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Control and Measurement of Three-Qubit Entangled States

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Science  04 Jun 2004:
Vol. 304, Issue 5676, pp. 1478-1480
DOI: 10.1126/science.1097522

Abstract

We report the deterministic creation of maximally entangled three-qubit states—specifically the Greenberger-Horne-Zeilinger (GHZ) state and the W state—with a trapped-ion quantum computer. We read out one of the qubits selectively and show how GHZ and W states are affected by this local measurement. Additionally, we demonstrate conditional operations controlled by the results from reading out one qubit. Tripartite entanglement is deterministically transformed into bipartite entanglement by local operations only. These operations are the measurement of one qubit of a GHZ state in a rotated basis and, conditioned on this measurement result, the application of single-qubit rotations.

Quantum information processing rests on the ability to deliberately initialize, control, and manipulate a set of quantum bits (qubits) forming a quantum register (1). Carrying out an algorithm then consists of sequences of quantum gate operations that generate multipartite entangled states of this quantum register. Eventually, the outcome of the computation is obtained by measuring the state of the individual qubits. For the realization of some important algorithms such as quantum error correction (15) and teleportation (6), a subset of the quantum register must be read out selectively, and subsequent operations on other qubits must be conditioned on the measurement result. The capability of entangling a scalable quantum register is the key ingredient for quantum information processing as well as for many-party quantum communication. Whereas entanglement with two or more qubits has been demonstrated (713), our experiment allows the deterministic generation of three-qubit entangled states and the selective readout of an individual qubit followed by local quantum operations conditioned on the readout.

The experiments are performed in an elementary ion-trap quantum processor (14, 15). For the investigation of tripartite entanglement (1618), we trap three 40Ca+ ions in a linear Paul trap. Qubits are encoded in a superposition of the S1/2 ground state and the metastable D5/2 state (lifetime τ ≈ 1.16 s). Each ion-qubit is individually manipulated by a series of laser pulses on the S ≡ S1/2 (mj = –1/2) to D ≡ D5/2 (mj = –1/2) quadrupole transition near 729 nm, with the use of narrow-band laser radiation tightly focused onto individual ions in the string. The entire quantum register is prepared by Doppler cooling, followed by sideband ground-state cooling of the center-of-mass vibrational mode (ω = 2π × 1.2 MHz). The ions' electronic qubit states are initialized in the S state by optical pumping.

Three qubits can be entangled in only two inequivalent ways, represented by the Greenberger-Horne-Zeilinger (GHZ) state, Math, and the W state, Math (17). The W state can retain bipartite entanglement when any one of the three qubits is measured in the {|S〉, |D〉} basis, whereas for the maximally entangled GHZ state a measurement of any one qubit destroys the entanglement. We synthesize the GHZ state with a sequence of 10 laser pulses and the W state with a sequence of five laser pulses (19).

Full information on the three-ion entangled states is obtained by state tomography (20, 21) using a charge-coupled device camera for the individual detection of ions. The pulse sequences generate three-ion entangled states within less than 1 ms. Determining all 64 entries of the density matrix with an uncertainty of less than 2% requires about 5000 experiments, corresponding to 200 s of measurement time.

In the experimental results for the absolute values of the density matrix elements of GHZ and W states, ρ|GHZ〉 and ρ|W〉 (Fig. 1, A and B), the off-diagonal elements are observed with heights nearly equal to those of the corresponding diagonal elements and with the correct phases (19). Fidelities of 72% for ρ|GHZ〉 and 83% for ρ|W〉 are obtained. The fidelity is defined as |〈Ψidealexpideal〉|2, where Ψideal denotes the ideal quantum state and ρexp is the experimentally determined density matrix. All sources of imperfection have been investigated independently (14) and the measured fidelities are consistent with the known error budget. Note that for the W state, coherence times greater than 200 ms were measured (exceeding the synthesis time by almost three orders of magnitude), whereas for the GHZ state only times of ∼1 ms were found. This is because the W states are a superposition of three states with the same energy. Thus, the dephasing due to magnetic field fluctuations is much reduced in contrast to a GHZ state that is maximally sensitive to such perturbations. Similar behavior has been observed with Bell states (21, 22).

Fig. 1.

(A) Absolute values of the density matrix elements of the experimentally obtained GHZ quantum state. The off-diagonal elements for SSS and DDD indicate the quantum correlation clearly. (B) Absolute values of the density matrix of the W state. Off-diagonal elements and diagonal elements are at equal height. (C) GHZ state after measuring the first qubit only. The GHZ state coherences have fully disappeared as compared with the results shown in (A). The state is thus fully described by a classical mixture. (D) W state after measuring the first qubit. Only the coherences involving the first qubit have disappeared, whereas two-ion Bell-type entanglement persists between the second and the third qubit. The state thus contains quantum correlations even after a local projective measurement.

