Abstract
Decoherence, often caused by unavoidable coupling with the environment, leads to degradation of quantum coherence^{1}. For a multipartite quantum system, decoherence leads to degradation of entanglement and, in certain cases, entanglement sudden death^{2,3}. Tackling decoherence, thus, is a critical issue faced in quantum information, as entanglement is a vital resource for many quantum information applications including quantum computing^{4}, quantum cryptography^{5}, quantum teleportation^{6,7,8} and quantum metrology^{9}. Here, we propose and demonstrate a scheme to protect entanglement from decoherence. Our entanglement protection scheme makes use of the quantum measurement itself for actively battling against decoherence and it can effectively circumvent even entanglement sudden death.
Main
One way to cope with decoherence is to make use of entanglement distillation protocols by which a pure maximally entangled state may be obtained from multiple copies of partially decohered states^{4,10,11,12,13,14}. Note, however, that it is impossible to obtain an entangled state from copies of fully decohered (that is, separable) states by applying entanglement distillation^{15}. Another method to deal with decoherence is to rely on the socalled decoherencefree subspace^{16,17}. The decoherencefree subspace, however, requires the interaction Hamiltonian to have an appropriate symmetry, which might not always be present. The quantum Zeno effect may also be used to suppress decoherence^{18,19} as well as to generate entanglement^{20} under some specific situations.
Our scheme for protecting entanglement from decoherence is based on the fact that weak quantum measurement can be reversed. The reversibility of weak quantum measurement was originally discussed in the context of quantum error correction^{21} and was demonstrated for a single superconducting qubit and a single photonic qubit^{22,23,24}. Recently, it was shown that weak measurement and quantum measurement reversal can effectively suppress amplitudedamping decoherence for a single qubit^{25,26}. Here, we experimentally demonstrate a scheme for protecting entanglement from amplitudedamping decoherence using weak measurement and quantum measurement reversal. The scheme can reduce or even completely nullify the effect of decoherence as evidenced by increased concurrence of the twoqubit system.
Consider a twolevel quantum system (S) whose computational bases are 0〉_{S} and 1〉_{S}. The environment (E) is initially at 0〉_{E}. Amplitudedamping decoherence, in which a particular computational basis state is irreversibly and probabilistically transferred to the other, results from statedependent coupling of the system qubit to the environment and is described by the following quantum map,
where 0 ≤ D ≤ 1 is the magnitude of the decoherence and (ref. 4). Amplitudedamping decoherence is highly relevant for many practical qubit systems. For instance, amplitudedamping decoherence is caused by photon loss for the vacuum–singlephoton qubit, by spontaneous decay for the atomic energy level qubit and by zerotemperature energy relaxation for the superconducting qubit.
We now investigate how the decoherence map of equation (1) affects a twoqubit entangled state. In particular, we consider a quantum communication scenario depicted in Fig. 1a where Alice prepares a twoqubit entangled state ϕ〉=α00〉_{S}+β11〉_{S} (α α^{*}+β β^{*}=1) and distributes the qubits to Bob and Charlie through quantum channels with decoherence D_{1} and D_{2}. The discussion, however, applies equally well to other types of twoqubit entangled state and physical qubit of stationary nature.
Although the initial state was a pure twoqubit entangled state ϕ〉, owing to decoherence, Bob and Charlie now share the twoqubit quantum state ρ_{d} given as
where ( k=1,2). The effect of decoherence D_{1} and D_{2} on the initial state ϕ〉 can then be investigated by evaluating concurrence C_{d} (which quantifies the amount of entanglement) of the shared quantum state ρ_{d} (ref. 27), which is calculated to be
Note that C_{d}=Λ_{d} if Λ_{d}>0. It is clear from equation (2) that the stronger the decoherence, D_{k}→1 (k=1,2), the smaller the concurrence C_{d}. When the decoherence is strongest D_{k}=1 (k=1,2), concurrence C_{d} becomes zero, meaning that the twoqubit system has become fully separable. Furthermore, it is interesting to point out that, for a particular initial twoqubit state ϕ〉=α00〉_{S}+β11〉_{S} with β≥α, the decoherence causes entanglement sudden death (ESD) in which C_{d}=0 although decoherence is not at its maximum: C_{d}=0 if decoherence (ref. 3).
