Abstract
Recently, there are tremendous developments on the number of controllable qubits in several quantum computing systems. For these implementations, it is crucial to determine the entanglement structure of the prepared multipartite quantum state as a basis for further information processing tasks. In reality, evaluation of a multipartite state is in general a very challenging task owing to the exponential increase of the Hilbert space with respect to the number of system components. In this work, we propose a systematic method using very few local measurements to detect multipartite entanglement structures based on the graph state—one of the most important classes of quantum states for quantum information processing. Thanks to the close connection between the Schmidt coefficient and quantum entropy in graph states, we develop a family of efficient witness operators to detect the entanglement between subsystems under any partitions and hence the entanglement intactness. We show that the number of local measurements equals to the chromatic number of the underlying graph, which is a constant number, independent of the number of qubits. In reality, the optimization problem involved in the witnesses can be challenging with large system size. For several widely used graph states, such as 1D and 2D cluster states and the Greenberger–Horne–Zeilinger state, by taking advantage of the area law of entanglement entropy, we derive analytical solutions for the witnesses, which only employ two local measurements. Our method offers a standard tool for entanglementstructure detection to benchmark multipartite quantum systems.
Introduction
Entanglement is an essential resource for many quantum information tasks,^{1} such as quantum teleportation,^{2} quantum cryptography,^{3,4} nonlocality test,^{5} quantum computing,^{6} quantum simulation,^{7} and quantum metrology.^{8,9} Tremendous efforts have been devoted to the realization of multipartite entanglement in various systems,^{10,11,12,13,14,15,16,17,18,19,20} which provide the foundation for small and mediumscale quantum information processing in near future and will eventually pave the way to universal quantum computing. In order to build up a quantum computing device, it is crucial to first witness multipartite entanglement. So far, genuine multipartite entanglement has been demonstrated and witnessed in experiment with a small amount of qubits in different realizations, such as 14iontrapqubit,^{10} 12superconductingqubit,^{14} and 12photonqubit systems.^{17}
In practical quantum hardware, the unavoidable coupling to the environment undermines the fidelity between the prepared state and the target one. Taking the Greenberger–Horne–Zeilinger (GHZ) state for example, the stateoftheart 10superconductingqubit^{13} and the 12photon^{17} preparations only achieve the fidelity of 66.8% and 57.2%, respectively, which just exceed the threshold 50% for the certification of genuine entanglement. As the system size becomes larger, see for instance, Google’s a 72qubit chip (https://www.sciencenews.org/article/googlemovestowardquantumsupremacy72qubitcomputer) and IonQ’s a 79qubit system (https://physicsworld.com/a/ionbasedcommercialquantumcomputerisafirst/), it is an experimental challenge to create genuine multipartite entanglement. Nonetheless, even without global genuine entanglement as the target state possesses, the experimental prepared state might still have fewerbody entanglement within a subsystem and/or among distinct subsystems.^{21,22,23} The study of lowerorder entanglement, which can be characterized by the detailed entanglement structures,^{24,25,26} is important for quantum hardware development, because it might reveal the information on unwanted couplings to the environment and acts as a benchmark of the underlying system. Moreover, the certified lowerorder entanglement among several subsystems could be still useful for some quantum information tasks.
Considering an Npartite quantum system and its partition into m subsystems (m ≤ N), the entanglement structure indicates how the subsystems are entangled with each other. Each subsystem corresponds to a subset of the whole quantum system. For instance, we can choose each subsystem to be each party (i.e., m = N), and then the entanglement structure indicates the entanglement between the N parties. In some specific systems, such as distributed quantum computing,^{27} quantum networks^{28} or atoms in a lattice, the geometric configuration can naturally determine the system partition (see Fig. 1 for an illustration). In other cases, one might not need to specify the partition in the beginning. By going through all possible partitions, one can investigate higher level entanglement structures, such as entanglement intactness (nonseparability),^{23,26} which quantifies how many pieces in the Npartite state are separated.
