1. Introduction

Decoherence of quantum states occurs through unwanted coupling with the environment, which often negatively impacts quantum information processing and communication. Much efforts have been invested to control it or even suppress it. They include quantum error correction [1,2] and dynamical decoupling [3,4]. An arguably viable alternative is to encode information in a decoherence-free subspace (DFS) [5–8].

Such a DFS protocol is appropriate when the system-bath coupling allows for a certain degree of symmetry. Collective decoherence is a typical example. It is relevant when all qubits undergo an identical random unitary transformation. It manifests itself when particles are in much the same environment so that they are indistinguishable or when a shared reference frame is missing in communication [9–11]. In the all-important two-qubit system, the DFS for the collective decoherence consists of the singlet, $|{\Psi _-}\rangle = (|{\mathrm {HV}}\rangle - |{\mathrm {VH}}\rangle)/ \sqrt {2}$, where H(V) is the horizontal(vertical) polarization of photon. Notice that the two arguments in the ket represent photons specified using different degrees of freedom (wavelength or frequency in our experiment) other than polarization.

Importantly, quantum entanglement between such indistinguishable particles invokes arguments. If one attempts to monitor the evolution of entangled states as they go down a collective decoherence channel, individual photons must be labeled so that they are distinguishable. Due to the bosonic symmetry, however, a biphoton state must be symmetric upon exchange of photons. This in turn suggests that introducing asymmetry with another degree of freedom to the otherwise antisymmetric two-photon singlet can restore the lost bosonic symmetry and thereby permitting coupling to the same mode of the transmission channel such as a single mode fiber (SMF).

The experimental evidence for the DFS has been largely obtained by using the polarization degree of freedom of photon [7,8]. When it comes to fiber transmission, however, photon pairs slightly offset in time have been used to exploit the DFS [9,10]. Such a time-domain protocol seems robust and useful since noise in fibers is deemed only slowly-varying. However, difficulties arise in preparing and measuring the biphoton states [11]. This is because they rely on probabilistic processes like particles partitioning on a beam splitter and as such success rates remain low unless high-speed fiber switches are available.

In this work, wavelength division is explored as a distinct class of the DFS pertaining to the SMF. To this end, polarization-entangled photon pairs with a slight detuning, e.g., $|{\Psi _-}\rangle = (|{\mathrm {H}_{\lambda }\mathrm {V}_{\lambda +\Delta \lambda }}\rangle - |{\mathrm {V}_{\lambda }\mathrm {H}_{\lambda +\Delta \lambda }}\rangle)/ \sqrt {2}$ (subscripts for wavelengths dropped hereafter), were coupled to the SMF, and their decoherence due to birefringence induced randomly in the fiber was thoroughly investigated. Note here that our focus of interest is not so much on nonlocal correlation with the Bell inequality in mind as on the properties of the localized entangled states. Asymmetry in the frequency (or wavelength) degree of freedom enables the singlet state, which is antisymmetric upon particle exchange, to couple to an identical waveguide mode [12]. This allows us to study the DFS under collective decoherence caused by interactions with the dynamically varying environment. Moreover, our protocol has a great affinity for the modern fiber communication technology by design. In fact, it permits lossless add/drop of photons with different colors and even allows one to discriminate photon polarization states, being leveraged by the advanced wavelength division multiplexing/demultiplexing [13,14].

2. Results and discussion

2.1 Collective decoherence in a two-qubit system

We consider a two-qubit state transmitted through a collective decoherence channel. We start with an arbitrary two-qubit state such that [15]

Thus the input state is mapped to the Werner state [20],

Hence among the four Bell states ($|{\Phi _\pm }\rangle = (|{{\textrm {HH}}}\rangle \pm |{{\textrm {VV}}}\rangle)/\sqrt 2$, $|{\Psi _\pm }\rangle = (|{{\textrm {HV}}}\rangle \pm |{{\textrm {VH}}}\rangle)/\sqrt 2$), only the singlet $|{\Psi _-}\rangle$ $(z=1)$ remains unchanged while the triplets, $|{\Psi _+}\rangle$ and $|{\Phi _\pm }\rangle$, eventually decohere to the identical separable Werner state, $\rho _{\textrm W}(-1/3)$.

The schematic experimental setup is shown in Fig. 1. The photon source is shown on the left with the state preparation unit. The violet-blue output from a cw diode-laser (center wavelength of 405 nm) is loosely focused onto a 3-mm-long noncollinear type-II BBO crystal. Two orthogonally-polarized correlated photons conserving the energy and momentum are generated by spontaneous parametric down-conversion and launched into the phase-matching directions.

