Experimental violation of multipartite Bell inequalities with trapped ions
We report on the experimental violation of multipartite Bell inequalities by entangled states of trapped ions. First we consider resource states for measurement-based quantum computation of between 3 and 7 ions and show that all strongly violate a Bell-type inequality for graph states, where the criterion for violation is a sufficiently high fidelity. Second we analyze GHZ states of up to 14 ions generated in a previous experiment using stronger Mermin-Klyshko inequalities, and show that in this case the violation of local realism increases exponentially with system size. These experiments represent a violation of multipartite Bell-type inequalities of deterministically prepared entangled states. In addition, the detection loophole is closed.
pacs:03.65.Ud, 03.67.Lx, 03.67.Bg, 37.10.Ty
Introduction — How strong can physical correlations be? Bell inequalities set a bound on the possible strength of non-local correlations that could be explained by a theory based on some fundamental assumptions known as "local realism". Quantum mechanics predicts the existence of states which violate Bell’s inequality, rendering a description of these states by a local hidden variable (LHV) model impossible. While first discovered for bipartite systems in a two-measurement setting Be64 (), Bell inequalities have been extended to multi-measurement settings and multipartite systems, leading to a more profound violation for larger systems of different kinds We01 (); Mermin (); Sc05 (); Gu05 (); Ca08 ().
In particular, it was shown that all graph states violate local realism, where the possible violation increases exponentially with the number of qubits for certain types of states Gu05 (); Sc05 (); Ca08 (). Graph states He04 (); He06 () are a large class of multiqubit states that include a number of interesting, highly entangled states, such as the 2D cluster states Ra01b () or the GHZ states. They serve as resources for various tasks in quantum information processing, including measurement-based quantum computation (MBQC) Ra01 (); Br09 () or quantum error correction CSS (). The results of Gu05 (); Sc05 () (see also Di97 ()) provide an interesting connection between the usability of states for quantum information processing and the possibility to describe them by a LHV model.
Here we experimentally demonstrate the violation of multi-partite Bell-type inequalities for graph states generated with trapped ions. First we consider a range of graph states that find application in MBQC and observe strong violations in all cases. Second, for a different class of graph states, we investigate the scaling of the multi-partite Bell violation with system size and confirm an exponential increase: that is the quantum correlations in these systems become exponentially stronger than allowed by any LHV model.
To be more precise, in the first part of our work we consider graph states that allow one to perform single-qubit and two-qubit gates in MBQC, as well as resource states for measurement-based quantum error correction La13 (). That is, we demonstrate that not only the codewords of quantum error correction codes violate local realism Di97 (), but also the resource states for encoding and decoding and other computational tasks. In this part we make use of general Bell-type inequalities derived for all graph states in Ref. Gu05 (). We show that the Bell observable simply corresponds to the fidelity of the state, i.e. a violation is guaranteed by a sufficiently high fidelity. This allows the many previous experiments that quote fidelities to be reanalyzed to see if a Bell violation has been achieved.
For the purpose of investigating the scaling of Bell violations we consider a sub-class of graph states, for which stronger inequalities are available Mermin (); Sc05 (); Ca08 (), e.g. the Mermin-Klyshko inequalities for -qubit GHZ states Mermin (). We show that these Mermin-Klyshko inequalities Mermin () are violated by GHZ states from 2 to 14 qubits generated in previous experiments Mo11 (). In fact, we confirm an (exponentially) increasing violation with system size.
Multi-partite Bell violations for smaller system sizes were previously obtained with photons ExpPhotons (). Here specific 4-photon states encoding up to 6 qubits were considered. For trapped ions only two-qubit systems have previously been shown to violate a Bell inequality ExpIons (). Here we deal with larger systems and states with a clear operational meaning in measurement-based quantum information processing, where each qubit corresponds to a separate particle. Finally, our detection efficiency is such that we close the detection loophole.
Background — Graph states are defined via the underlying graph , which is a set of vertices and edges , that is . One defines an operator for very vertex , where and denote Pauli spin- operators. denotes the neighborhood of vertex and is given by all vertices connected to vertex by an edge. The graph state is the unique quantum state which fulfills for all , i.e. it is the common eigenstate of all operators . An equivalent definition starts with associating a qubit in state with every vertex and applying a controlled phase (CZ) gate between every vertices connected by an edge, with . Graph states have important applications in the context of measurement-based quantum computation as resource states Ra01 (); Br09 () and quantum error correction CSS ().