Multipartite entanglementstructure detection is generally a challenging task. Naively, one can perform state tomography on the system. As the system size increases, tomography becomes infeasible due to the exponential increase of the Hilbert space. Entanglement witness,^{29,30,31} on the other hand, provides an elegant solution to multipartite entanglement detection. In literature, various witness operators have been proposed to detect different types of quantum states, generally requiring a polynomial number of measurements with respect to the system size.^{32,33} Interestingly, a constant number of local measurement settings are shown to be sufficient to detect genuine entanglement for stabilizer states.^{34,35} Compared with genuine entanglement, multipartite entanglement structure still lacks a systematic exploration, due to the rich and complex structures of Npartite system. Recently, positive results have been achieved for detecting entanglement structures of GHZlike states with two measurement settings^{26} and the entanglement of a specific 1D cluster state of the 16qubit superconducting quantum processor ibmqx5 machine from the IBM cloud.^{36} Unfortunately, it remains an open problem of efficient entanglementstructure detection of general multipartite quantum states.
In this work, we propose a systematic method to witness the entanglement structure based on graph states. Note that the graph state^{37,38} is one of the most important classes of multipartite states for quantum information processing, such as measurementbased quantum computing,^{39,40} quantum routing and quantum networks,^{28} quantum error correction,^{41} and Bell nonlocality test.^{42} It is also related to the symmetryprotected topological order in condensed matter physics.^{43} Typical graph states include cluster states, GHZ state, and the states involved in the encoding process of the 5qubit Steane code and the concatenated [7,1,3]CSScode.^{38}
The main idea of our entanglementstructure detection method runs as follows. First, with the close connection between the maximal Schmidt coefficient and quantum entropy, we upperbound the fidelity of fully and biseparable states. These bounds are directly related to the entanglement entropy of the underlying graph state with respect to certain bipartition. Then, inspired by the genuine entanglement detection method,^{34} we lowerbound the fidelity between the unknown prepared state and the target graph state, with local measurements corresponding to the stabilizer operators of the graph state. Finally, by comparing theses fidelity bounds, we can witness the entanglement structures, such as the (genuine multipartite) entanglement between any subsystem partitions, and hence the entanglement intactness.
Our detection method for entanglement structures based on graph states is presented in Theorems 1 and 2, which only involves k local measurements. Here, k is the chromatic number of the corresponding graph, typically, a small constant independent of the number of qubits. For several common graph states, 1D and 2D cluster states and the GHZ state, we construct witnesses with only k = 2 local measurement settings, and derive analytical solutions to the optimization problem. These results are shown in Corollaries 1–4. The proofs of propositions and theorems are left in Methods, and the proofs of Corollaries 1–4 are presented in Supplementary Methods 1–4.
Results
Definitions of multipartite entanglement structure
Let us start with the definitions of multipartite entanglement structure. Considering an Nqubit quantum system in a Hilbert space \({\cal{H}} = {\cal{H}}_{2}^{ \otimes N}\), one can partition the whole system into m nonempty disjoint subsystems A_{i}, i.e., \(\{ N\} \equiv \{ 1,2, \ldots ,N\} = \mathop {\bigcup}\nolimits_{i = 1}^{m} {A_i}\) with \({\cal{H}} = \mathop { \otimes }\nolimits_{i = 1}^{m} {\cal{H}}_{A_i}\). Denote this partition to be \({\cal{P}}_{m} = \{ A_{i}\}\) and omit the index m when it is clear from the context. Similar to definitions of regular separable states, here, we define fully and biseparable states with respect to a specific partition \({\cal{P}}_{m}\) as follows.
Definition 1
An Nqubit pure state, \(\left \mathrm{\Psi}_{f} \right\rangle \in \mathcal{H}\), is \(\mathcal{P}\)fully separable, iff it can be written as
An Nqubit mixed state ρ_{f} is \(\cal{P}\)fully separable, iff it can be decomposed into a convex mixture of \(\cal{P}\)fully separable pure states
with p_{i} ≥ 0, ∀i and \(\mathop {\sum}\nolimits_i {p_i} = 1\).
Denote the set of \(\cal{P}\)fully separable states to be \(S_{f}^{\cal{P}}\). Thus, if one can confirm that a state \(\rho \ \notin \ S_{f}^{\cal{P}}\), the state ρ should possess entanglement between the subsystems {A_{i}}. Such kind of entanglement could be weak though, since it only requires at least two subsystems to be entangled. For instance, the state \(\left {\mathrm{\Psi }} \right\rangle = \left {{\mathrm{\Psi }}_{A_{1}A_{2}}} \right\rangle \otimes \mathop {\prod}\nolimits_{i = 3}^{m} {\left {{\mathrm{\Psi }}_{A_{i}}} \right\rangle }\) is called entangled nevertheless only with entanglement between A_{1} and A_{2}. It is interesting to study the states where all the subsystems are genuinely entangled with each other. Below, we define this genuine entangled state via \(\cal{P}\)biseparable states.
