_{i j k}$ reduces to a fixed matrix with no parameter to optimize.

Another possible construction is of the so-called diagonal-type entanglement witness [52], $W = D - d P_{d}^{+},$ (11)where $D = \sum_{i, j = 1}^{d} d_{i j} | i ⟩ ⟨ i | \otimes | j ⟩ ⟨ j |$ is a $d \times d$ diagonal matrix, and $P_{d}$ is a $d$ -dimensional projector corresponding to a maximally correlated state $| φ ⟩ = \sum_{i} x_{i} | i i ⟩$ , satisfying $d P_{d}^{+} = F^{Γ}$ . Here, $F$ is a so-called flip (or swap) operator defined as follows $F ψ \otimes ϕ = ϕ \otimes ψ$ and $⟨ ψ \otimes ϕ | F | ψ \otimes ϕ ⟩ = | ⟨ ψ | ϕ ⟩ |^{2}$ [52].

We consider $d = 3$ and $D_{[a b c]} = \sum_{i = 1}^{3} | i ⟩ ⟨ i | \otimes [(a + 1) | i ⟩ ⟨ i | + b | i + 1 ⟩ ⟨ i + 1 | + c | i + 2 ⟩ ⟨ i + 2 |],$ (12)where we add $\mod 2$ and $a, b, c \geq 0$ . Then, $W_{[a b c]} = D_{[a b c]} - 3 P_{3}^{+}$ is an entanglement witness [53] iff: $a < 2$ , $a + b + c \geq 2$ , and if $a < 1$ , then, additionally, $b c > {(1 - a)}^{2}$ . It is worth mentioning that the class $W_{[a b c]}$ contains indecomposable entanglement witnesses iff $b \neq c$ , which can be used to detect bound entangled states.

In our experiment, a family of three-dimensional bipartite states is used to demonstrate the above entanglement witnesses, $| ϕ ⟩ = c_{0} | 00 ⟩ + c_{1} | 11 ⟩ + c_{2} | 12 ⟩ + c_{3} | 21 ⟩ + c_{4} | 22 ⟩,$ (13)where $c_{0} = \cos 2 θ_{1}$ , $c_{1} = \sin 2 θ_{1} \cos 2 θ_{2} \cos 2 θ_{3}$ , $c_{2} = - \sin 2 θ_{1} \cos 2 θ_{2} \sin 2 θ_{3}$ , $c_{3} = - \sin 2 θ_{1} \sin 2 θ_{2} \cos 2 θ_{3}$ , and $c_{4} = \sin 2 θ_{1} \sin 2 θ_{2} \sin 2 θ_{3}$ . We fix $θ_{2}, θ_{3} = π / 8$ and vary $θ_{1}$ from 0 to $π / 8$ corresponding to the logarithmic negativity of the states varying from 0 to 1. Then, a similar optimization procedure is performed on the measurement results of $M$ over the class of entanglement witnesses.

The experimental results of $Tr (W_{i j k} ρ)$ and $Tr (W_{[abc]} ρ)$ are shown in Figs. 3(a) and 3(b), respectively. The negativities of the states in Fig. 3(c) increase from $0.1116 \pm 0.0147$ to $0.9477 \pm 0.0095$ with $θ_{1}$ ; the values of $Tr (W_{i j k} ρ)$ and $Tr (W_{[a b c]} ρ)$ decrease from $0.0001 \pm 0.0052$ to $- 0.2262 \pm 0.0041$ and from $0.0190 \pm 0.0134$ to $- 0.9005 \pm 0.0166$ , respectively. This indicates the validity of the proposed extensions of the entanglement witnesses to higher-dimensional bipartite systems.

Figure 3.Experimental results for higher-dimensional bipartite systems. (a) Values of the entanglement witness $Tr (W_{i j k} ρ)$ . (b) $Tr (W_{[a b c]} ρ)$ as functions of the parameter $θ_{1}$ of the states in Eq. (13). The solid curve indicates the theoretical predictions, and symbols are for the experimental results. (c) Logarithmic negativities of the states $E_{N} (ρ)$ versus the state parameter $θ_{1}$ . Error bar indicates the statistical uncertainty which is obtained via the Monte Carlo simulation.

