Experimental Detection of Entanglement Polytopes via Local Filters

Entanglement polytopes result in finitely many types of entanglement that can be detected by only measuring single-particle spectra. With high probability, however, the local spectra lie in more than one polytope, hence providing no information about the entanglement type. To overcome this problem, we propose to additionally use local filters. We experimentally demonstrate the detection of entanglement polytopes in a four-qubit system. Using local filters we can distinguish the entanglement type of states with the same single particle spectra, but which belong to different polytopes.

One central question regarding entanglement is that how |Ψ N may be entangled and how to detect that feature in practice. An obvious fact is that the parameters needed to specify |Ψ N grows exponentially with N. A natural idea to eliminate some of the free parameters is to take that two states |Ψ N and |Φ N have similar entanglement features if they can be connected by some single-qubit operations, for instance, local unitary (LU) transformation [2][3][4] or stochastic local operation combined with classical communication (SLOCC) [5][6][7].
For N = 2, the Schmidt decomposition tells us that |Ψ 2 = λ 0 |00 + λ 1 |11 up to LU, with λ 0 ≥ λ 1 and λ 2 0 + λ 2 1 = 1. Different λ 0 ∈ [1/ √ 2, 1] corresponds to different LU classes of entanglement, which are in fact infinitely many. Up to SLOCC, however, there is only one class of entangled states which contains the EPR pair with λ 0 = 1/ √ 2. For N = 3, up to SLOCC, there are only two types of entanglement: the GHZ-type state and the W -type state [5]. These two types can be distinguished by a quantity called 3-tangle, which is however not a single-copy observable and hence cannot be directly measured in experiment [8] (that is, to get the value of 3-tangle, one either needs to measure jointly on multiple copies of the states, or needs a state tomography [9]).
For any N > 3, SLOCC no longer results in a finite number of entanglement types [6,7]. Despite the efforts of studying SLOCC classification of entanglement for N > 3 systems, the exponential growth of parameters with N for describing |Ψ N makes it hopeless to extract clear physical meanings of these classifications. It is highly desired to coarse-grain these classes such that we can grasp the key features of each entanglement type. The concept of entanglement polytopes provides an elegant idea to meet this need [10], where for each N there exists only finite number of types. More importantly, the polytopes are directly detectable in experiments via measuring only single-particle spectra of each qubit [10][11][12].
In this work, we experimentally demonstrate the detection of entanglement polytopes in a four-qubit system. Unfortunately, it turns out that different entanglement polytopes form a nested hierarchy [10][11][12], and they may have a large overlap. If the vector of local spectra of a state |Ψ N lies in an overlapping region, then we cannot uniquely identify the polytope that |Ψ N belongs to (see e.g. a recent experiment in which the states are chosen to be in non-overlapping regions [13]).
It turns out that for a randomly chosen three-qubit pure state |Ψ 3 , the probability that the vector of local spectra of |Ψ 3 lies in the overlapping region of the W and GHZ polytopes is ≈ 94%. In general, for a randomly chosen state N-qubit state |Ψ N , with high probability the vector of local spectra falls in some overlapping region of polytopes (we include a more detailed discussion of these probabilities in the Appendix). To overcome this difficulty, we use local filters (see, e.g. [14][15][16][17][18]) to effectively distinguish states with the same single particle spectra, but which belong to different polytopes.

