ISSN 1054-660X, Laser Physics, 2006, Vol. 16, No. 7, pp. 1140–1144.

QUNTUM INFORMATION AND QUANTUM COMPUTATION

© MAIK “Nauka /Interperiodica” (Russia), 2006. Original Text © Astro, Ltd., 2006.

Preparation of Bell States with Controlled White Noise A. Ling, P. Y. Han, A. Lamas-Linares, and C. Kurtsiefer Department of Physics, National University of Singapore, Singapore, 117542 e-mail: [email protected] Received October 8, 2005

Abstract—We report two methods for producing Bell States with an arbitrary amount of white noise. White noise in this context refers to controlled admixtures of unpolarized light. Our methods differ from previous experiments in that we use the minimum necessary elements for generating a Bell state by c-w spontaneous parametric down conversion. We also investigated the spectral properties of a mixed state and show that one of our methods introduces irreversible noise into the Bell state, making a permanent mixed state. PACS numbers: 03.65.–w, 03.67.–a DOI: 10.1134/S1054660X06070206

1. INTRODUCTION Quantum information is a field that uses the nonclassical properties of physical systems to solve complex problems of computation and communication [1]. Often these quantum properties use the correlation functions that exist between separated physical systems [2]. This nonclassical correlation is known as entanglement. When two systems share such quantum correlations, they are said to be in an entangled state. In fact, quantum information often uses specific entangled states. Moreover, it requires the states to be pure and maximally entangled. Bell states are pure and maximally entangled states and are useful for a variety of applications. For example, in quantum key distribution (QKD), the Ekert91 [3] protocol uses Bell states to grow a secret key between two distant parties. In optics, a Bell state refers to a pair of photons whose polarizations are (anti)correlated. This means that the polarization states of the two photons are always orthogonal with respect to each other no matter what measurement basis is used. To detect Bell states, one would first look for a pair of photons within a welldefined time window, since Bell state photons are highly correlated in time. If a pair is detected in coincidence, then a check is made to see if their polarizations are (anti)correlated. Satisfying these conditions means that photons sharing a Bell state have been detected and sundry information processes can be carried out based on this detection. The singlet state is an example of a Bell state whose photon polarizations are always (anti)correlated. It can be expressed as 1 |Φ〉 = ------- ( |H A〉 |V A〉 – |V A〉 |H A〉 ), 2

(1)

1 = ------- ( | A+〉 |B–〉 – | A–〉 |B+〉 ). 2

(2)

In a realistic implementation, the photons would interact with the environment and their polarizations would change in arbitrary ways, or they could get mixed with stray light leading to mistaken correlations at the detectors. These arbitrary changes lead to the loss of ideal anticorrelations between the pairs of photons. This loss in the ideal anticorrelation is called noise. For entanglement-critical QKD protocols, the security depends on the lack of noise. In order to maintain the integrity of the secure key exchange, noise must be viewed as evidence of an eavesdropper. This makes it desirable to study noisy states. Theoretically, the disturbance due to an eavesdropper is modeled as white noise, since this is easy to express analytically. A Werner state [4] is an example of a Bell state plus white noise. It can be expressed as r ρ W ( r ) ≡ ( 1 – r ) |Φ〉 〈Φ| + --- , 4

(3)

where r is the noise admixture. However, real noise is often polarized and not white. To test the theoretical claims of security protocols based on white noise, it is desirable to make idealized sources of noise. Experimental realizations of Werner states have been reported previously in [5–8]. The common feature of these experiments was that they all used a technique called temporal decoherence. Temporal decoherence occurs when one photon in a Bell state is delayed beyond the coherence time. Thus, when coincidences are looked for, the polarization correlations are less than ideal. Previous experiments have all utilized quartz plates of varying thicknesses to achieve the timing delay. However, this is cumbersome, because one needs a large collection of quartz plates to input an arbitrary amount of noise. There has been at least one proposal to generate Bell states without quartz plates [9]. We introduce two more methods that avoid the use of quartz plates and present experimental measurements. The first method uses

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temporal decoherence, but this is achieved by manipulating the detection apparatus directly. The second method uses blackbody radiation.