Definition 2
An Nqubit pure state, \(\left {{\mathrm{\Psi }}_s} \right\rangle \in \cal{H}\), is \(\cal{P}\)biseparable, iff there exists a subsystem bipartition \(\{ A,\bar A\}\), where \(A = \mathop {\bigcup}\nolimits_i {A_i}\), \(\bar A = \{ N\} /A \ \ne \ \emptyset\), the state can be written as,
An Nqubit mixed state ρ_{b} is \(\cal{P}\)biseparable, iff it can be decomposed into a convex mixture of \(\cal{P}\)biseparable pure states,
with p_{i} ≥ 0, ∀i and \(\mathop {\sum}\nolimits_i {p_i} = 1\), and each state \(\left {{\mathrm{\Psi }}_b^i} \right\rangle\) can have different bipartitions.
Denote the set of biseparable states to be \(S_{b}^{\cal{P}}\). It is not hard to see that \(S_{f}^{\cal{P}} \subset S_{b}^{\cal{P}}\).
Definition 3
A state ρ possesses \(\cal{P}\)genuine entanglement iff \(\rho \ \notin \ S_{b}^{\cal{P}}\).
The three entanglementstructure definitions of \(\cal{P}\)fully separable, \(\cal{P}\)biseparable, and \(\cal{P}\)genuinely entangled states can be viewed as generalized versions of regular fully separable, biseparable, and genuinely entangled states, respectively. In fact, when m = N, these pairs of definitions are the same.
Following the conventional definitions, a pure state Ψ_{m}〉 is mseparable if there exists a partition \({\cal{P}}_m\), the state can be written in the form of Eq. (1). The mseparable state set, S_{m}, contains all the convex mixtures of the mseparable pure states, \(\rho _{m} = \mathop {\sum}\nolimits_{i} {p_{i}} \left {{\mathrm{\Psi }}_{m}^{i}} \right\rangle \left\langle {{\mathrm{\Psi }}_{m}^{i}} \right\), where the partition for each term \(\left {{\mathrm{\Psi }}_m^i} \right\rangle\) needs not to be same. It is not hard to see that S_{m+1} ⊂ S_{m}. Meanwhile, define the entanglement intactness of a state ρ to be m, iff ρ ∉ S_{m+1} and ρ ∈ S_{m}. Thus, as ρ ∉ S_{m+1}, the intactness is at most m, i.e., the nonseparability can serve as an upper bound of the intactness. When the entanglement intactness is 1, the state is genuinely entangled; and when the intactness is N, the state is fully separable. See Fig. 2 for the relationships among these definitions.
By definitions, one can see that if a state is \({\cal{P}}_m\)fully separable, it must be mseparable. Of course, an mseparable state might not be \({\cal{P}}_m\)fully separable, for example, if the partition is not properly chosen. In experiment, it is important to identify the partition under which the system is fully separated. With the partition information, one can quickly identify the links where entanglement is broken. Moreover, for some systems, such as distributed quantum computing, multiple quantum processor, and quantum network, natural partition exists due to the system geometric configuration. Therefore, it is practically interesting to study entanglement structure under partitions.
Entanglementstructure detection method
Let us first recap the basics of graph states and the stabilizer formalism.^{37,38} In a graph, denoted by G = (V, E), there are a vertex set V = {N} and a edge set E ⊂ [V]^{2}. Two vertexes i, j are called neighbors if there is an edge (i, j) connecting them. The set of neighbors of the vertex i is denoted as N_{i}. A graph state is defined on a graph G, where the vertexes represent the qubits initialized in the state of \(\left + \right\rangle = (\left 0 \right\rangle + \left 1 \right\rangle )/\sqrt 2\) and the edges represent a ControlledZ (CZ) operation, \({\mathrm{CZ}}^{\{ i,j\} } = \left 0 \right\rangle _i\left\langle 0 \right \otimes {\mathbb{I}}_j + \left 1 \right\rangle _i\left\langle 1 \right \otimes Z_j\), between the two neighbor qubits. Then the graph state can be written as,
Denote the Pauli operators on qubit i to be X_{i}, Y_{i}, Z_{i}. An Npartite graph state can also be uniquely determined by N independent stabilizers,
which commute with each other and S_{i}G〉 = G〉, ∀i. That is, the graph state is the unique eigenstate with eigenvalue of +1 for all the N stabilizers. Here, S_{i} contains identity operators for all the qubits that do not appear in Eq. (6). As a result, a graph state can be written as a product of stabilizer projectors,
The fidelity between ρ and a graph state G〉 can be obtained from measuring all possible products of stabilizers. However, as there are exponential terms in Eq. (7), this process is generally inefficient for large systems. Hereafter, we consider the connected graph, since its corresponding graph state is genuinely entangled.