 

Fig. 1. Schematic setup. Red (purple) arrows indicate the signal (idler) photon paths. Solid (dashed) lines indicate the horizontally (vertically) polarized photons. The first half waveplate (HWP) enables entanglement concentration by distributing photons in two spatial modes according to their colors. Fan allows random fiber bending. PBS, polarization beam splitter; HWP, half waveplate; QWP, quarter waveplate; SPCM, single-photon counting module; SMF, single mode fiber.

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The entanglement concentration technique [21] is used to create perfectly polarization-entangled two-color photons: The first HWP rotates the photon polarization by $90^\circ$ so that photons are distributed to either of the two phase-matching directions according to their polarizations within the BBO crystal. For state preparation, we used a half wave plate (HWP) and a quarter wave plate (QWP) after a short SMF, and the path length difference was adjusted by use of a linear translation stage.

Polarization-entangled photon pairs were localized at a 20-m-long SMF (Thorlabs 780HP) by dichroic mirror coupling using a tilted bandpass filter (Thorlabs FBH810-10, FWHM 10 nm) (Fig. 2). The measured transmission center of the filter was approximately 805 nm and a tilt angle $\approx$ 15$^{\textrm o}$ from the normal incidence of the photon beam was used. Its reflectance spectrum roughly matched the inverted transmittance one and as such it acted like a notch filter. However, due to the frequency correlation between photons centered at 810nm, only photons with a wavelength of approximately 815nm in coincidence with the transmitted photons were extracted by post-selection. As a result, the bandpass filter acts essentially as a dichroic mirror. A model collective decoherence and hence depolarization channel was created by randomly bending and twisting the SMF incessantly under a turbulent dynamic air flow using a blower. A linear polarizer was set before and after the SMF to examine the polarization status of the photon pairs by measuring photon counts every 20 ms using an avalanche photodiode operated in the geiger mode (Perkin Elmer SPCM-AQR-14-FC). Figures 2(a) and 2(b) show the time evolution of photon count (a) with and (b) without the air flow. One can see that the two-photon polarizations vary wildly while largely tracking each other when the decoherence channel is activated (Fig. 2(a)), which contrasts with the case when the blower is deactivated (Fig. 2(b)). The normalized cross-correlation function of Fig. 2(a)(b) is shown in Fig. 2(c)(d). The coincidental cross-correlation amounts to 1.23 in Fig. 2(c), which indicates that the two-photon polarization has been randomized in a collective manner. To the contrary, the normalized cross-correlation in Fig. 2(d) is nearly leveled off around unity, which shows that fluctuations associated with the normalized count rate of Fig. 2(b) are most likely due to the uncorrelated noise inherent to the detector and/or the photon source.

 

Fig. 2. Normalized count rate of photons with a slight detuning per every 20 ms by setting a polarizer before and after an SMF. Compared are (a) when the SMF is subject to random bending and twisting under a turbulent air flow with a blower and (b) when the blower is off. (c) Cross-correlation of (a). (d) Cross-correlation of (c).

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Coincidence count during a 7.8-ns time gate in the diagonal basis ($|{\pm }\rangle = \{|{\textrm H}\rangle \pm |{\textrm V}\rangle\}/\sqrt 2$) is plotted in Fig. 3 (a) as a function of the path length difference, $x$ (see Fig. 1). Clear beats of 100-% visibility are a signature of high-purity two-color polarization-entangled state [22–24]. Such oscillatory behaviors are traced back to a wavelength-dependent phase shift the photons undergo. From the beat frequency, the wavelength detuning is found to be about 10 nm. Figures 3(b) and 3(c) show the two-photon beats observed at the output port of the SMF decoherence channel. The following two polarization-entangled biphoton states were input-coupled to the fiber to see if two-photon beats with a high initial visibility is negatively impacted after fiber transmission,

Here $\epsilon (x) = \eta \, {\textrm {exp}}[-(x/d)^{2}]$ with $d$ being determined by the pass band while $\eta$ is a measure of the quality of entanglement corresponding to the visibility of two-photon beats ($\approx 0.95$ in Fig. 3). It is also noted that

At $x=0$, $\rho _\Psi (x)$ and $\rho _\Phi (x)$ are the maximally entangled states $|{\Psi _-}\rangle$ and $|{\Phi _+}\rangle$ for $\eta =1$, respectively. Here we introduce the expected value $\langle {{*}}\rangle _{\bot (\parallel )}$ for projection measurements in the orthogonal (parallel) two-photon basis since it is the angle between the two basis vectors as opposed to a particular choice of the basis that counts after the depolarizing channel. For $\rho _\Psi (x)$, we find

Likewise for $\rho _\Phi (x)$, we should have

Thus $\rho _\Psi (x)$ is predicted to continue to oscillate as a function of $x$ even after the transmission along the depolarization channel, as opposed to $\rho _\Phi (x)$ that is reduced to a constant depending on the polarization. Conversely, this indicates that $\rho _\Psi (0) = {|\Psi _-}\rangle \langle\Psi_-|$ for $\eta = 1$. These are consistent with the experimental results shown in Fig. 3.