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Counts per 100 ms 900 800

2. THE EXPERIMENTAL SETUP 2.1. The Singlet Source To create an arbitrary Werner state, there must first be a source capable of producing Bell states. To obtain our singlet states, we use a Type-II spontaneous parameteric down conversion (SPDC) technique first reported in 1995 [10]. We use an Argon Ion laser at 351 nm to pump a BBO crystal that has been cut for noncollinear phase matching. With noncollinear SPDC, the two photons of the singlet state travel in two different directions. By mode matching techniques, we are able to collect photons of narrow bandwidth into single-mode fibers. The bandwidth of the light collected was 4.74 ± 0.04 nm. The efficiency of our source was found to be 700 coincidences per mW of pump power. We proceeded to characterize the singlet states by performing a polarization correlation analysis. Because optical fibers are birefringent, they arbitrarily rotate the polarization of the photons they carry. The single-mode fibers were made neutral to polarization by bat-ear controllers. The polarization correlation measurement consisted of a half-wave plate followed by a polarizing beam splitter whose transmission arm is coupled to a single photon counter. In this way, we are able to project any polarization basis to be detected. The photon counters were Silicon Avalanche Photodiodes that were passively quenched. The polarization correlation measurement was carried out for horizontal (H) and vertical (V) linear polarizations, as well as for the halfway H and halfway V orientation, denoted by 45°. To select a basis, the halfwave plate in one arm is rotated to project either H or 45° into the detector. The half-wave plate in the other arm is then rotated one full revolution. This causes the coincidences collected to vary sinusoidally. The visibilI max – I min ity is defined as ----------------------, where the Is are the maxiI max + I min mum and minimum count rates observed. For ideal polarization correlations, the visibility is 100%. In the linear basis, the visibility was measured to be 99.3 ± 0.3%, while in the 45° orientation the visibility was measured to be 96.4 ± 0.3%, showing that we had a high-quality source of entangled photons. 2.2. Getting Noise Via the Time Window It is possible to induce white noise in our measured state by manipulating the time window of the measurement apparatus. The collection efficiency is defined as the ratio of coincidence counts to single counts. Our collection setup has an efficiency of 25%; although this is a bright source of entangled photons, it still leaves a lot of photons that are not registered as coincidences. LASER PHYSICS

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700 600 500 400

FWHM

300 200 100 699.8

704.6

0 694 696 698 700 702 704 706 708 710 Wavelength, nm Fig. 1. Typical spectrum of down-converted light collected by mode matching. From the fitted curve, the central wavelength is found at 702.2 nm with a FWHM of 4.74 nm.

Sometimes, a pair of single photons will be registered as a valid pair. Such pairs are known as accidental coincidences. The rate of accidental coincidences is proportional to the rate of singles and the time window width. In most cases, these accidentals are to be suppressed; this is done by using narrow time windows (on the order of 5 ns). To induce noise, we could simply increase the rate of detected accidental coincidences. If we were to increase the time window width, the probability of detecting an accidental coincidence increases. Essentially, the system is detecting photons from different pairs. Because the polarizations of different pairs of photons are not correlated, accidental pairs lead to noise. Our time window width is controlled by a variable capacitor, allowing us to vary the width continuously. For each value of the time window, we performed a polarization correlation test. Figure 3a shows the measured visibilities as a function of time window values. We are limited to a maximum value of the time window because of our electronic components. 2.3. Inducing Noise by a Blackbody In our second method, we return our nominal time window of 5 ns. An incandescent light bulb is then used to mix thermal light with the collected down conversion light. An increase in noise is obtained by increasing the power delivered to the light bulb. Although increasing the power to the light bulb would change the temperature of the filament and change the spectrum of the thermal light, this problem can be overcome by using inter-

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LING et al. Coincidences in 100 ms HV ±45 800

Visibility, % 100 (a)

90

600

80 400 70 200 60 50

100 150 200 250 300 350 Wave plate 2 orientation, deg

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Fig. 2. Visibilities in two measurement orientations. The error bars are smaller than the dots used.

ference filters. We use interference filters with a FWHM of 5-nm bandwidth (slightly larger than the bandwidth of light we collect). This ensures that the noise we collect is restricted to a sufficiently narrow bandwidth and does not vary significantly. The light bulbs are placed at any suitable position of the photon transmission channel. The resulting mixed light is then checked for polarization correlations. The change in visibility with power is shown in Fig. 3b. We are able to cause the visibility to drop to 0% by this method, as the limit is the maximum power the light bulb can sustain. The drop in visibility varies linearly with power for low power, but near the zero visibility region the variation is much slower. A possible reason for this is that the light bulb might be heating up and its blackbody spectrum is shifting, causing the intensity in this bandwidth to be dropping. The noise admixture of a Werner state, r, and the visibility, v, are related simply as v = 1 – r. To confirm that these methods do create a Werner state, we performed state tomography and calculated the density matrix of each state. In the next section, we will show that the noise admixture calculated from those density matrices vary with visibility as predicted. 3. DENSITY MATRIX OF WERNER STATES Of several methods to do quantum state tomography of a qubit [11], there exists at least one method to perform minimal tomography easily [12]. Quantum state tomography on a two-photon system would produce a 16-element density matrix, and minimal tomography allows us to take only 16 readings to perform a complete tomography.