Now, we propose a systematic method to detect entanglement structures based on graph states. First, we give fidelity bounds between separable states and graph states as the following proposition.
Proposition 1
Given a graph state G〉 and a partition \({\cal{P}} = \{ A_{i}\}\), the fidelity between G〉 and any \(\cal{P}\)fully separable state is upper bounded by
and the fidelity between G〉 and any \(\cal{P}\)biseparable state is upper bounded by
where \(\{ A,\bar A\}\) is a bipartition of {A_{i}}, and S(ρ_{A}) = −Tr[ρ_{A} log_{2} ρ_{A}] is the von Neumann entropy of the reduced density matrix \(\rho _A = {\mathrm{Tr}}_{\bar A}(\left G \right\rangle \left\langle G \right)\).
The bound in Eq. (9) is tight, i.e., there always exists a \(\cal{P}\)biseparable state to saturate it. The bound in Eq. (8) may not be tight for some partition \(\cal{P} = \{ A_{\it{i}}\}\) and some graph state G〉. In addition, we remark that to extend Theorem 1 from the graph state to a general state Ψ〉, one should substitute the entropy in the bounds of Eqs. (8) and (9) with the minentropy S_{∞}(ρ_{A}) = −logλ_{1} with λ_{1} the largest eigenvalue of \(\rho _A = {\mathrm{Tr}}_{\bar A}(\left \Psi \right\rangle \left\langle \Psi \right)\).
Next, we propose an efficient method to lowerbound the fidelity between an unknown prepared state and the target graph state. A graph is kcolorable if one can divide the vertex set into k disjoint subsets \({\bigcup} {V_l} = V\) such that any two vertexes in the same subset are not connected. The smallest number k is called the chromatic number of the graph. (Note that the colorability is a property of the graph (not the state), one may reduce the number of measurement settings by local Clifford operations.^{38}) We define the stabilizer projector of each subset V_{l} as
where S_{i} is the stabilizer of G〉 in subset V_{l}. The expectation value of each P_{l} can be obtained by one local measurement setting \(\mathop { \otimes }\nolimits_{i \in V_l} X_i\mathop { \otimes }\nolimits_{j \in V/V_l} Z_j\). Then, we can propose a fidelity evaluation scheme with k local measurement settings, as the following proposition.
Proposition 2
For a graph state \(\left G \right\rangle \left\langle G \right\) and the projectors P_{l} defined in Eq. ( 10 ), the following inequality holds,
where A ≥ B indicates that (A − B) is positive semidefinite.
Note that Proposition 2 with k = 2 case has also been studied in literature.^{34} Combining Propositions 1 and 2, we propose entanglementstructure witnesses with k local measurement settings, as presented in the following theorem.
Theorem 1
Given a partition \({\cal{P}} = \{ A_{i}\}\), the operator \(W_{f}^{\cal{P}}\) can witness non\(\cal{P}\)fully separability (entanglement),
with \(\langle W_{f}^{\cal{P}}\rangle \ge 0\) for all \(\cal{P}\)fully separable states; and the operator \(W_{b}^{\cal{P}}\) can witness \(\cal{P}\)genuine entanglement,
with \(\langle W_{b}^{\cal{P}}\rangle \ge 0\) for all \(\cal{P}\)biseparable states, where \(\{ A,\bar A\}\) is a bipartition of {A_{i}}, \(\rho _{A} = {\mathrm{Tr}}_{\bar A}(\left G \right\rangle \left\langle G \right)\), and the projectors P_{l} is defined in Eq. (10).
