Measurement based quantum computation

Once the cluster state is obtained it allows to demonstrate the principle of one of its main applications, namely measurement based quantum computation. To this end, a detailed explanation of the measurements and the correspondingly implemented computation will be given. Thereby the notation used in [180] is partly adopted. This allows an interpretation of the results obtained in Sec. 3.1.4 as four different computations.

In conventional quantum computation the information is encoded in qubits that are pro- cessed by the application of logical one- and two-qubit gates within a quantum circuit. In contrast, in a one-way quantum computer the logical qubits on which the computation is carried out do not need to coincide with the actual physical qubits forming the cluster state. The processing of the former is achieved by single qubit measurements of the latter according to an algorithm specific pattern. The cluster state serves thereby as a substrate for any kind of computation. Indeed, due to its entanglement, the individual physical qubits do not carry any information and the logical qubits are encoded non-locally in its correlations. Finally, the result of a computation is found in form of the specific state of particular read-out qubits. As the entanglement of the cluster state is destroyed by the single-qubit measurements dur- ing the process of computation, the state can be used only once. Therefore this scheme is often also referred to as one-way quantum computation in contrast to conventional quantum computation which is reversible.

With respect to such a computation two types of single qubit measurements need to be distinguished: Measurements in the computational basis remove or disconnect single qubits from the cluster resulting in a cluster state of a lower qubit number. Thus, these types of measurements can be used to shape or structure the cluster. Measurements in the basis

B(α) ={| ±αi}with| ±αi= √1

2(|0i ±exp(iα)|1i) apply a rotation ’z(α)≡•(0, α,0,0) = exp(−iαz

2 ) (cp. Eqn. (1.27)) on the logical qubit followed by ahadgate. Together with the entanglement of the cluster, these measurements implement the logical operations required for a computation. In the following, the result for each type of measurement is presented.

As an example for the first type of measurement, qubit d is removed from the cluster resulting in the states

|C3i±abc = 1 √ 2(|00± iabc+|11∓ iabc) = √1 2(|φ +_i ab|0ic∓ |φ−iab|1ic) (3.24) for the outcomes + or−, respectively. The starting cluster state |C4idiffers from|C4liby a

3.3 Cluster state 89 (a) |pi • rα z h |ψ(α, β)i |pi z rβ z h      (b)

Figure 3.11: Scheme for measurement based quantum computation carried out on a four- qubit Horse-Shoe cluster, |Chs

4 i. (a) The logical input qubits are encoded in the state of physical qubits b and c. The computation is performed by projective measurements in the bases B(α) andB(β). Qubits aandd carry the logical output state, |ψ(α, β)i_a,d. (b) The corresponding quantum circuit for the logical qubits. Acphasegate is applied to the initial state|pi ⊗ |pi, followed by the rotations’z(α),’z(β) and ahadgate.

hadrotation on this qubit, therefore a measurement in the computational basis corresponds tox instead ofz.

In the experiment the three-qubit cluster states are observed with fidelitiesF_C+

3 = 0.756±

0.028 and F_C−

3 = 0.753±0.026. The fidelities are again determined from a measurement of

the corresponding stabilizer operators and the values are significantly greater than 1₂ what allows to proof genuine three-qubit entanglement of the experimental states [145]. Three qubit cluster states belong to theghz-class. Using Eqn. (1.38) the obtained fidelities are just at the limit to prove this unambiguously.

In order to demonstrate an example of a measurement of the second type, it is convenient to consider a cluster state which equals the four-qubit linear cluster in its mathematical form, but which has a slightly different graphical representation, see Fig. 3.11(a). Referring to its graph it was named ”Horse-Shoe” cluster,|Chs

4 i in [180],

|Chs 4 i=

1

2(|H+H+i+|H−V− i+|V +V− i+|V −H+i). (3.25) The Horse-Shoe cluster equals also the state|Ci up to a Hadamard transformation on the photons in modesaand d.