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1.4 1.6 1.8 Power per bulb, W

Fig. 3. Figure 3a shows the change in visibility with increasing time window. The solid line is the predicted change in visibility for increasing time window. It takes into account the reduced efficiency of the collection apparatus as the time window increases. Figure 3b shows the measured change in visibility with changing bulb power. The error bars are smaller than the marks.

The general density matrix of a Werner state looks like ⎛ ⎜ ⎜ ⎜ ⎜ ρ = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

r --4

0

2–r 0 ----------4 r–1 0 ----------2 0

0

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0 r–1 ----------2 2–r ----------4 0

⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟. ⎟ 0⎟ ⎟ r⎟ --- ⎟ 4⎠

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PREPARATION OF BELL STATES WITH CONTROLLED WHITE NOISE (a)

r value

Noise admixture (r) vs. polarization correlations

100

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708 706 λ1, nm

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40 700 20 0

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40 60 80 100 Polarization correlations, visibility

1 2 r = 1 + --- [ ρ 11 + ρ 44 – ρ 22 – ρ 33 ] + --- [ ρ 23 + ρ 32 ]. (5) 3 3 Knowing this, we performed state tomography on the measured state for a range of visibilities. From the constructed density matrices, we computed the r value and plotted them against the visibility. This plot is shown in Fig. 4. The measured data follows the theoretical prediction perfectly, showing that our methods of introducing noise do produce Werner states. 4. REMOVING NOISE In most artificial methods of producing Werner states, the noise can be removed in trivial ways. If a quartz plate was used to introduce timing delays, another quartz plate of the same thickness oriented properly at another place in the transmission line would reverse the delay. Similarly, in our method of achieving a Werner state by widening the time window, we could reverse the Werner state by narrowing the time window. A Werner state created by thermal light, however, would be difficult if not impossible to reverse. This can be understood by looking at the spectral properties of the mixed state [13]. In particular, one would need to see if thermal photons can be distinguished from SPDC photons. To gain a better understanding of these spectral properties, we investigated the relationship between photon pair coincidences and their wavelength dependence. Such a measurement is done by passing the light from each arm of the polarization correlation LASER PHYSICS

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120 80 40

708

Fig. 4. The noise admixture calculated from the density matrices is shown. The straight line is the predicted change in the r value with visibility.

0

706 λ1, nm

The r values represent the amount of noise in the system. The ideal singlet state will have r = 0 and only the central terms of the density matrix are nonzero. Thus, the value of r is easily extracted from the density matrix by the following equation:

701 703 λ 2, nm (b)

704 702 700 698 697

699

701 703 λ 2, nm

705

Fig. 5. Spectral dependence of coincidence intensity is revealed by taking the joint spectrum of photon pairs. Figure 5a shows the joint spectrum from a pure singlet state, while Fig. 5b shows the joint spectrum from a Werner state at 46% visibility. The resolution of the monochromators was 0.3 nm.

measurement into a monochromator. Using the monochromators, we are able select pairs of wavelengths and measure the coincidence rate at each. This allowed us to scan the range of wavelengths over which SPDC occurs and build up a map of coincidence intensities. A similar technique for obtaining spectral information of down-converted photon pairs has been reported independently by Kim and Grice [14]. Two such maps are presented in Fig. 5. In both maps, we see two strong lines of coincidence corresponding to two pump wavelengths, 351.1 and 351.4 nm. These are argon ion excitation wavelengths and show that one is able to obtain a high-quality singlet state even though the cw pump has two wavelengths. In both maps, no bandwidth filters were used. In Fig. 5b, the map of a Werner state is seen to have a strong background due to thermal light and is very clearly mixed in with light from the SPDC source.

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Using filters of very narrow bandpass, one could hopefully filter out most but never all of the thermal light. Thus, we have created an irreversible Werner state. 5. CONCLUSIONS In summary, we have presented two simple ways to create Werner states from a continuous-wave SPDC source. The main observation we made was that white noise in a Werner state can be related to accidental coincidences. The first method was to detune the timing precision of the measurement apparatus, leading to a high accidental count rate. The second method of obtaining accidental coincidences was to mix light from the SPDC source with light from a blackbody. Both methods create Werner states whose polarization correlations were continuously variable. That the states were indeed Werner states was confirmed independently by performing state tomography and reconstructing their density matrix. From these density matrices, we showed that the noise admixture present is directly propotional to their polarization correlation (as predicted by theory). Lastly, we inspected the spectral properties of the pure singlet state and a Werner state. From the spectral map of the Werner state created by mixing thermal and down-converted light, we see that their spectral information is not fully distinguishable. To conclude, we have created idealized noisy states that are not easily reversible.

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