The proposed entanglementstructure witnesses have several favorable features. First, given an underlying graph state, the implementation of the witnesses is the same for different partitions. This feature allows us to study different entanglement structures in one experiment. Note that the witness operators in Eqs. (12) and (13) can be divided into two parts: The measurement results of P_{l} obtained from the experiment rely on the prepared unknown state and are independent of the partition; The bounds, \(1 + \min {\mkern 1mu} (\max )_{\{ A,\bar A\} }2^{  S(\rho _A)}\), on the other hand, rely on the partition and are independent of the experiment. Hence, in the data postprocessing of the measurement results of P_{l}, we can study various entanglement structures for different partitions by calculating the corresponding bounds analytically or numerically.
Second, besides investigating the entanglement structure among all the subsystems, one can also employ the same experimental setting to study that of a subset of the subsystems, by performing different data postprocessing. For example, suppose the graph G is partitioned into three parts, say A_{1}, A_{2}, and A_{3}, and only the entanglement between subsystems A_{1} and A_{2} is of interest. One can construct new witness operators with projectors \(P_{l}^{\prime}\), by replacing all the Pauli operators on the qubits in A_{3} in Eq. (10) to identities. Such measurement results can be obtained by processing the measurement results of the original P_{l}. Then the entanglement between A_{1} and A_{2} can be detected via Theorem 1 with projectors \(P_{l}^{\prime}\) and the corresponding bounds of the graph state \(\left {G_{A_{1}A_{2}}} \right\rangle\). Details are discussed in Supplementary Note 1.
Third, when each subsystem A_{i} contains only one qubit, that is, m = N, the witnesses in Theorem 1 become the conventional ones. It turns out that Eq. (13) is the same for all the graph states under the Npartition \({\cal{P}}_{N}\), as shown in the following corollary. Note that, a special case of the corollary, the genuine entanglement witness for the GHZ and 1D cluster states, has been studied in literature.^{34}
Corollary 1
The operator \(W_{b}^{{\cal{P}}_{N}}\) can witness genuine multipartite entanglement,
with \(\langle W_{b}^{{\cal{P}}_{N}}\rangle \ge 0\) for all biseparable states, where P_{l} is defined in Eq. (10) for any graph state.
Fourth, the witness in Eq. (12) can be naturally extended to identify nonmseparability, by investigating all possible partitions \({\cal{P}}_{m}\) with fixed m. In fact, according to the definition of mseparable states and Eq. (8), the fidelity between any mseparable state ρ_{m} and the graph state G〉 can be upper bounded by \({\mathrm{max}}_{{\cal{P}}_{m}}{\mathrm{min}}_{\{ A,\bar{A}\} }2^{  S(\rho _{A})}\), where the maximization is over all possible partitions with m subsystems. As a result, we have the following theorem on the nonmseparability.
Theorem 2
The operator W_{m} can witness nonmseparability,
with 〈W_{m}〉 ≥ 0 for all mseparable states, where the maximization takes over all possible partitions \({\cal{P}}_{m}\) with m subsystems, the minimization takes over all bipartition of \({\cal{P}}_{m}\), \(\rho _A = {\mathrm{Tr}}_{\bar A}(\left G \right\rangle \left\langle G \right)\), and the projector P_{l} is defined in Eq. (10).
The robustness analysis of the witnesses proposed in Theorems 1 and 2 under the white noise is presented in Methods. It shows that our entanglementstructure witnesses are quite robust to noise. Moreover, the optimization problems in Theorems 1 and 2 are generally hard, since there are exponentially many different possible partitions. Surprisingly, for several widely used types of graph states, such as 1D, 2D cluster states, and the GHZ state, we find the analytical solutions to the optimization problem, as shown in the following section.
Applications to several typical graph states
In this section, we apply the general entanglement detection method proposed above to several typical graph states, 1D, 2D cluster states, and the GHZ state. Note that for these states the corresponding graphs are all 2colorable. Thus, we can realize the witnesses with only two local measurement settings. For clearness, the vertexes in the subsets V_{1} and V_{2} are associated with red and blue colors respectively, as shown in Fig. 3. We write the stabilizer projectors defined in Eq. (10) for the two subsets as,
The more general case with kchromatic graph states is presented in Supplementary Note 1.
