Special issue: Quantum information technology

Progress in Informatics, No. 8, pp.27–31, (2011) 27

Research Paper

Generation and characterization of boundentanglement in optical qubits

Fumihiro KANEDA1, Ryosuke SHIMIZU2, Yasuyoshi MITSUMORI3,Hideo KOSAKA4, and Keiichi EDAMATSU5

1,3,4,5Research Institute of Electrical Communication, Tohoku University2PRESTO, Japan Science and Technology Agency (JST)

ABSTRACTFor efficient quantum information processing (QIP), pure and strong entanglement in qubitshas been thought to be indispensable. However, it was recently pointed out that bound entan-gled states, which involve undistillable entanglement between qubits, also have great potentialfor QIP. In this paper, we report the efficient generation of a four-qubit bound entangled statereferred to as the Smolin state using photon polarization qubits. We observed the unlockablebound entanglement which is the specific property of the Smolin state.

KEYWORDSQuantum information, entanglement, quantum state preparation

1 IntroductionEntanglement plays a crucial role in quantum in-

formation processing (QIP) [1]. It has been thoughtthat pure and strongly entangled states are essential inQIP [2], [3]. In contrast, a class of the multipartite en-tanglement referred to as bound entanglement [4] hasbeen considered the weak and useless entanglement forany QIPs, since the bound entanglement cannot be dis-tilled into pure entanglement under local operations andclassical communication (LOCC). However, it is inter-esting that the bound entanglement in a certain kind ofbound entangled state can be activated when two of theparties coming together. The process is called “unlock-ing” [5]. It is also possible that two independent boundentangled states cooperatively distill the entanglement.The process is called “superactivation” [6]. These in-teresting properties of the bound entangled states haveattracted attention in QIP applications, e.g., remote in-formation concentration [7], secure QKD [8], and con-vertivility of multipartite pure entangled states [9].

Despite the theoretical interests, the experimental re-

Received October 31, 2010; Revised December 2, 2010; Accepted December28, 2010.1) kaneda@quantum.riec.tohoku.ac.jp, 2) r-simizu@pc.uec.ac.jp,3) mitumori@riec.tohoku.ac.jp, 4) kosaka@riec.tohoku.ac.jp,5) eda@riec.tohoku.ac.jp

alization of the bound entangled states is still a chal-lenging task. To realize the bound entangled states us-ing photons, we need (1) the high generation and de-tection efficiencies of multiple photon pairs, which de-crease exponentially with increasing number of pho-tons concerned, and (2) precise and stable control ofthe entanglement in multiple photon states, e.g., entan-glement of the polarization states of the photons. Thesuccessful demonstrations of the four-qubit bound en-tangled state (Smolin state [5]) were reported recentlyby Amselem et al. [10] and by Lavoie et al. [11]. How-ever, the generation rate of the Smolin state was not suf-ficiently high for further demonstration of the valuableQIP protocols based on bound entanglement. Thus, ef-ficient generation of multi-qubit state is indispensable.In this paper, we report the generation and characteri-zation of the four-qubit bound entangled Smolin state[5] encoded in photon polarization qubits, with muchhigher efficiency than ever reported.

2 The bound-entangled Smolin state

The Smolin state is an equal statistical mixture ofpairs of the four Bell states. Its density matrix ρs is

DOI: 10.2201/NiiPi.2011.8.3

c©2011 National Institute of Informatics

28 Progress in Informatics, No. 8, pp.27–31, (2011)

given by

ρs =

4∑i=1

|φiAB〉〈φi

AB| ⊗ |φiCD〉〈φi

CD |, (1)

where |φi〉 ∈ {|φ±〉, |ψ±〉} are the two-qubit Bell states.The Bell states encoded in photon polarization aregiven by

|φ±〉 = 1√2

(|HH〉 ± |VV〉) (2)

|ψ±〉 = 1√2

(|HV〉 ± |VH〉) . (3)

Here, |H〉 and |V〉 indicate the single photon states hav-ing horizontal and vertical polarizations, respectively.As shown in Eq. (1), the Smolin state is separableacross the bipartite cut AB|CD. Since the state has asymmetry with respect to the exchange of any two par-ties, it is separable with respect to any two-two bipartitecuts. This implies that no entanglement can be distilledbetween any two parties.