Entanglement Polytopes -
This SLOCC equivalence relation partitions all N-qubit pure states into SLOCC equivalent classes, called the SLOCC orbits.
For an N-qubit state |Ψ N , each of the single-particle reduced density matrices ρ i for i = 1, . . . , N has two eigenvalues λ α i , λ β i that are normalized, i.e. λ α i + λ β i = 1. It suffices to consider the maximum eigenvalue of ρ i , i.e. λ max . . , λ max N ) then corresponds to a point in R N . It has been shown that for the closure of an N-qubit SLOCC orbitŌ of Ψ N (i.e. some M i in (1) possibly non-invertible), all the points λ of all N i=1 M i |Ψ N ∈Ō form a polytope in R N , called the entanglement polytope ofŌ. Moreover, for any finite N, there are only finitely many polytopes. This provides a natural classification of entanglement for N-qubit states, which 'coarse-grains' the infinitely many SLOCC orbits (for N > 3). Since the entanglement polytope for an SLOCC orbit is fully determined by the local spectra of the states in the orbit, this offers an appealing experimental approach for identifying the entanglement type for an N-qubit system, for which only measurements on single particles are required.
In practice, for a state |Ψ N , while a point λ may clearly distinguish its entanglement type, if λ is in an overlapping region of two polytopes, we fail to get information on which entanglement type the state belongs to. In the N = 3 case, for instance, this means that a point λ ∈ P W fails to distinguish the W -type entanglement from the GHZ-type entanglement. Unfortunately, for a randomly chosen pure state of three qubits, with ≈ 94% probability the corresponding λ falls into P W .
Luckily, one can apply local filter operations N i=1 M i to the system to 'move around' λ , with the hope that λ ends up in a non-overlapping area of polytopes. As demonstrated in the three-qubit case, this step becomes crucial in practice when using the polytope method for detecting entanglement types, as the probability of overlapping is high.
Four-qubit Polytopes -Our experiments demonstrate the detection of entanglement types of four-qubit states. In this case, there are infinitely many SLOCC orbits. The full poly-tope, containing λ for any four-qubit state, denoted by P full , is spanned by the vertices Up to permutation of the qubits, there are 6 other polytopes inside P full , which may also be mutually overlapping. Similar as in the N = 3 case, for a randomly chosen four-qubit state, the chance that λ lies in an overlapping region is high (for details, see the Appendix). Therefore, we have to apply local operations to 'move around' λ . The proposed experiment is given by the diagram in Fig. 1. Here ϑ i for i = 1, 2 denotes a unitary local transformation U ϑ i of the form and γ denotes a non-unitary local transformation Fig. 1, two of the qubits encounter non-unitary local transformations: qubit 1 is measured in some basis and postselected, resulting in λ max 1 = 1; qubit 4 is going through a filter operation given by A γ . In the most general case, one can also apply local filter operations (or measurements) on the other qubits. However, a single filter (or measurement) may already suffice to 'move around' λ to non-overlapping regions of the polytopes, depending on the input state |Ψ (i) .
Experimental setup -In our experiments, two different four-qubit states |Ψ (1) and |Ψ (2) are prepared, where and |Ψ (2) is the four-qubit GHZ state The qubits are encoded by horizontal |H and vertical |V polarization. The goal is to determine the entanglement type for each of the input state using the polytope method. For both |Ψ (1) and |Ψ (2) , we have λ = ( 1 2 , 1 2 , 1 2 , 1 2 ). That is, local spectra do not tell them apart, hence local filter operations are needed to 'move around' λ .
The SLOCC orbit of the four-qubit GHZ state |Ψ (2) correspond to the full polytope P full . However, the state |Ψ (1) corresponds to a smaller polytope P s ⊂ P full with vertices The smaller polytope P s is characterized by the additional constraint and all permutations of it. Our experimental setup for the states |Ψ (1) and |Ψ (2) is shown in Fig. 2. A 390 nm femto-second pump light, frequency-doubled from a 780 nm mode-locked Ti:sapphire pulsed laser (with the pulse width about 150 fs and repetition rate 76 MHz) was used to pump the respective downconverter. For the preparation of |Ψ (1) , a 2 mm type-II phasematched BBO crystal is used as down-converter to produce two pairs of entangled photons [19], and two 1 mm BBO crystals are used to compensate the birefringence of o-light and e-light in the 2 mm BBO. HWP1 rotates the polarization of the photons in path '2' (horizontal to vertical and vertical to horizontal). Then after the beam splitters (BS), the above two pairs of entangled photons are transformed into the state |Ψ (1) . In mode '6', we use two beam displacers and three half wave plates (HWPs, HWP3 is used for balancing the optical length of the two beams between partdisplacers) to construct the local filter A γ . For the four-qubit GHZ state |Ψ (2) shown in the right part of Fig. 2, a cascaded sandwich beamlike BBO entangled source [20] is used. A PBS combines the photons from mode '1' and '2'. We will get the four-qubit GHZ state |Ψ (2) if there is one photon in each of the modes '3', '4', '5', and '6' [21].
Results -We first perform full quantum state tomography to reconstruct the density matrix of |Ψ (1) and |Ψ (2) , the fidelity of which are 0.9422 ± 0.0036 and 0.9001 ± 0.0038. Then we collect data from each mode to obtain the corresponding single-qubit density matrix and calculate their local spectra for both states. As shown in Table I, the local spectra of |Ψ (1) and |Ψ (2) are almost identical, so we can not distinguish their entanglement polytopes.
To distinguish the entanglement polytopes of |Ψ (1) and state λ max  together with f ( λ ) for the states Ψ (1) and Ψ (2) . The uncertainties inside the brackets are obtained by Monte Carlo simulation (1000 steps).  |Ψ (2) , we then try to move λ out of the smaller polytope P s using local filters, as illustrated in Fig. 1. We fix ϑ 2 = −π/8, and then measure the first qubit in the computational basis. By post-selection we have λ max 1 = 1. For each setting of ϑ 1 and γ, we perform tomography of the qubits 2, 3, and 4 to determine the values of λ max 2 , λ max 3 , and λ max 4 (see Table II). The smaller polytope P s is characterized by f ( λ ) ≥ 1.  (1) and |Ψ (2) respectively. Experimental data 'a ∼ e' is for |Ψ (1) while ' f ' is for |Ψ (2) . Error bars are too small to identify (see Table II).
The results are illustrated in Fig. 3 and 4. In Fig. 3, the data is shown in the three-dimensional polytope for λ max 2 , λ max 3 , λ max 1 = 1. The smaller polytope P s be-  Table II).
Since the first photon of the four-qubit state is post-selected and the last photon goes through a non-unitary filter, the probability of succes for the experiment is 0.2917, 0.2222, 0.1667, 0.1768, 0.5, and 0.5 for our experimental data 'a ∼ f ,' respectively.
Because the birefringence of the o-light and the e-light in the BBO (down-converter) cannot be compensated completely, and because of the high-term noise from the SPDC process and some mode mismatch, we do not obtain the pure states |Ψ (1) and |Ψ (2) , but some noisy version of them. Nonetheless, the errors in our experiment are mainly due to the time uncertainty of the photon pairs generated in the BBO. The coincidence counts obey a Poisson distribution, the parameters of which we estimate from the experimental data. Then we perform Monte Carlo simulation to estimate the errors indicated in Tables I and II.
Summary -We experimentally demonstrate the detection of entanglement polytopes in a four-qubit system. We use local filters to effectively distinguish states with the same singleparticle spectra, but which belong to different polytopes. This provides a new tool to experimental detection of entanglement in a multi-qubit system using only local operations.