We start with a 1D cluster state C_{1}〉 with stabilizer projectors in the form of Eq. (16). Consider an example of tripartition \({\cal{P}}_{3} = \{ A_{1},A_{2},A_{3}\}\), as shown in Fig. 3a, there are three ways to divide the three subsystems into two sets, i.e., \(\{ A,\bar A\}\) = {A_{1}, A_{2}A_{3}}, {A_{2}, A_{1}A_{3}}, {A_{3}, A_{1}A_{2}}. It is not hard to see that the corresponding entanglement entropies are \(S(\rho _{A_{1}}) = S(\rho _{A_{3}}) = 1\) and \(S(\rho _{A_{2}}) = 2\). Note that in the calculation, each broken edge will contribute 1 to the entropy, which is a manifest of the area law of entanglement entropy.^{44} According to Theorem 1, the operators to witness \({\cal{P}}_{3}\)entanglement structure are given by,
where the two projectors P_{1} and P_{2} are defined in Eq. (16) with the graph of Fig. 3a.
Next, we take an example of 2D cluster state C_{2}〉 defined in a 5 × 5 lattice and consider a tripartition, as shown in Fig. 3b. Similar to the 1D cluster state case with the area law, the corresponding entanglement entropies are \(S(\rho _{A_{1}}) = S(\rho _{A_{3}}) = 5\) and \(S(\rho _{A_{2}}) = 4\). According to Theorem 1, the operators to witness \({\cal{P}}_{3}\)entanglement structure are given by,
where the two projectors P_{1} and P_{2} are defined in Eq. (16) with the graph of Fig. 3b. Similar analysis works for other partitions and other graph states.
Now, we consider the case where each subsystem A_{i} contains exactly one qubit, \({\cal{P}}_{N}\). Then, witnesses in Eq. (12) become the conventional ones, as shown in the following Corollary.
Corollary 2
The operator \(W_{f,C}^{{\cal{P}}_{N}}\) can witness nonfully separability (entanglement),
with \(\langle W_{f,C}^{{\cal{P}}_{N}}\rangle \ge 0\) for all fully separable states, where the two projectors P_{1} and P_{2} are defined in Eq. (16) with the stabilizers of any 1D or 2D cluster state.
Here, we only show the cases of 1D and 2D cluster states. We conjecture that the witness holds for any (such as 3D) cluster states. For a general graph state, on the other hand, the corollary does not hold. In fact, we have a counter example of the GHZ state shown in Fig. 3c. It is not hard to see that for any GHZ state, the entanglement entropy is given by,
Then, Eqs. (12) and (13) yield the same witnesses. That is, the witness constructed by Theorem 1 for the GHZ state can only tell genuine entanglement or not.
Following Theorem 2, one can fix the number of the subsystems m and investigate all possible partitions to detect the nonmseparability. The optimization problem can be solved analytically for the 1D and 2D cluster states, as shown in Corollary 3 and 4, respectively.
Corollary 3
The operator \(W_{m,C_{\mathrm{1}}}\) can witness nonmseparability,
with \(\langle W_{m,C_1}\rangle \ge 0\) for all mseparable states, where the two projectors P_{1} and P_{2} are defined in Eq. (16) with the stabilizers of a 1D cluster state.
In particular, when m = 2 and m = N, \(W_{m,C_{\mathrm{1}}}\) becomes the entanglement witnesses in Eqs. (14) and (19), respectively.
Corollary 4
The operator \(W_{m,C_{\mathrm{2}}}\) can witness nonmseparability for N ≥ m(m − 1)/2,
with \(\langle W_{m,C_2}\rangle \ge 0\) for all mseparable states, where the two projectors P_{1} and P_{2} are defined in Eq. (16) with the stabilizers of a 2D cluster state.
We remark that the witnesses constructed in Corollaries 1, 2, and 3 are tight. Take the witness \(W_{m,C_{\mathrm{1}}}\) in Corollary 3 as an example. There exists an mseparable state ρ_{m} that saturates \({\mathrm{Tr}}(\rho _mW_{m,C_1}) = 0\). In addition, as m ≤ 5, the witness \(W_{m,C_{\mathrm{2}}}\) in Corollary 4 is also tight. Detailed discussions are presented in Supplementary Methods 1–4.