5. CONCLUSION

We have reported an experimental demonstration of a method for construction of entanglement witnesses from a limited fixed set of local measurements ( $M$ ). The method required a classical optimization on the statistics of the results after performing the measurements within $M$ and can be executed without knowing a priori information about the specific forms of the witnesses. Such a method is suitable for a scenario where only a limited local measurement resource (but not a full state tomography) is available. We tested the performance of the method on pure and mixed two-qubit entangled states, which were motivated by realistic experimental conditions, and we confirmed the method worked well for both cases, indicating that even a limited set of local measurements can be used for quick entanglement detection. Possible generalizations to higher-dimensional bipartite systems have been considered in our paper, whereas a similar analysis for the multipartite case still remains open to further study. The experimental accessibility of our method made it suitable for further development of other entanglement witnesses on higher-dimensional systems.

Acknowledgment

Acknowledgment. G. Z. performed the experiments with contributions from C. Z., K. W., and L. X.; P. X. designed the experiments, analyzed the results, and wrote the paper.

APPENDIX A: EXPERIMENTALLY REALIZING NOISY CHANNELS

M_{1} = \sqrt{1 - p} σ_{0}

M_{1} = | 0 ⟩ ⟨ 0 | + \sqrt{1 - p} | 1 ⟩ ⟨ 1 |

APPENDIX B: IMPERFECTIONS IN OUR EXPERIMENT

The systematic shift between experimental results and theoretical predictions is caused by imperfection of state preparation, whereas the error bars are estimated through the statistical uncertainty of the photon numbers. Ideally, we prepare pure two-qubit entangled states and mixed two-qubit entangled states. Mixed entangled states emerge from four types of quantum noisy channels, i.e., Pauli channels, dephasing channels, depolarizing channels, and amplitude damping channels. However, in our experiment, the state preparation is not perfect, and the fidelity of the initial pure entangled state is about 97%, which is obtained by the state tomography.

| Φ^{+} ⟩ = \frac{1}{\sqrt{2}} (| 00 ⟩ + | 11 ⟩),

Figure 4.Concurrence and negativity of the states versus the noisy parameter $p$ of the Pauli channel. The blue line indicates the theoretical predictions obtained by the assumption of the perfect pure initial-state $| Φ^{+} ⟩$ , and the red line indicates the theoretical predictions obtained by the assumption of the mixed initial-state $ρ_{w}$ . Symbols are for the experimental results. Error bar indicates the statistical uncertainty, which is obtained via the Monte Carlo simulation method.

Figure 5.(a) Entanglement witness value as a function of the state parameter $α$ for the pure states and of parameter $p$ of the quantum noisy channels for the mixed states. (b) Negativities and (c) concurrence versus state parameter $α$ or noisy parameter $p$ . The solid curves indicate the theoretical predictions, and the symbols are for the experimental results. Error bar indicates the statistical uncertainty, which is obtained via the Monte Carlo simulation method.

APPENDIX C: HIGHER-DIMENSIONAL BIPARTITE SYSTEMS

Below, we discuss the possible generalizations to higher-dimensional bipartite systems. Similarly, the goal is to detect entanglement of bipartite qudit systems when only a limited fixed set of local measurements can be performed.

d

d \times d

C

| ψ ⟩ ⟨ ψ |

W_{i j k} = {| φ_{i} ⟩}_{j k} {⟨ φ_{i} |}_{j k}^{Γ},

W = D - d P_{d}^{+},

d = 3

d = 3

Figure 6.Experimental setup for two-qutrit systems. The two-qutrit states are generated in the state preparation module, consisting of a set of SPDC entangled photon sources, two BDs, and two HWPs by employing the spatial and polarization modes of the photons. The local measurements and the state tomographic measurements are executed by the measurement modules, consisting of a sequence of HWPs, QWPs, a BD, and a PBS.

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