A. Entanglement polytopes
Two quantum states |ψ 1 and |ψ 2 are said to be equivalent with respect to SLOCC if there exists a sequence of local operations and classical communication that converts the state |ψ 1 into |ψ 2 with non-zero probability p 1→2 > 0, and another protocol for the conversion of |ψ 2 into |ψ 1 that succeeds with probability p 2→1 > 0. As we only require the success probabilities to be non-zero, it is sufficient to consider one branch of the protocol that has non-zero success probability. Thus we can, for example, where the matrices M i correspond to the combination of all operations performed on particle i. In [22] it was shown that the matrices M i in (5) Results on the classification of pure four-qubit states with respect to SLOCC can be found in [23][24][25][26]. Note that in the literature, sometimes the scaling factor λ is incorrectly ignored. Unlikely the situation for local unitary transformations, polynomial invariants of the group SL(d) ⊗n yield only a necessary condition for SLOCC equivalence of two quantum states.
Proposition 1 Let f 1 , . . . , f m be homogeneous polynomial invariants of the group SL(d) ⊗n . If the normalized states |ψ 1 and |ψ 2 are in the same SLOCC class, then there exists a non-zero constant λ ∈ C such that In the case of four qubits, we have four polynomial invariants B 0000 , D 0000 , and F 0000 of degree 2, 4, 4, and 6, respectively, see [27].
In order to get a necessary and sufficient criterion to decide SLOCC equivalence, one may consider covariants. Two vectors are in the same orbit of the group SL(d) ⊗n if and only if all covariants agree. Again, one has to take care of the scaling parameter λ to apply this criterion. In the case of four qubits, there are 170 covariants, see [28].
It has been shown that the points corresponding to the (sorted) spectra of the single-particle reduced density matrices of pure quantum states in the closure of an orbit under SLOCC transformations form a so-called entanglement polytope, see [29,30]. The vertices of the polytope correspond to the covariants that do not vanish.

B. Four-qubit Polytopes
In the case of four qubits, there are 7 different 4-dimensional polytopes up to permutation of the qubits [29]. Lower-dimensional polytopes correspond to states that partially factorize. When we also consider the permutations, the polytopes P 3 and P 6 come in 6 different versions, while the polytopes P 1 and P 2 split into 4 different subtypes. The vertices of all polytopes are listed in Table III. The polytope P 5 is contained in all other polytopes, and the polytope P 7 = P full is the largest polytope containing all states. Fig. 5 illustrates how the polytopes are contained in each other.
In the experiment, we investigate the four-qubit state and the four-qubit GHZ state The qubits are encoded by horizontal |H and vertical |V polarization. Evaluating the covariants from [28] for the states |Ψ (1) and |Ψ (2) we find that the corresponding polytopes are P s = P 4 and the full polytope P full = P 7 , respectively. The polytope P 4 is obtained from P 7 by removing the vertex (1/2, 1/2, 1/2, 1) and all its permutations. The discriminating inequalities are and all its permutations.

C. Volume of the Polytopes
In the case of three qubits, we have only two three-dimensional polytopes P W ⊂ P GHZ corresponding to the SLOCC class containing the W -state and the GHZ-state, respectively. Picking a pure three-qubit state with respect to the Haar measure at random, the resulting distribution of the eigenvalues of the local density matrices has been computed in [31]. From this one finds that the volume of the sub-polytope P W is 203/216 ≈ 93.98%. Hence the probability for a random three-qubit state to have a local spectra corresponding to a point outside the polytope P W is only 13/216 ≈ 6.02%. For four qubits, we computed the local spectra of 10 6 random pure states and determined which of the polytopes contains the vector of local spectra. The results are summarized in Table IV. While the polytope P 4 corresponding to the state |Ψ (1) of our experiment is fairly low in the hierarchy of polytopes (see Fig. 5), the local spectra of only 9522 out of one million random states violate the discriminating inequalities (10). Hence, the chance for a random four-qubit state to have a local spectrum that lies outside of P 4 is only about 0.95%. This clearly indicates that one has to apply local filters in order to get information about the entanglement polytopes.