3 ExperimentTo explore the multipartite entanglement including

the bound entanglement, we have developed the exper-imental system optimized for efficient generation andfast evaluation of the multi-photon states [12]. The ex-perimental setup for the generation and characterizationof the Smolin state is sketched in Fig. 1. We used thethird harmonics (wavelength = 343 nm, average power= 200 mW, pulse duration = 250 fs) of the mode-lockedYb laser (Amplitude Systems, t-Pulse 500) as a pumpsource for SPDC using two Type-II beta barium borate(BBO) crystals. The pump source has higher pulse en-ergy (∼60 nJ) than those of typical multi-photon gen-eration systems based on the second harmonics of Ti-Sapphire lasers (10∼20 nJ) (e.g., Ref. [13]). Moreover,the center wavelength (686 nm) of the SPDC pumpedby our source is close to the spectral region where Si-avalanche photodiodes (Si-APD) have maximum quan-tum efficiency (∼65%). Each photon produced by theSPDC was passed through a band-pass filter (BPF, cen-ter wavelength = 686 nm, FWHM = 1 nm) for spectralfiltering, and was led to a single-mode fiber (SMF) forspatial filtering. The filtering processes make photonsindistinguishable except for their polarization. EachBBO crystal emits photon pairs having polarization en-tanglement in the Bell state |ψ+〉. When the two BBOcrystals simultaneously produce the pair of Bell states,the resulting four-photon state is given by

ρ(ψ+) = |ψ+〉AB〈ψ+ | ⊗ |ψ+〉CD〈ψ+|. (4)

The liquid crystal valuable retarder (LCVR) adds ar-bitrary phase shifts between two orthogonal polariza-

Fig. 1 Experimental setup for the generation of the Smolinstate. LCVR: liquid crystal variable retarder, POL: polar-izer, BPF: band-pass filter, SMF: single mode fiber, APD:avalanche photodiode detector.

tions; it can add π phase shift between H and V polar-izations, or, it can flip the two polarizations. In this way,the state |ψ+〉 generated by each BBO crystal can beconverted into any of the four Bell states by using twoLCVRs. Thus, by synchronously controlling the fourLCVRs, we can prepare the Smolin state (1), i.e., ran-dom and uniform statistical mixture of the four states,ρ(ψ+), ρ(ψ−), ρ(φ+), and ρ(φ−). The retardation of eachLCVR is set by a pseudo-random number generator op-erating at a rate of 3 Hz. The polarizers, each of whichconsists of a quarter wave plate, a half wave plate and apolarization beam splitter, project the photons into anypolarization states. The photons passed through the po-larizers were detected by the single-photon avalanchephotodiode detectors (APDs). Then, the signal pulsesfrom the APDs are recorded by a multifold coincidencecounter. The typical single, two-fold and four-fold co-incidence rates were 1×105, 2×104, and 4×101 sec−1,respectively. To carry out the quantum state tomogra-phy, we recorded the four-fold coincidence signals for256 combinations of the polarization projection mea-surements. The recording time for each projection mea-surement was four minutes, and the corresponding totalmeasurement time was ∼18 hours. For the tomogra-phy process, the maximally likelihood method [14] wasused to avoid reconstruction of unphysical density ma-trices.

4 Results and discussionFig. 2 (a) and (b) show the real part of the recon-

structed density matrix ρexp and that of the Smolin state

ρs, respectively. The fidelity F =(Tr√√

ρsρexp√ρs

)2

of the reconstructed matrix ρexp to the ideal Smolin stateρs was 0.876. This value is comparable with those of

Generation and characterization of bound entanglement in optical qubits 29

Fig. 2 Real part of the density matrices. (a) measuredstate ρexp and (b) Smolin state ρs.

the previous experiments [10] of the Smolin state gen-eration. It is noteworthy that the total measurementtime of our experiment is almost 10 times shorter thanthose of the experiments (∼10 days). This comes fromthe high multi-photon counting rate of our experimentalsystem.

From the reconstructed density matrix ρexp, we eval-uated the separability of the generated state across thebipartite cuts AB|CD, AC|BD, and AD|BC in terms ofthe negativity [15], which quantifes the distillable en-tanglement of bipartite cuts under LOCC. The negativ-ity of the density matrix which is composed of two sub-systems ρab is given by

N(ρab, a) =||ρTa

ab|| − 1

2, (5)

Table 1 The negativity values for the two-two bipar-tite cuts.