Discussion
In this work, we propose a systematic method to construct efficient witnesses to detect entanglement structures based on graph states. Our method offers a standard tool for entanglementstructure detection and multipartite quantum system benchmarking. The entanglementstructure definitions and the associated witness method may further help to detect novel quantum phases, by investigating the entanglement properties of the ground states of related Hamiltonians.^{43}
The witnesses proposed in this work can be directly generalized to stabilizer states,^{6,45} which are equivalent to graph states up to local Clifford operations.^{38} It is interesting to extend the method to more general multipartite quantum states, such as the hypergraph state^{46} and the tensor network state.^{47} Meanwhile, the generalization to the neural network state^{48} is also intriguing, since this kind of ansatz is able to represent broader quantum states with a volume law of entanglement entropy,^{49} and is a fundamental block for potential artificial intelligence applications. In addition, one may utilize the proposed witness method to detect other multipartite entanglement properties, such as the entanglement depth and width,^{50,51} as mseparability in this work. Moreover, one can also consider the selftesting scenario, such as (measurement) deviceindependent settings,^{52,53,54} which can help to manifest the entanglement structures with less assumptions on the devices. Furthermore, translating the proposed entanglement witnesses into a probabilistic scheme is also interesting.^{55,56}
Methods
Proof of Proposition 1
Proof. First, let us prove the \(\cal{P}\)biseparable state case in Eq. (9). Since the \(\cal{P}\)biseparable state set \(S_{b}^{\cal{P}}\) is convex, one only needs to consider the fidelity 〈Ψ_{b}G〉^{2} of the pure state Ψ_{b}〉 defined in Eq. (3). It is known that the maximal value of the fidelity equals to the largest Schmidt coefficient of G〉 with regard to the bipartition \(\{ A,\bar {A}\}\),^{57} i.e.,
with the Schmidt decomposition \(\left G \right\rangle = \mathop {\sum}\nolimits_{i = 1}^d {\sqrt {\lambda _i} } \left {{\mathrm{\Phi }}_i} \right\rangle _A\left {{\mathrm{\Phi }}_i^\prime } \right\rangle _{\bar A}\) and λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{d}. For general graph state G〉, the spectrum of any reduced density matrix ρ_{A} is flat, i.e., λ_{1} = λ_{2} = ⋯λ_{d}, with d being the rank of ρ_{A}.^{58} As a result, one has
To get an upper bound, one should maximize \(2^{  S(\rho _{A})}\) on all possible subsystem bipartitions and then get Eq. (9).
Second, we prove the \(\cal{P}\)fully separable state case in Eq. (8). Similarly, we only need to upperbound the fidelity of the pure state Ψ_{f}〉 defined in Eq. (1), due to the convexity property of the \(\cal{P}\)fully separable state set \(S_{\mathrm{f}}^{\cal{P}}\). From the proof of Eq. (9) above, we know that the fidelity of the \(\cal{P}\)biseparable state satisfies the bound 〈Ψ_{b}G〉^{2} ≤ \(2^{  S\left(\rho _{A}\right)}\), given a subsystem bipartition \(\{ A,\bar {A}\}\). It is not hard to see that these bounds all hold for Ψ_{f}〉, since \(\mathrm{S}_{f}^{\cal{P}} \subset S_{b}^{\cal{P}}\). Thus, one can obtain the finest bound via minimizing over all possible bipartitions and finally get Eq. (8).
The entanglement entropy S(ρ_{A}) equals the rank of the adjacency matrix of the underlying bipartite graph, which can be efficiently calculated. Details are discussed in Supplementary Note 1. While the optimization problems can be computationally hard due to the exponential number of possible bipartitions, one can solve it properly as the number of the subsystems m is not too large. In addition, we can always have an upper bound on the minimization by only considering specific partitions. Analytical calculation of the optimization is possible for graph states with certain symmetries, such as the 1D and 2D cluster states and the GHZ state.
Proof of Proposition 2
Proof. As shown in Main Text, a graph state G〉 can be written in the following form
Accordingly, Eq. (11) in Proposition 2 becomes,
Note that the projectors P_{l} commute with each other, thus we can prove Eq. (26) for all subspaces which are determined by the eigenvalues of all P_{l}. For the subspace where the eigenvalues of all P_{l} are 1, the inequality (1 + k − 1) − k ≥ 0 holds. For the subspace where only one of P_{l} takes value of 0, the inequality (0 + k − 1) − (k − 1) ≥ 0 holds. Moreover, for the subspace in which there are more than one P_{l} taking 0, the inequality also holds. As a result, we finish the proof.
Proofs of Theorems 1 and 2
Proof of Theorem 1
Proof. The proof is to combine Propositions 1 and 2. Here we only show the proof of Eq. (12), and one can prove Eq. (13) in a similar way. To be specific, one needs to show that any \(\cal{P}\)fully separable state satisfies \(\langle W_{f}^{\cal{P}}\rangle \ge 0\), that is,
Here the first and the second inequalities are right on account of Propositions 2 and 1, respectively.