Having tripartite entangled states available as a resource, we make use of individual ion addressing to read out only one of the three ions' quantum state while preserving the coherence of the other two. Qubits are protected from being measured by transferring their quantum information into superpositions of levels that are not affected by the detection—that is, by a light-scattering process. In Ca+, an additional Zeeman level D′ ≡ D5/2 (m = –5/2) can be used for this purpose. Thus, after the state synthesis, we apply two π pulses on the S-D′ transition of ions 2 and 3, moving any S population of these ions into their respective D′ level. The D and D′ levels do not couple to the detection light at 397 nm (Fig. 2). Therefore, ion 1 can be read out by the electron shelving method as usual (15). After the selective readout, a second set of π-pulses on the D′-S transition transfers the quantum information back into the original computational subspace {S, D}.

Fig. 2.

Selective readout of ion 1: Ions 2 and 3 are protected from measurement by transfer into dark states. Only the relevant levels of the three Ca+ ions are shown.

For a demonstration of this method, GHZ and W states are generated and qubits 2 and 3 are mapped onto the {D, D′} subspace. Then, the state of ion 1 is projected onto S or D by scattering photons for a few microseconds on the S-P transition. In a first series of experiments, we did not distinguish whether ion 1 was projected into S or D. After remapping qubits 2 and 3 to the original subspace {S, D}, the tomography procedure is applied to obtain the full density matrix of the resulting three-ion state. As shown in Fig. 1C, the GHZ state is completely destroyed; that is, it is projected into a mixture of |SSS〉 and |DDD〉. In contrast, for the W state, the quantum register remains partially entangled as coherences between ions 2 and 3 persist (Fig. 1D). Related experiments have been carried out with mixed states in nuclear magnetic resonance (13) and with photons (11).

In a second series of experiments with the W state, we deliberately determined the first ion's quantum state before tomography: The ion string is now illuminated for 500 μs with light at 397 nm, and its fluorescence is collected with a photomultiplier tube (Fig. 3A). Then the state of ion 1 is known, and we subsequently apply the tomographic procedure to ions 2 and 3 after remapping them to their {S, D} subspace. Depending on the state of ion 1, we observe the two density matrices presented in Fig. 3, B and C (19). Whenever ion 1 was measured in D, ions 2 and 3 were found in a Bell state Math, with a fidelity of 82%. If the first qubit was observed in S, the resulting state is |DD〉 with fidelity of 90%. This is a characteristic signature of Math: In one-third of the cases, the measurement projects qubit 1 into the S state, and consequently the other two qubits are projected into D. With a probability of 2/3, however, the measurement shows qubit 1 in D, and the remaining quantum register is found in a Bell state (17). Experimentally, we observe the first ion in D in 65 ± 2% of the cases.

Fig. 3.

(A) Histogram of photon counts within 500 μs for ion 1 and threshold setting. (B and C) Density matrix of ions 2 and 3 conditioned on the previously determined quantum state of ion 1. The absolute values of the reduced density matrix are plotted for ion 1 measured in the S state (B) and ion 1 measured in the D state (C). Off-diagonal elements in (B) show the remaining coherences.

The GHZ state can be used to deterministically transform tripartite entanglement into bipartite entanglement by means of only local measurements and one-qubit operations. For this, we first generate the GHZ state Math. In a second step, we apply a π/2 pulse to ion 1, with phase 3π/2, rotating a state |S〉 to Math and |D〉 to Math, respectively. The resulting state of the three ions is Math. A measurement of the first ion, resulting in |D〉 or |S〉, projects qubits 2 and 3 onto the state Math or the state Math, respectively. The corresponding density matrix is plotted in Fig. 4A. With the information on the state of ion 1 available, we can now transform this mixed state into the pure state Math by only local operations. Provided that ion 1 is found in |D〉, we perform an appropriate rotation (19) on ion 2 to obtain Math. In addition, we flip the state of ion 1 to reset it to |S〉. Figure 4B shows that the bipartite entangled state Math is produced with fidelity of 75% (19). This procedure can also be regarded as an implementation of a three-spin quantum eraser, as proposed in (23).

Fig. 4.

(A) Real part of the density matrix elements of the system after ion 1 of the GHZ state Embedded Image has been measured in a rotated basis. (B) Transformation of the GHZ state Embedded Image into the bipartite entangled state Embedded Image by conditional local operations. Note the different vertical scaling of (A) and (B).

Our results show that selectively reading out a qubit of the quantum register indeed leaves the entanglement of all other qubits in the register untouched. Even after such a measurement has taken place, single-qubit rotations can be performed with high fidelity. Such techniques represent a step toward the one-way quantum computer (24). The implementation of unitary transformations conditioned on measurement results allows for the realization of deterministic quantum teleportation and of active quantum error correction algorithms. With further improvements of the gate fidelity, it will be possible to realize two different protocols for quantum error correction—either to perform a strong measurement on the ancilla qubit and to rotate the target qubit correspondingly or, alternatively, to rotate the target qubit conditional upon the still-unknown ancilla quantum state by conditional gate operations and to reset the ancilla afterward. It remains to be determined which of the two methods is favorable for our setup.

Supporting Online Material

www.sciencemag.org/cgi/content/full/304/5676/1478/DC1

Materials and Methods

Tables S1 to S3

References

References and Notes

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