In the simple one-way quantum computation scheme presented here, the logical input qubits are encoded in the state of the two physical qubits on the left hand side (b and c

in Fig. 3.11(a)) before the entangling operation (phase gate) acts on the qubits that are connected in the graph, i.e., between the pairsa,bandb,cand c,d. After this initialization, the computation is performed by application of projective measurements on all qubits apart from the pair on the right hand side (in Fig. 3.11(a),aand d). The qubitsaanddcarry the logical output state at the end of the computation. For a better illustration logical qubits are denoted for the rest of this section as

|0i, |pi= √1

2(|0i+|1i),

|₁i, |mi= _√1

2(|0i − |1i), (3.26)

So the actual physical computation carried out on the Horse-Shoe cluster reads as follows, trb,c ³ (1⊗|αihα| ⊗|βihβ| ⊗1)· |Chs 4 i ´ =|ψ(α, β)i_a,d, (3.27) where the result is encoded in the state |ψ(α, β)i_a,d (see Fig. 3.11(a)). On the level of the logical qubits this is equivalent to

(had⊗had)·(’z(α)⊗’z(β))·cphase·(|pi ⊗ |pi) =|ψ(α, β)i, (3.28) depicted in form of a quantum circuit in Fig. 3.11(b). As the qubitsb and c are connected, acphaseoperation acts on the logical input qubits prior to the single qubit rotations. This demonstrates how the entanglement in the cluster can be used to emulate non-local two-qubit operations required for universal computations.

In the experiment qubits band care measured in thexxbasis what corresponds to rota- tions on the logical qubits according to four sets of angles{α, β}={0,0},{0, π},{π,0},{π, π}. For the usage of|C4iinstead|C4hsi, the output states are obtained up to another Hadamard transformation, which cancels with thehad gate acting in the computation. Thus, the re- sulting transformation on the logical input state is ((’z(α)⊗’z(β))·cphase) leading to

|ψ(0,0)i = √1 2(|0pi+|1mi), (3.29a) |ψ(0, π)i = _√1 2(|0mi+|1pi), (3.29b) |ψ(π,0)i = √1 2(|0pi − |1mi), (3.29c) |ψ(π, π)i = _√1 2(|0mi − |1pi). (3.29d)

These are the states |φf+_{i, |ψ}f+_{i, |φ}f−_{i, |ψ}f−_{i, and as the logical output states in this in-} stance are identical to the physical states of qubits a and d, the experimental result of this computation is the same as the one presented in Fig. 3.5 and Tab. 3.2.

3.3.4 Discussion

The realized computation scheme corresponds to the one presented in [180] achieving one output state with a slightly higher fidelity of 0.84±0.03 despite of the lower fidelity of the cluster state itself. In contrast to [180], here four different single qubit rotations on the logical qubits have been implemented and the output states are all obtained at comparable quality. This proves the functionality of the small one-way computation for different computational settings. A possible next step is the implementation of the scheme in a way that different input states can be used. This can, however, also be solved by realizing a bigger cluster state that allows to first transform the logical qubits into the input state as part of the cluster scheme. Very important is further the feed-forward of measurement results which allows to make the computation deterministic. This route is followed in the group of Anton Zeilinger and was applied for the first time in [181].

The successful observation of the four-photon cluster state presented above is another proof for the applicability of thecphasegate. Together with the results discussed in Sec. 3.1 and Sec. 3.2 it shows that the gate can be used for entangling as well as disentangling qubits within tasks involving more than just two qubits. This makes the implementedcphasegate an indispensable tool in future multi-photon quantum information applications.

3.4 Conclusion 91

3.4

Conclusion

This chapter has presented three multi-photon applications of the cphase gate introduced in Chap. 2. The first two used the gate for a complete Bell state measurement. In this con- text for the first time, a teleportation and entanglement swapping protocol was performed where all four Bell states are distinguished by means of linear optics only. The teleported polarization states showed fidelities clearly above the classical bound. The quality of the im- plemented teleportation and the achievement of an efficient quantum channel was confirmed by reconstruction of the quantum process matrix. Running the entanglement swapping pro- tocol yielded high fidelities and states which were entangled strong enough to violate a Bell inequality.

The complete Bell state measurement was further used for a direct measure of concurrence. The underlying method based on measurements on multiple identically prepared quantum systems was already demonstrated before, however under the very restrictive and unrealistic assumption of having pure states. Here, for the first time, a generalization of the method for mixed states was tested accounting for the imperfections of state-of-the art technology which is used in current set-ups. Thereby it was particularly important to consider the influence of non-ideal gate operation. This allowed a realistic lower bound estimation on the concurrence of the prepared states.

In the last application thecphasegate was used to entangle the photons of two Bell pairs. This enabled the successful observation of a four photon cluster state with high fidelity. As the properties of the observed state were analyzed in detail elsewhere, the treatment of the results in this chapter was focused on a proof-of-principle demonstration of measurement based quantum computation. The two types of single-qubit measurements commonly applied in such computation schemes have been demonstrated. This comprised in particular the entangling operation of two encoded logical qubits.