In Gu05 () it was shown that all graph states give rise to a Bell inequality and that the graph state saturates it. Thus neither the correlations nor the quantum information processing that exploits these correlations can be accounted for by a LHV model. The inequality is constructed in the following way. One aims at writing down an operator (specifying certain correlations in the system) such that the expectation value for all LHV models is bounded by some value , while certain quantum states yield an expectation value larger than . Let denote the stabilizer Go96 () of a graph state . It is the group of the products of the operators and is given by with where denotes a subset of the vertices of . For the state corresponding to the empty graph, the generators of the stabilizer group are given by , and the stabilizer group is given by all possible combinations of and on the different qubits. For we have . Notice that for any non-trivial graph states (i.e. graph states with a non-empty edge set ), these operators are simply transformed via since , where , i.e. the stabilizing operators of the graph state specified above.
The normalized Bell operator is defined as , and we have (where, in quantum mechanics, for density matrix ). Let where the maximum is taken over all LHV models. For any non-trivial graph state Gu05 (). The maximization is generally hard to perform, but has been explicitly carried out for graph states with small in Gu05 (). The basic idea is to assign a fixed value ("hidden variable") or to each Pauli operator , and determine (numerically) the setting that yields a maximum value of . This then also provides an upper bound on all LHV models. The corresponding Bell inequality reads
which is non-trivial whenever . For the states one finds Gu05 (), while we show in Sup () that and (see figure 1 for the different states). For fully connected graphs corresponding (up to a local basis change) to -qubit GHZ states , we obtain for (see Sup ()).
Any graph state fulfills , since the state is a eigenstate of all operators appearing in the sum that specifies . Hence it follows that the graph state maximally violates the graph Bell inequality (1), .
A straightforward calculation shows that the normalized Bell operator equals the projector onto the graph state: . This can be seen directly for the empty graph by noting that , and writing out the product for which yields all combinations of and . The result for a general graph state follows by transforming each operator via , together with . Thus, the expectation value equals the fidelity , where denotes the density matrix of the experimentally obtained graph state. As it is common practice to report on the fidelity this provides a simple way of reinvestigating earlier experiments.
In addition, this provides a possibility for measuring the fidelity of a graph state by measuring the stabilizers, which add up to . Although this method has the same exponential scaling behavior as full state tomography, it requires significantly fewer measurement settings.
Results: Graph states for MBQC — The first group of graph states that we consider are resources for MBQC and are shown in figure 1. The four-qubit box cluster represents the smallest element of the 2D cluster (family) required to implement arbitrary quantum algorithms Ra01b (); Ra01 (); Br09 (). The four-qubit linear cluster state can be used to demonstrate a universal quantum logic gate set for MBQC La13 (); Wa05 (). The graph states allow for the demonstration of an -qubit measurement-based quantum error correction code La13 ().
Except for , all of these states were generated in a system of trapped ions and their application to MBQC was demonstrated in our recent paper La13 (). In that work, and in particular its accompanying supplementary material, one can find information on the experimental techniques used to prepare the states. In summary, qubits are encoded into the electronic state of Ca ions held in a radio-frequency linear Paul trap: each ion represents one qubit. After preparing each qubit into the electronic and motional ground state, graph states are generated deterministically and on demand using laser pulses which apply qubit-state dependent forces to the ion string. Additional details relevant to Bell inequality measurements are now described. The ions are typically 6 m apart and it takes approximately 500 s to generate the states. Individual qubits can be measured in any basis with near unit fidelity in 5 s. The state belongs to the same family as the error correction graphs, i.e. , and was thus generated using exactly the method described in La13 ().
For each -qubit graph state shown in figure 1 we experimentally estimate each of the expectation values that are required to estimate . If this final number is larger than allowed by LHV models then the multi-partite Bell inequality is violated. The experimental uncertainty in each is the standard quantum projection noise that arises from using a finite number of repeated measurements to estimate an expectation value.
We note that the full density matrices for a subset of the graph states shown in figure 1 were presented in La13 (). We do not extract the data from these matrices but directly measure the observables in each case. No previous characterization of the states and has been done.