 

Fig. 3. Coincidence count in the orthogonal (blue open circle) and parallel basis (red closed circle) (a) before the entry to and (b) and (c) after the propagation through an SMF decoherence and hence depolarization channel. Note that the results in (b),(c) remain unchanged regardless the choice of a set of two-photon orthogonal and parallel basis. Beat fringes of a high visibility for $\rho _{\Psi }$ in (b) reflect the robustness of $|{\Psi _-}\rangle$ at x=0, which compares with $\rho _{\Phi }$ in (c) where fringes are apparently missing.

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Figure 4 shows the real part of the experimentally reconstructed density matrix $\rho$ by quantum state tomography using the maximum likelihood method before (a,c) and after (b,d) the decoherence channel. For each photon path, we performed polarization measurements necessary for quantum state tomography using the following basis set. In the path reflected off the DM, a motorized HWP was rotated in 5$^{\textrm o}$ steps from 0$^{\textrm o}$ to 90$^{\textrm o}$ both with and without an QWP at 45$^{\textrm o}$ in front of the PBS. Thus the measurement basis changed roughly as follows: $|{\textrm H}\rangle \to |{\textrm +}\rangle \to |{\textrm V}\rangle \to |{\textrm -}\rangle \to |{\textrm H}\rangle$ or $|{\textrm R}\rangle \to |{\textrm +}\rangle \to |{\textrm L}\rangle \to |{\textrm -}\rangle \to |{\textrm R}\rangle$. In the other transmission path, we used six measurement bases implemented by manually switching from one to one as follows: $|{\textrm H}\rangle, |{\textrm V}\rangle, |{\textrm R}\rangle, |{\textrm L}\rangle, |{\textrm +}\rangle$ and $|{\textrm -}\rangle$. Coincidence was measured for 5 seconds for each pair of the 19$\times$2$\times$6 bases in total. The fidelity error bars were drawn using the bootstrap method. We randomly selected 228 new bases from the above measurement bases, allowing overlap, and generated 10$^3$ estimated matrices with their fidelity values. Notice that the error bar indicates the 68%-interval centered on the median of the population. $|{\Psi _-}\rangle$ (Fig. 4(a)) with fidelity $F = \sqrt { \langle {\Psi _-}|{{\rho }|{\Psi _-} } }\rangle = 0.958 \pm 0.009$ was sent through the SMF decoherence channel. Figure 4(b) shows that $|{\Psi _-}\rangle$ remains essentially unchanged with $F = 0.951 \pm 0.009$. In contrast, $|{\Phi _+}\rangle$ (Fig. 4(c)) with $F = \sqrt {\langle {{\Phi _+}|\rho |{\Phi _+}} }\rangle = 0.956\pm 0.004$ decohered to $\rho _{\textrm W}(-1/3)$ (Fig. 4(d)) with $F = {\textrm {Tr}}\sqrt {\sqrt {\rho _W(-\frac {1}{3})}\rho \sqrt {\rho _W(-\frac {1}{3})}} = 0.995 \pm 0.003$, leaving no discernible trace of two-photon beats.

 

Fig. 4. Quantum state tomography before (a), (c) and after (b), (d) the SMF collective decoherence channel. Note that only the real part is shown. Fidelity $(F)$ with the theoretically predicted state is found to be $0.958\pm 0.009$ (a), $0.951\pm 0.009$ (b), $0.956\pm 0.008$ (c), and $0.995\pm 0.023$ (d).