To summarize, the analysis of data obtained within the same set-up from three differ- ent perspectives constitutes together an interesting proof that a universal two photon gate based on linear optics only can be successfully applied in tasks involving more than just two qubits. This is a further demonstration that meanwhile linear optics gates are no longer feasible just in principle but have reached a level of functionality and simplicity which allows their implementation in quantum information applications. If this is combined with recently developed active feed-forward techniques it might additionally open up new vistas for linear optics quantum computation.

Chapter 4

Discriminating multi-partite

entangled states

In Chap. 1 it was mentioned that in the case of more than two qubits it is necessary to distinguish not only between separable and entangled but also between the different kinds of multi-partite entanglement. It was shown that witness operators provide a tool to distinguish the different degrees of separability from each other. Whereas for three qubits witness oper- ators are still suited to distinguish ghz- from w-type entanglement, there is no instructive method to do something similar for more than three qubits.

This chapter addresses the problem of experimentally discriminating different types of four qubit entanglement. For this purpose characteristic Bell operators are introduced which are shown to be suitable for this task [187]. Finally, it will be demonstrated that, provided additional information about the state space is available, these characteristic operators can be chosen without the constraint of excluding local realistic descriptions of the measurement results.

4.1

The problem of state discrimination

Entanglement is the crucial resource for quantum information processing and as such the ”currency” to pay with in almost all applications. For two-partite quantum states measures have been developed that uniquely specify the value of this resource. In contrast, forn-partite states the picture changes significantly. First, it has to be distinguished not only between fully separable or entangled, but also between genuinen-partite, bi-, and tri- separable entangled states, etc. (see Sec. 1.2.2). Second, even states with the same level of separability are different in the sense that they have, for example, different Schmidt rank [188] or that they cannot be transformed into each other, e.g., by, lu or, more generally, by slocc [18, 19]. From an experimental point of view, classifying states according to the latter property is very reasonable, as states from onesloccclass are suited for the same multi-party quantum communication applications. Thus, for the usage of multi-partite states it is of importance to know not only theamount but also the type of entanglement contained in a particular state. In other words, the valueand the type of the ”currency” is what matters.

Tools to detect the entanglement of a state exist, most prominently entanglement wit- nesses [43]. An alternative method, relying on the correlations between results obtained by local measurements, are Bell inequalities. Being originally devised to test fundamental issues

of quantum physics they allow to distinguish entangled from separable two-qubit quantum systems [67, 189]. Bell inequalities, meanwhile extended to three- and more partite quantum states [190–195], can thus serve as witness for both entanglement and the violation of local realism. Recently it was observed that for each graph state all non-vanishing correlations (or even a restricted number thereof) form a Bell-inequality, which is maximally violated only by the respective quantum state [196–198]. In particular, the Bell inequality for the four-qubit cluster state is not violated at all byghzstates [196]. Naturally, several questions arise: Is it in general possible to apply such Bell inequalities for the discrimination of particular states from other classes of multi-partite entangled states, if so, can they also be constructed and ap- plied for non-graph states, and finally, are there other operators that allow to experimentally discriminate entanglement classes.

In this chapter these problems are addressed starting from Bell inequalities. A way is presented to construct Bell operators [38, 199] that arecharacteristicfor a particular quantum state. Thereby the following definitions are used:

Definition 4.1.1 (Bell operator) A Hermitian operator ‚ is called Bell operator if there exists a constantβlr∈Rand a state such that| h‚iavg| ≤βlr for all local realistic theories

andβlr <tr(‚).

Thereby h‚iavg = E(B) is the weighted mean of measurement results in the local realistic

theory for an observable B which is represented by the operator ‚ in the quantum theory. This is analogous to Eqn. (1.14) and the considerations on page 12.

Definition 4.1.2 (Characteristic operator) A Hermitian operator „ : H → H is called characteristic for a state |Xi ∈H, iff

„|Xi=λX|Xi, dim(ker(„−λX1H)) = 1 and |λX|= sup({|λ|:λ∈sp(„)}), i.e., iff |Xi is non-degenerate eigenvector of „ with maximal eigenvalue. Thereby, sp(„) denotes the spectrum of „.

With respect to experimental applications it is further desirable that the expectation value of each derived operator can be obtained by a minimal number of measurement settings. Under certain conditions, the initial requirement that the constructed operators have to be characteristic and in addition Bell operators can be relaxed, which allows further reduction of the number of settings. Comparison of the experimentally obtained expectation value with the maximal expectation values for states from other entanglement classes enables the distinction of an observed state from other multi-party entangled states.