The results are summarized in table 1 and clearly show that all experimentally generated states violate their graph state inequalities by many tens of standard deviations. Recall that is equal to the state fidelity. For comparison, table 1 also presents the state fidelity measured in another way — by reconstructing the density matrix via full quantum state tomography and using . This approach is much more measurement-intensive, requiring the estimation of expectation values and was therefore not carried out for the 7-qubit state . The fidelities derived in these different ways are seen to overlap to within 1 standard deviation. In the supplementary material we give an explicit example of how the experimental value of for one graph state () was derived.
|Graph||qubits||Fidelity ()||Multipartite Bell inequality||relative violation|
Results: scaling of violation with system size — In the second part of our work we are interested in investigating the scaling of the violation of multi-partite Bell inequalities with the system size. Table 1 presents the relative violation observed for the graph state inequalities, defined as the ratio of the quantum mechanical expectation value of the Bell observables and the maximal reachable value in a LHV model (). From this it is clear that while all the generated MBQC graph states violate their inequalities, the size of the violation does not change significantly with the size of the graph state. However, there is another class of Bell inequalities, the Mermin-Klyschko (MK) inequalities Mermin (), for which the quantum mechanical violation is predicted to increase exponentially with qubit number. The MK inequalities apply to the GHZ states , which are (up to local unitary operations) equivalent to graph states corresponding to a fully connected graph (see figure 2).
The MK Bell operator Mermin () can be defined recursively by
and starts with footnoteNorm (). The are given by scalar products of three dimensional unit vectors and the vector consisting of the three Pauli operators, i.e. . The operator is obtained from by exchanging all the and . Within a LHV model one can only reach Mermin (). This can be seen intuitively by assigning specific values or to each of the operators , which implies that the recursive relation reduces to or where for all possible choices. It follows that in this case, and similarly for all LHV models.
Quantum mechanics allows a violation of the MK inequality by ; by comparison to the maximum allowed LHV value , one sees that the violation scales exponentially with the system size. Note that the MK inequality achieves the highest violation for any inequality with two observables per qubit We01 (). The observables can be significantly simplified by choosing the same measurement directions for all qubits, e.g. and for all . It can then be shown that Mermin ()
with . The determination of then reduces to determining two specific off-diagonal elements in the density matrix . The states which violate the MK inequality maximally are then given by , leading to . Notice that the local observables can be adjusted in such a way that GHZ states with arbitrary phase maximally violate the corresponding MK-inequality, i.e. the relevant quantity for a violation is given by the absolute value of the coherences .
GHZ states of the form for up to qubits have previously been prepared using trapped ions Mo11 () (again 1 qubit is encoded per ion). In that work the state fidelities were estimated via measurements of the logical populations and , and the coherences . From this information both the graph state Bell observable and the MK Bell observable can now be calculated.
The relative violations , defined as for the MK inequalities and for the graph inequality, are presented graphically in figure 3. An exponential scaling is apparent for the relative violation of the MK inequalities, i.e. by using larger systems a stronger violation of non-locality can be observed. We now show that the violation of the MK inequalities with larger systems can be more robust to noise than for smaller systems. This can be illustrated as follows. Assume the preparation of a noisy -qubit GHZ state, where imperfections and decoherence is modeled in such a way that each qubit is effected by single qubit depolarizing noise , i.e. . Even though the state can be shown straightforwardly to have an exponentially small fidelity, one nevertheless encounters a violation of the MK inequality even for a large amount of local depolarizing noise. To be specific, one finds that (the off-diagonal elements are simply suppressed by this factor), leading to . That is, as long as , one encounters a violation of the MK inequality for large enough . This means that MK inequalities can tolerate almost noise per qubit. The graph inequalities for GHZ states demand a fidelity larger than 0.5 Sup (), requiring the noise per qubit to reduce exponentially with system size.
Conclusion and outlook — We have demonstrated the violation of multi-partite Bell inequalities for graph states which are resources in MBQC, thereby confirming a connection between applicability of states as resources for quantum information processing and violation of LHV models. In addition, we show that the data in a previous experiment is sufficient to identify an exponentially increasing Bell violation with system size Mo11 (). Given the fact that our set-up can readily be scaled up to a larger number of ions, this opens the possibility to demonstrate LHV violations for large-scale systems.
Acknowledgements — This work was supported by the Austrian Science Fund (FWF): P25354-N20, P24273-N16 and SFB F40-FoQus F4012-N16.