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2.2 Quantum communication in collective dephasing channel

Finally, we attempt secure quantum communication using the two-color protocol. To encode quantum bits subject to arbitrary collective decoherence, more than three photons are needed [8]. Instead we consider here only the channel noise due to collective dephasing, which adds a random phase shift $\phi$ such that $|{\textrm {H}}\rangle \to |{\textrm H}\rangle$ and $|{\textrm V}\rangle \to e^{i \phi } |{\textrm V}\rangle$. This implies the PMF transmission where the H (V) polarization of light is aligned with the slow (fast) axis of the PMF. It should be noted that while the implementation of the BB84 protocol requires a 2-dimensional DFS, there is only the singlet 1-dimensional one with all noise in the SMF considered. A two-dimensional DFS can be used by limiting the noise to the phase noise using the PMF. The corresponding 2-dimensional DFS is such that

Note that the dephasing channel adds an overall phase shift alone to the above states in such a way that $e^{i\phi }(\alpha |{\textrm {HV}}\rangle+\beta |{\textrm {VH}}\rangle)$. One quickly notices that states in this space are nonseparable except when $\alpha = 0$ and $\beta = 0$. We utilize this two-photon DFS for quantum key distribution by implementing the BB84 protocol [25,26]. $\{|{{\textrm {HV}}}\rangle, |{{\textrm {VH}}}\rangle\}, \{{|{\Psi _+\rangle}, |{\Psi _-}}\rangle\}$ are chosen as the non-orthogonal BB84 basis sets. Briefly, (i) the sender (Alice) randomly chooses the encoding basis and sends a two-photon state through the quantum channel, (ii) the receiver (Bob) measures coincidence count using a randomly chosen basis. (iii) Instead of Bell-state measurements [27], coincidence count is recorded in the diagonal basis $|{\pm }\rangle$, which can distinguish between ${|{\Psi _+}}\rangle$ and ${|{\Psi _-}}\rangle$. (iv) Finally, Alice and Bob compare their results over a classical channel and retain the matched pairs to generate the raw keys eventually.

Figure 5 shows the schematic setup. For demonstrative purposes, stationary state preparation and detection were chosen. Polarization-maintaining (PM) fiber (Thorlabs PM780-HP) was put in a turbulent air flow to emulate a collective dephasing channel. The polarization mode dispersion was compensated for by an orthogonal mechanical splice of fibers at the position of the cross mark. First we block one path and insert a linear polarizer in the other to test the conventional one-photon BB84 protocol. The results are shown in Fig. 6(a) for the computational (H,V) basis and Fig. 6(b) for the diagonal basis. In this phase-noise-only channel, bit error rate (BER) amounts to 25% due to the failure in the diagonal basis where no secure key can be generated. Next we examined the two-photon DFS, the result of which is shown in Fig. 6(c) for computational while Fig. 6(d) for diagonal basis. For an entangled state, even in the diagonal basis, the BER remains as low as 8.3% (Fig. 6(d)). The total average BER is 5.4%, which is low enough to generate secure keys. The BE source is presumably due to the imperfection associated with the state preparation.

 

Fig. 5. Experimental setup to implement the entanglement-enabled BB84 quantum key distribution protocol in a polarization-maintaining (PM) fiber. DM, dichroic mirror; PBS, polarization beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; SPCM, single-photon counting module. The fan induces birefringence fluctuations in the PM fiber.

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Fig. 6. Comparison of BB84 protocols: (a),(b) one-photon and (c),(d) two-photon. Qantum bit error rates are (a) $5.66\pm 0.04\ {\% }$, (b) $45.2\pm 0.12\ {\% }$, (c) $2.48\pm 0.15\ {\% }$, (d) $8.34\pm 0.17\ {\% }$, respectively. Different colors correspond to different input states as indicated.

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3. Conclusion

A distinct class of DFS was proposed by using the frequency degree of freedom of photons in an attempt to demonstrate entanglement-enhanced quantum communication. Quantum state tomography and two-photon beats showed that such one-dimensional DFS does exist in single-mode fibers. The fidelity of the polarization-entangled singlet Bell state $|{\Psi _-}\rangle$ at the input of a fiber depolarization channel subject to random bending remained almost unchanged even after propagation. This compares with the triplet $|{\Phi _+}\rangle$ that eventually decohered. Additionally, DFS-enabled dephasing-tolerant transmission of biphoton qubits in a polarization-maintaining fiber was demonstrated at a BER of 5.4% by implementing the two-photon BB84 protocol. Our frequency-based DFS is provably extensible to multi-particle entanglement as it has a natural affinity for the fiber communication technology by design. As such, it offers prospects for noise-tolerant photonic quantum information processing and even for imaging robust against scattering-fluctuation blur when it is properly coupled with the spatial mode of photons.

Admittedly, loss increases as the square of the communication distance in a two-photon BB84 protocol. However, in view of a typical loss rate of 0.2 dB/km for the 1.5-$\mu$m fiber communication, for example, a net transmission loss amounts to no more than 4 dB over 10 km. This seems low enough to make our two-photon protocol efficient and as such useful for short-range quantum communication including end-user quantum network [28] and handheld QKD devices [29,30].