Proof of Theorem 2
Proof. With Eq. (8) one can bound the fidelity from any \(\cal{P}\)fully separable state to a graph state G〉. The mseparable state set S_{m} contains all the state ρ_{m} which can be written as the convex mixture of pure mseparable state, \(\rho _m = \mathop {\sum}\nolimits_i {p_i} \left {{\mathrm{\Psi }}_m^i} \right\rangle \left\langle {{\mathrm{\Psi }}_m^i} \right\), where the partition for each constitute \(\left {{\mathrm{\Psi }}_m^i} \right\rangle\) needs not to be the same. Hence one can bound the fidelity from ρ_{m} to a graph state G〉 by investigating all possible partitions, i.e.,
where the maximization takes over all possible partitions \({\cal{P}}_{m}\) with m subsystems, the minimization takes over all bipartition of \({\cal{P}}_{m}\). Then like in Eq. (27), by combing Eqs. (11) and (28) we finish the proof.
The optimization problem in Theorem 2 over the partitions is generally hard, since there are about m^{N}/m! possible ways to partition N qubits into m subsystems. For example, when N is large (say, in the order of 70 qubits), the number of different partitions is exponentially large even with a small separability number m. Surprisingly, for several widely used types of graph states, such as 1D, 2D cluster states, and the GHZ state, we find the analytical solutions to the optimization problem, as shown in Corollaries in main text.
Robustness of entanglementstructure witnesses
In this section, we discuss the robustness of the proposed witnesses in Theorems 1 and 2. In practical experiments, the prepared state ρ deviates from the target graph state G〉 due to some nonnegligible noise. Here we utilize the following white noise model to quantify the robustness of the witnesses.
which is a mixture of the original state G〉 and the maximally mixed state with coefficient p_{noise}. We will find the largest p_{limit}, such that the witness can detect the corresponding entanglement structure when p_{noise} < p_{limit}. Thus p_{limit} reflects the robustness of the witness.
Let us first consider the entanglement witness \(W_{f}^{\cal{P}}\) in Eq. (12) of Theorem 1. For clearness, we denote \(C_{{\mathrm{min}}} = \min _{\{ A,\bar {A}\} }2^{  S(\rho _{A})}\). Insert the state of Eq. (29) into the witness, one gets,
where n_{l} = V_{l} is the qubit number in each vertex set with different color, and in the second equality we employ the facts that \({\mathrm{Tr}}(P_l) = 2^{N  n_l}\) and Tr(P_{l}G〉〈G) = 1. Let the above expectation value less than zero, one has
Similarly, for the \(\cal{P}\)genuine entanglement witness and the nonmseparability witness in Eqs. (13) and (15), we have,
where we denote the optimizations \(\max _{\{ A,\bar {A}\} }2^{  S(\rho _{A})}\) and \(\max _{{\cal{P}}_{m}}\min _{\{ A,\bar {A}\} }2^{  S(\rho _{A})}\) as C_{max} and C_{m}, respectively.
Moreover, it is not hard to see that all the coefficients C_{min}, C_{max}, and C_{m} are not larger than 0.5. Thus, for any entanglementstructure witness, one has
As a result, our entanglementstructure witness is quite robust to noise, since the largest noise tolerance p_{limit} is just related to the chromatic number of the graph.
Data availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
Code availability
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Acknowledgements
We acknowledge Y.C. Liang for the insightful discussions. This work was supported by the National Natural Science Foundation of China Grant Nos. 11875173 and 11674193, and the National Key R&D Program of China Grant Nos. 2017YFA0303900 and 2017YFA0304004, and the Zhongguancun Haihua Institute for Frontier Information Technology. Xiao Yuan was supported by the EPSRC National Quantum Technology Hub in Networked Quantum Information Technology (EP/M013243/1).
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Y.Z. and X.M. initialized the project. Y.Z., Q.Z., and X.Y. developed the idea and formulated the problem as it is presented. X.M. supervised the project. All authors contributed to deriving the results and writing the paper.
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Correspondence to Xiongfeng Ma.
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Zhou, Y., Zhao, Q., Yuan, X. et al. Detecting multipartite entanglement structure with minimal resources. npj Quantum Inf 5, 83 (2019) doi:10.1038/s4153401902009
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