ρexp ρs ρ(ψ+)

N(ρ, AB) 0.06 ± 0.01 0 0

N(ρ, AC) 0.11 ± 0.02 0 1.5

N(ρ, AD) 0.12 ± 0.02 0 1.5

where ρTa

ab represents the partial transpose with respectto the subsystem a, and ||ρTa

ab|| is the trace norm of ρTa

ab.For instance, the separability across the bipartite cutAB|CD is quantified as the N(ρexp, AB). The negativ-ity values for the three bipartite cuts of the ρexp are pre-sented in Table 1, together with those of the Smolinstate ρs and the state ρ(ψ+) in (3). The Smolin state ρs

has zero negativity for all the three bipartite cuts, whilethe state ρ(ψ+) has finite values, i.e., finite distillable en-tanglement, for AC|BD and AD|BC cuts. For ρexp, thenegativity values are all close to zero, indicating that ithas almost no distillable entanglement as we expect forthe Smolin state.

Furthermore, to quantify the entanglement withoutthe one-three separability and three-party entangle-ment, we calculated the expectation value of the sta-bilizer witness Tr{Wρexp}, where W = I⊗4−σ⊗4

x −σ⊗4y −

σ⊗4z and I is the two-dimensional identity operator, and

σx, σy, and σz are the Pauli operators [10], [16]. Weobtained Tr{Wρexp} = −1.51 ± 0.01; the values for theideal Smolin state and the ρ(ψ+) is−2. Note that no sep-arable, triseparable and one-three separable states (e.g.,A|B|C|D, A|B|CD, and A|BCD) have negative values ofthe witness. The negative witness value, therefore, in-dicates that ρexp has a considerable amount of entan-glement, while it is almost undistillable as indicated bythe negativity described above. These analyses clearlydemonstrate that the state we generated has bound en-tanglement.

Finally, it is worth discussing the reason for the de-graded fidelity of the Smolin state we prepared. Fromthe reconstructed density matrix of each Bell state wegenerated, we expected the fidelity of the preparedSmolin state to be 0.980, assuming perfectly equal sta-tistical mixture of pairs of the four Bell states. Thus,the degraded fidelity of the reconstructed density ma-trix ρexp possibly originates from insufficient statisticsof our measurements. To improve the fidelity, andthus the negativities and witness of the prepared Smolinstate, longer measurement time with more random anduniform distribution between ρ(ψ+), ρ(ψ−), ρ(φ+), andρ(φ−) would be necessary.

30 Progress in Informatics, No. 8, pp.27–31, (2011)

5 ConclusionWe have experimentally demonstrated the efficient

generation and characterization of the four-qubit boundSmolin state, encoded in photon-polarization qubits.We reconstructed the density matrix of the genetatedstate by full quantum state tomography with ∼10 timesshorter measurement time than that of the previousdemonstration of bound entanglement. The analyses ofthe reconstructed density matrix showed that the gen-erated state has almost no distillable entanglement be-tween any two parties, while the state still contains aconsiderable amount of entanglement. These resultsdemonstrate the bound entanglement involved in thegenerated state. Further experiments to demonstrateQIP protocols using the Smolin state are in progress.

This work was supported by a Grant-in-Aid for Cre-ative Scientific Research (17GS1204) from the JapanSociety for the Promotion of Science.

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Fumihiro KANEDAFumihiro KANEDA is a Ph.D. stu-dent in Department of Electronic En-gineering, Graduate School of Engi-neering, Tohoku University.

Ryosuke SHIMIZURyosuke SHIMIZU is an associateprofessor of Center for Frontier Sci-ence and Engineering, University ofElectro-Communications, and a re-searcher at PRESTO, Japan Scienceand Technology Agency.

Yasuyoshi MITSUMORIYasuyoshi MITSUMORI is an assis-tant professor of Research Institute ofElectrical Communication, TohokuUniversity.

Generation and characterization of bound entanglement in optical qubits 31

Hiedo KOSAKAHiedo KOSAKA is an associate pro-fessor of Research Institute of Elec-trical Communication, Tohoku Uni-versity.

Keiichi EDAMATSUKeiichi EDAMATSU is a Professorof Research Institute of ElectricalCommunication, Tohoku University.He received B.S., M.S., and D.S. de-grees in Physics from Tohoku Uni-versity. He was a Research Associate

in Department of Applied Physics, Faculty of Engineer-ing, Tohoku University, a Visiting Associate in Nor-man Bridge Laboratory of Physics, California Insti-tute of Technology, and an Associate Professor in Di-vision of Materials Physics, Graduate School of Engi-neering Science, Osaka University. He is a member ofthe Physical Society of Japan and Optical Society ofAmerica. His current research interests are solid-statephotophysics, quantum optics, and quantum informa-tion/communication science.