Abstract
Quantum technology relies on the utilization of resources, like quantum coherence and entanglement, which allow quantum information and computation processing. This achievement is however jeopardized by the detrimental effects of the environment surrounding any quantum system, so that finding strategies to protect quantum resources is essential. NonMarkovian and structured environments are useful tools to this aim. Here we show how a simple environmental architecture made of two coupled lossy cavities enables a switch between Markovian and nonMarkovian regimes for the dynamics of a qubit embedded in one of the cavity. Furthermore, qubit coherence can be indefinitely preserved if the cavity without qubit is perfect. We then focus on entanglement control of two independent qubits locally subject to such an engineered environment and discuss its feasibility in the framework of circuit quantum electrodynamics. With uptodate experimental parameters, we show that our architecture allows entanglement lifetimes orders of magnitude longer than the spontaneous lifetime without local cavity couplings. This cavitybased architecture is straightforwardly extendable to many qubits for scalability.
Introduction
Entangled states are not only an existing natural form of compound systems in the quantum world, but also a basic resource for quantum information technology^{1,2,3}. Due to the unavoidable coupling of a quantum system to the surrounding environment, quantum entanglement is subject to decay and can even vanish abruptly, a phenomenon known as earlystage disentanglement or entanglement sudden death^{4,5,6,7,8,9,10,11,12,13}. Harnessing entanglement dynamics and preventing entanglement from disappearing until the time a quantum task can be completed is thus a key challenge towards the feasibility of reliable quantum processing^{14,15}.
So far, a lot of researches have been devoted to entanglement manipulation and protection. A pure maximally entangled state can be obtained from decohered (partially entangled mixed) states^{16,17,18,19,20} provided that there exist a large number of identically decohered states, which however will not work if the entanglement amount in these states is small. In situations where several particles are coupled to a common environment and the governing Hamiltonian is highly symmetric, there may appear a decoherencefree subspace that does not evolve in time^{21,22,23}: however, in this decoherencefree subspace only a certain kind of entangled state can be decoupled from the influence of the environment^{24,25}. The quantum Zeno effect^{26} can also be employed to manipulate decoherence process but, to prevent considerable degradation of entanglement, special measurements should be performed very frequently at equal time intervals^{24,25}. By encoding each physical qubit of a manyqubit system onto a logical one comprising several physical qubits^{27,28,29,30,31}, an appropriate reversal procedure can be applied to correct the error induced by decoherence after a multiqubit measurement that learns what error possibly occurred. Yet, as has been shown^{31}, in some cases this method can indeed delay entanglement degradation but in other cases it leads to sudden disentanglement for states that otherwise disentangle only asymptotically. The possibility to preserve entanglement via dynamical decoupling pulse sequences has been also theoretically investigated recently for finitedimensional or harmonic quantum environments^{32,33,34,35} and for solid state quantum systems suffering random telegraph or 1/f noise^{36,37}, but these procedures can be demanding from a practical point of view.
In general, environments with memory (socalled nonMarkovian) suitably structured constitute a useful tool for protecting quantum superpositions and therefore the entanglement of composite systems^{8,38,39,40}. It is nowadays wellknown that independent qubits locally interacting with their nonMarkovian environments can exhibit revivals of entanglement, both spontaneously during the dynamics^{38,41,42,43,44} and ondemand by local operations^{45,46}. These revivals, albeit prolonging the utilization time of entanglement, however eventually decay. In several situations, the energy dissipations of individual subsystems of a composite system are responsible for disentanglement. Therefore, methods that can trap system excitedstate population would be effective for entanglement preservation. A stationary entanglement of two independent atoms can be in principle achieved in photonic crystals or photonicbandgap materials^{47,48} if they are structured so as to inhibit spontaneous emission of individual atoms. This spontaneous emission suppression induced by a photonic crystal has been so far verified experimentally for a single quantum dot^{49} and its practical utilization for a multiqubit assembly appears far from being reached. Quantum interference can also be exploited to quench spontaneous emission in atomic systems^{50,51} and hence used to protect twoatom entanglement provided that three levels of the atoms can be used^{52}. Since the energy dissipations originate from excited state component of an entangled state, a reduction of the weight of excitedstate by prior weak measurement on the system before interacting with the environment followed by a reversal measurement after the timeevolution proves to be an efficient strategy to enhance the entanglement^{53,54,55}. However, the success of this measurementbased strategy is always conditional (probability less than one)^{53,54,55}. It was shown that steadystate entanglement can be generated if two qubits share a common environment^{24,56}, interact each other^{57} and are far from thermal equilibrium^{58,59,60,61,62}. It has been also demonstrated that nonMarkovianity may support the formation of stationary entanglement in a nondissipative pure dephasing environment provided that the subsystems are mutually coupled^{63}.