Let us now describe our scheme, depicted in Fig. 1b, to protect entanglement from decoherence by making use of weak measurement and quantum measurement reversal. Before the system qubits undergo decoherence (that is, coupling to the environment), they are subject to weak measurement (M_{wk}), which partially collapses the state towards 0〉_{S} (refs 22, 23, 24). The twoqubit weak measurement can be written as a nonunitary quantum operation
where p_{1} and p_{2} are the weak measurement strengths^{24}. As the computational basis state 0〉_{S} does not couple to the environment as shown in equation (1), the system qubits after the weak measurement are less vulnerable to decoherence.
After the decoherence quantum channel, Bob and Charlie carry out quantum measurement reversal operations on the qubits. The twoqubit reversing measurement (M_{rev}) is a nonunitary operation that can be written as
where ${p}_{{\text{r}}_{\text{1}}}$ and ${p}_{{\text{r}}_{\text{2}}}$ are the strengths of the reversing measurement. The optimal reversing measurement strength that gives the maximum amount of entanglement of the twoqubit state ρ_{r} is calculated to be , where ( k=1,2; refs 25, 26). Assuming that the reversing measurement is optimal, the twoqubit state now shared by Bob and Charlie (after the sequence of weak measurement, decoherence and reversing measurement) is given as
where . The concurrence C_{r} of the twoqubit state ρ_{r} shared by Bob and Charlie is then calculated to be
where C_{r}=Λ_{r} if Λ_{r}>0.
We can draw two important conclusions from the result in equation (3). First, C_{r} is always larger than C_{d}, which means that weak measurement and quantum measurement reversal indeed can be used for protecting entanglement from decoherence. It is possible to achieve ρ_{r}→ϕ〉 provided that the strength of the weak measurement p_{k} and that of the corresponding optimal reversing measurement ${p}_{{\text{r}}_{\text{k}}}$ (k=1,2) are sufficiently strong. Second, even for the ESD condition (that is, for the initial state ϕ〉 with β>α, C_{d}=0 if decoherence ), by applying the entanglement protection protocol with proper weak measurement () and the corresponding optimal reversing measurement, Bob and Charlie are able to share some entanglement.
In Fig. 2, we show how C_{d} (Λ_{d}) and C_{r} (Λ_{r}) behave for two particular initial states under different decoherence, weak measurement and corresponding optimal reversing measurement. It is clear that decoherence affecting the two qubits independently and at different magnitudes can be circumvented by exploiting weak measurement and quantum measurement reversal.
Let us now describe the experimental demonstration of protecting entanglement from decoherence using weak measurement and quantum measurement reversal for an entangled twoqubit system. The experimental setup consisting of three principle sections (weak measurement, decoherence and reversing measurement) is schematically shown in Fig. 3. Details of the experimental setup are described in the Methods section. For a clear demonstration, we consider identical decoherence and weak measurement for both qubits, that is, D_{1}=D_{2}=D and p_{1}=p_{2}=p. In our experiment, the system qubits are realized with the singlephoton polarization state; 0〉_{S} and 1〉_{S} refer to horizontal and vertical polarization, respectively.
We first examine the effect of decoherence D on the initial twoqubit entangled state ϕ〉=α00〉_{S}+β11〉_{S} without introducing the weak measurement and the reversing measurement. The resulting twoqubit state ρ_{d} is reconstructed with quantum state tomography^{24,26} and concurrence C_{d} is then evaluated. Figure 4a shows the experimental data for three input state conditions (α=β, α>β and α<β) as a function of decoherence D. In all cases, the data show loss of entanglement due to decoherence. Note that, for α<β, ESD occurs as expected in equation (2).
We now test whether our entanglement protection scheme, based on weak measurement and quantum measurement reversal, can indeed circumvent decoherence. To demonstrate the scheme’s ability to protect entanglement even under severe decoherence, we chose D=0.6, at which ESD is demonstrated for the initial state with α<β as shown in Fig. 4a. Figure 4b shows concurrence C_{r} of the twoqubit state ρ_{r} as a function of weak measurement strength pfor two sets of initial states: α=β and α<β. For each weak measurement strength p, the optimal reversing measurement strength was chosen. For the maximally entangled twoqubit state (α=β), the weak measurement and the reversing measurement indeed suppress twoqubit decoherence so that C_{r}>C_{d} and C_{r}→1 as p→1. Furthermore, as demonstrated for a nonmaximally entangled state (α<β), our scheme can circumvent even ESDcausing conditions: for the weak measurement strength p larger than a certain value, the twoqubit state ρ_{r}exhibits a positive concurrence C_{r}, which means that Bob and Charlie can now share entanglement even through quantum channels of severe decoherence.