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Appendix A appendix
Appendix B Examples of Bell operators (theory)
For illustration, we explicitly provide some of the Bell operators, both for graph state inequalities and MK inequalities. As an example, for the graph , i.e. the linear cluster state of four qubits, the normalized Bell operator is given by
For the graph , i.e. the box cluster state of four qubits, the normalized Bell operator is given by
For the four-qubit GHZ state and the corresponding MK inequalities the Bell operator is given by
Appendix C A complete experimental example
In this section we provide details on how the Multipartite Bell inequality , given in the table in the main text, was measured for one of the graph states. Specifically we choose the 4-qubit box cluster shown in figure 4. As described in the main text, the most well known method to prepare clusters states is to initialize each physical qubit in the state and then to apply CP gates between every pair of qubits with a connecting edge: in this case qubit pairs 1&2, 1&3, 2&4 and 3&4. Note that
In our experiments we prepare all our graph states in a different way, which is equivalent to the method using CP gates up to single qubit rotations: i.e. the states are equivalent up to a local change of basis. In summary we begin by initializing all qubits into and applying pairwise entangling operations generated by Hamiltonians of the form , where the subscript refers to the Mølmer-Sørensen interaction on which our qubit interactions are based MS99 (). For more experimental details on the state generation see the supplementary material of La13 () where laser pulse sequences can be found. Note that the 4-qubit box cluster is not presented in La13 (), however the laser pulse sequence is identical to that for all the error-correction states . In fact : rotating the box cluster diagram in figure 4 by 45 degrees in either direction (so that it becomes a diamond) makes it clear that it belongs to the same family of states.
As stated, experimentally we do not prepare , but ideally a locally rotated state for which we will use the label . This state is given by
which is equivalent to the state made with CP gate once it is corrected by the following single-qubit correction rotations. qubit 1: HXZ, qubit 2: HX, qubit 3: HX, qubit 4 HXZ, where H is the Hadamard, and X and Z are standard Pauli operators.
For the experimentally generated 4-qubit box cluster , the normalized Bell operator is given by
The experimentally observed expectation values for all 16 observables are presented in table 2. The average values of the last column is and is the normalized Bell operator we observe for this state. All uncertainties are one standard deviation and derive from the intrinsic uncertainty in using a finite number of measurements to estimate expectation values.
|1||Z Z Z Z||1.000||0.8700|
|2||- Z X X||1.000||0.8700|
|3||Z Y Y||1.000||0.8500|
|5||X Z X||1.000||0.8300|
|6||X X Y Y||1.000||0.8100|
|7||X Y X Y||1.000||0.7800|
|8||-X Z X||1.000||0.8500|
|9||Y Z Y||1.000||0.8700|
|10||Y X Y X||1.000||0.7300|
|11||Y Y X X||1.000||0.8800|
|12||Y Z Y||1.000||0.8300|
|14||- X X Z||1.000||0.8600|
|15||Y Y Z||1.000||0.8600|
Appendix D Values for for
A bound for , where is the graph underlying the five qubit state which we used to demonstrate quantum error correction, can be found in the following way. First one notes that is equivalent to the state in figure 5b) up to local Clifford (LC) operations. The two graph states have the same rank indices and are thus equivalent up to local unitary operations He04 (). The fact that they are both graph states then implies the LC equivalence. The local Clifford operations do not change the value of . The graph state is build from a four qubit GHZ state and a single qubit graph , connected by an edge. Application of Lemma 3 in Gu05 () then gives a bound on :
In a similar way one can bound the value ,
Appendix E Values for for GHZ states
The values for for GHZ states with up to ten qubits have been derived numerically in Gu05 (). Here we illustrate how one can simplify the numerical procedure and provide the values for GHZ states with twelve and fourteen qubits. In addition we show that a fidelity larger than one half is required for all GHZ states in order to violate the graph state inequality.
The generators for GHZ states are given (up to irrelevant local Clifford operations) by , , …, . The Bell operator contains all products of the generators, as described in the main text. In Gu05 () it is shown that one can restrict to LHV models which assign to all measurements. For GHZ states one can then show by simply multiplying the generators that one only has to check the following operators: for stabilizers with an odd number of generators:
and for stabilizers with an even number of generators:
and all the permutations of the qubits in both cases.
Since each of the operators and contain only or operators, they can be optimized independently. For the operators it is easy to see that they contribute maximally by assigning to all measurement outcomes. Their total contribution to is then given by , where the factor comes from the normalization in the definition of and the sum comes from the total number of operators . The optimization for the operators is done numerically and we find and . For even one can confirm . We leave it as a conjecture that this expression holds for arbitrary even .
The contribution from the operators puts a lower bound on and thus, via the relation , on the fidelity . Consequently, a necessary requirement for any GHZ state to violate the Bell type inequality derived in Gu05 () is that the fidelity is greater than one half.