Error filtration and entanglement purification for quantum communication

The key realization that led to the emergence of the new field of quantum information processing is that quantum mechanics, the theory that describes microscopic particles, allows the processing of information in fundamentally new ways. But just as in classical information processing, errors occur in quantum information processing, and these have to be corrected. A fundamental breakthrough was the realization that quantum error correction is in fact possible. However, most work so far has not been concerned with technological feasibility, but rather with proving that quantum error correction is possible in principle. Here we describe a method for filtering out errors and entanglement purification which is particularly suitable for quantum communication. Our method is conceptually new, and, crucially, it is easy to implement in a wide variety of physical systems with present-day technology and should therefore be of wide applicability.

Figure 1

Implementation of error filtration using multiple optical fibers. A source $\left(S_{1\nu }\right)$ produces a single photon that is coupled into an optical fiber. The photon is split into two using a fiber coupler (C). Note that an arbitrary state in a two-dimensional space (a qubit) can be prepared in this way by changing the coupling ratio of the coupler and by modifying the phase ${\varphi }_A$. In order to protect the state against noise, each basis state is multiplexed into two transmission states using $50∕50$ couplers. A single qubit is thus encoded into four transmission states, each traveling through a separate fiber. Because the photon cannot jump from one fiber to the other, it will only be affected during transmission by phase noise. Furthermore the noise in the different fibers will be independent. Hence the noise is of the type (independent phase noise on the different transmission channels) studied in the main text. The decoding is the reverse of the encoding procedure: two transmission states are combined into one receiver state using fiber couplers. This is done in such a way that in the absence of noise, constructive interference occurs and the photon always emerges in the receiver states. Due to the noise, the photon may not emerge in the receiver states, in which case it is discarded. But if the photon emerges in the receiver states, then the noise has been filtered out. The receiver can then use the filtered state. For instance he may, as described in the text, test the quality of the filtered state by carrying out the measurement shown (D, single photon detector). The measurement basis is chosen by varying the phase ${\varphi }_B$ and the coupling ratio of the measurement coupler. Note, however, that the measurement step is included in the figure for illustration only—it is not part of the filtration protocol per se. The receiver can use the filtered signal for other purposes.

Figure 2

Implementation of error filtration using time bins propagating in optical fibers. A source produces a single photon in time bin $\mid 1⟩$. A first Mach-Zender (MZ) interferometer produces a state in a two-dimensional Hilbert space as follows: the fiber coupler C splits the time bin $\mid 1⟩$ into two pulses which follow the short and long arm of the interferometer. Then the switch (Sw), synchronized with the source, is used to direct the pulses exiting from the first MZ interferometer into the fiber leading to the encoder. In this way, a superposition of two time bins $(\mid 1⟩+e^{i{\varphi }_A}\mid 2⟩)∕\sqrt{2}$ is produced where the phase ${\varphi }_A$ encodes the quantum information that must be transmitted. A second MZ interferometer realizes the encoding part of the error filtration protocol: it multiplexes time bin 1 (2) exiting from the state preparation into time bins 1 and 3 (2 and 4). Thus after encoding, the qubit is a superposition of four time bins $(\mid 1⟩+\mid 3⟩+e^{i{\varphi }_A}\mid 2⟩+e^{i{\varphi }_A}\mid 4⟩)∕2$. During transmission, the state is affected by noise. If the time bins are sufficiently separated, then the probability that a photon jumps from one time bin to the other is negligible and the noise affecting each time bin will be essentially independent. Hence one is in the case of independent phase noise considered in the main text. The decoding operation is the reverse of the encoding operation. A first MZ interferometer projects the state onto the $\mid 1⟩+\mid 3⟩$, $\mid 2⟩+\mid 4⟩$ subspace. The receiver may then use the filtered state. For instance, he can carry out a measurement using a second MZ interferometer. The measurement basis is chosen by varying the phase ${\varphi }_B$.

Figure 3

Implementation of error filtration for entangled particles using multiple optical fibers. A source $\left(S_{2\nu }\right)$ produces two photons in the entangled state $({\mid 1⟩}_A{\mid 1⟩}_B+{\mid 2⟩}_A{\mid 2⟩}_B+{\mid 3⟩}_A{\mid 3⟩}_B+{\mid 4⟩}_A{\mid 4⟩}_B)∕2$. States ${\mid i⟩}_{A\left(B\right)}$ travel to receivers $A$ ($B$) through different optical fibers. As in Fig. 1, the transmission states will be affected by independent phase noise. The receivers filter out the noise by projecting the state onto the subspaces spanned by ${\mid 1⟩}_{A\left(B\right)}+{\mid 2⟩}_{A\left(B\right)}$, ${\mid 3⟩}_{A\left(B\right)}+{\mid 4⟩}_{A\left(B\right)}$ using $50∕50$ couplers. When the projections of both parties succeed, the noise has been filtered out. When the projection of either of the parties fails, the state is rejected. Note that this decoding operation is based on the Hadamard transform and is distinct from the decoding operation based on the Fourier transform considered in the main text. However, one can easily show (see Appendix ) that both methods filter out the same amount of noise.

Figure 4

Implementation of error filtration for entangled particles using time bins. A source $\left(S_{2\nu }\right)$ produces two photons in the entangled state $({\mid 1⟩}_A{\mid 1⟩}_B+{\mid 2⟩}_A{\mid 2⟩}_B+{\mid 3⟩}_A{\mid 3⟩}_B+{\mid 4⟩}_A{\mid 4⟩}_B)∕2$ where states ${\mid i⟩}_{A\left(B\right)}$ correspond to different time bins traveling through the same optical fiber. A possible such source, adapted from [20], is described in the inset: a laser (L) produces intense pulses of light which pass through two unbalanced Mach-Zender (MZ) interferometers so as to produce four coherent equally spaced pulses. The pulses impinge on a nonlinear crystal (NLC) thereby producing the entangled state by parametric down conversion. As in Fig. 2, the transmission states will be affected by independent phase noise. To filter out the noise, the photons are sent through MZ interferometers. A switch (Sw), synchronized with the source, sends time bins 2 and 4 through the long arm and time bins 1 and 3 through the short arm. Time bins 1 and 2 and time bins 3 and 4 then interfere. The state is kept if both photons exit through the lower branch, in which case it has been projected onto the subspace spanned by ${\mid 1⟩}_{A\left(B\right)}+{\mid 2⟩}_{A\left(B\right)}$, ${\mid 3⟩}_{A\left(B\right)}+{\mid 4⟩}_{A\left(B\right)}$. If either of the photons exit through the upper branch, the filtration has failed. As in Fig. 3, this example is based on the Hadamard transform.