As the weak measurement and the reversal measurement are nonunitary operations, our scheme naturally has less than unity success probability^{25,26}. As shown in Fig. 4b, the higher the concurrence C_{r}, the larger the weak measurement strength p. We can then explore the tradeoff relation between the success probability and weak measurement strength p. For a single qubit, the overall success probability can be calculated by averaging the statedependent success probability for all states on the Bloch sphere and is given by (refs 25, 26). For the twoqubit state, the success probability is calculated to be
We have measured by averaging the success probabilities for four twoqubit states, 00〉_{S}, 01〉_{S}, 10〉_{S} and 11〉_{S}. Owing to the symmetry, averaging over the four twoqubit states is sufficient to obtain experimentally. Figure 5 shows the experimentally obtained success probability as a function of weak measurement strength p. Clearly, the larger the weak measurement strength p, the less the success probability. In the asymptotic limit of p→1, concurrence C_{r} can be arbitrarily close to the initial value but .
We have demonstrated that weak measurement and quantum measurement reversal can indeed be useful for battling against decoherence. In particular, for amplitudedamping decoherence, we have shown that our protocol can distribute (protect) entanglement even through (from) severe decoherence. We have also studied the tradeoff relation between the success probability, concurrence and weak measurement strength. Although the demonstration in this work was done for twophoton polarization qubits, the protocol can easily be applied to other types of qubit, making weak measurement and quantum measurement reversal powerful tools for battling against decoherence. We thus believe that it should be possible to effectively handle decoherence in quantum information by combining the scheme for protecting entanglement from decoherence discussed in this paper and entanglement distillation.
Methods
State preparation.
First, the twoqubit maximally entangled state (ϕ〉=α00〉_{S}+β11〉_{S} with α=β) is generated using typeI frequencydegenerate spontaneous parametric downconversion from a 6mmthick βBaB_{2}O_{4} crystal pumped by a 405 nm diode laser^{28}. The photons are frequency filtered with a set of interference filters with a 5 nm passband. The nonmaximally entangled states (ϕ〉=α00〉_{S}+β11〉_{S} with ) are generated by preferential amplitude reduction of one of the basis states using a set of glass plates oriented at the Brewster angle^{13,24,26} (BPs).
Weak measurement and reversing measurement.
The weak measurement and the reversing measurement for the singlephoton polarization qubit are implemented with BPs and wave plates^{24}. As the BP probabilistically rejects vertical polarization (1〉_{S}) and completely transmits horizontal polarization (0〉_{S}), a singlephoton polarization qubit found behind a BP had been subject to weak measurement or partial collapse measurement towards the 0〉_{S}. The reversing measurement is designed to reverse the effect of weak measurement by making partial collapse measurement towards the 1〉_{S} and it can be implemented by adding 45° halfwave plates (HWPs) before and after the BPs. The weak measurement and the reversing measurement strength p and p_{r} can be varied by changing the number of BPs.
Amplitudedamping decoherence.
The decoherence map of equation (1) causes statedependent coupling of the system qubit (the singlephoton polarization state) to the environment qubit (the singlephoton path qubit) and is realized with an interferometer shown in Fig. 3 (ref. 26). The displaced Sagnac interferometer implements the coupling of the polarization qubit to the path qubit. The horizontal polarization 0〉_{S} entering the polarizing beam splitter (PBS) can be found only at the 0〉_{E} output mode. The vertical polarization 1〉_{S} at the input of the PBS can be found both at 0〉_{E} and 1〉_{E}output modes according to the angle θ of the HWP. The probability that the vertical polarization 1〉_{S} at the input of the PBS ends up at the 1〉_{E} output mode of the PBS corresponds to decoherence D in equation (1) such that . As we are interested only in the system qubit, we need to trace out the environment qubit once the coupling is done. We realize the tracing out of the environment qubit by incoherently mixing 1〉_{E}(horizontally polarized) and 0〉_{E} (vertically polarized) at another beam splitter (BS) with a pathlength difference sufficiently larger than the coherence length (∼140 μm) of the single photon.
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Acknowledgements
This work was supported by the National Research Foundation of Korea (20090070668 and 20110021452).
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YS.K. and JC.L. carried out the theoretical calculations, designed and carried out the experiment, analysed data and drafted the manuscript. O.K. carried out the experiment. YH.K. conceived the idea, designed the experiment, analysed data, wrote the manuscript and supervised the project.
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Kim, Y., Lee, J., Kwon, O. et al. Protecting entanglement from decoherence using weak measurement and quantum measurement reversal. Nature Phys 8, 117–120 (2012). https://doi.org/10.1038/nphys2178
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