Separated, independent twolevel quantum systems at thermal equilibrium, locally interacting with their own environments, are however the preferable elements of a quantum hardware in order to accomplish the individual control required for quantum information processing^{14,15}. Therefore, proposals of strategies to strongly shield quantum resources from decay are essential within such a configuration. Here we address this issue by looking for an environmental architecture as simple as possible which is able to achieve this aim and at the same time realizable by current experimental technologies. In particular, we consider a qubit embedded in a cavity which is in turn coupled to a second cavity and show that this basic structure is able to enable transitions from Markovian to nonMarkovian regimes for the dynamics of the qubit just by adjusting the coupling between the two cavities. Remarkably, under suitable initial conditions, this engineered environment is able to efficiently preserve qubit coherence and, when extended to the case of two noninteracting separated qubits, quantum entanglement. We finally discuss the effectiveness of our cavitybased architecture by considering experimental parameters typical of circuit quantum electrodynamics^{15,64}, where this scheme can find its natural implementation.
Results
Our analysis is divided into two parts. The first one is dedicated to the singlequbit architecture which shall permit us to investigate the dynamics of quantum coherence and its sensitivity to decay. The second part treats the twoqubit architecture for exploring to which extent the time of existence of quantum entanglement can be prolonged with respect to its natural disappearance time without the proposed engineered environment.
Singlequbit coherence preservation
The global system is made of a twolevel atom (qubit) inside a lossy cavity C_{1} which in turn interacts with another cavity C_{2}, as depicted in Fig. 1. The Hamiltonian of the qubit and two cavities is given by (ħ = 1)
where is a Pauli operator for the qubit with transition frequency ω_{0}, are the raising and lowering operators of the qubit, and the annihilation (creation) operators of cavities C_{1} and C_{2} which sustain modes with frequency ω_{1} and ω_{2}, respectively. The parameter κ denotes the coupling of the qubit with cavity C_{1} and J the coupling between the two cavities. We take ω_{1} = ω_{2} = ω and, in order to consider both resonant and nonresonant qubitC_{1} interactions, ω_{0} = ω+δ with δ being the qubitcavity detuning. Taking the dissipations of the two cavities into account, the density operator ρ(t) of the atom plus the cavities obeys the following master equation^{65}
where and Γ_{1} (Γ_{2}) denotes the photon decay rate of cavity C_{1} (C_{2}). The rate Γ_{n}/2 physically represents the bandwidth of the Lorentzian frequency spectral density of the cavity C_{n}, which is not a perfect singlemode cavity^{65}. A cavity with a high quality factor will have a narrow bandwidth and therefore a small photon decay rate. Weak and strong coupling regimes for the qubitC_{1} interaction can be then individuated by the conditions and ^{41,65}.
Let us suppose the qubit is initially in the excited state and both cavities in the vacuum states , so that the overall initial state is , where the first, second and third element correspond to the qubit, cavity C_{1} and cavity C_{2}, respectively. Since there exist at most one excitation in the total system at any time of evolution, we can make the ansatz for ρ(t) in the form
where with and with h(0) = 1 and . It is convenient to introduce the unnormalized state vector^{66,67}
where represents the probability amplitude of the qubit and (n = 1, 2) that of the cavities being in their excited states. In terms of this unnormalized state vector we then get
The timedependent amplitudes , , of Eq. (4) are determined by a set of differential equations as
The above differential equations can be solved by means of standard Laplace transformations combined with numerical simulations to obtain the reduced density operators of the atom as well as of each of the cavities. In particular, in the basis the density matrix evolution of the qubit can be cast as
where u_{t} and z_{t} are functions of the time t (see Methods).
An intuitive quantification of quantum coherence is based to the offdiagonal elements of the desired quantum state, being these related to the basic property of quantum interference. Indeed, it has been recently shown^{68} that the functional
where ρ_{ij}(t) (i ≠ j) are the offdiagonal elements of the system density matrix, satisfies the physical requirements which make it a proper coherence measure^{68}. In the following, we adopt C as quantifier of the qubit coherence and explore how to preserve and even trap it under various conditions. To this aim, we first consider the resonant atomcavity interaction and then discuss the effects of detuning on the dynamics of coherence.
Suppose the qubit is initially prepared in the state (with ), namely, C , then at time t > 0 the coherence becomes C . Focusing on the weak coupling between the qubit and the cavity C_{1} with κ = 0.24 Γ_{1}, we plot the dynamics of coherence in Fig. 2(a). In this case, the qubit exhibits a Markovian dynamics with an asymptotical decay of the coherence in the absence of the cavity C_{2} (with J = 0). However, by introducing the cavity C_{2} with a sufficiently large coupling strength, quantum coherence undergoes nonMarkovian dynamics with oscillations. Moreover, it is readily observed that the decay of coherence can be greatly inhibited by increasing the C_{1}C_{2} coupling strength J. On the other hand, if the coupling between the atom and the cavity C_{1} is initially in the strong regime with the occurrence of coherence collapses and revivals, the increasing of the C_{1}C_{2} coupling strength J can drive the nonMarkovian dynamics of the qubit to the Markovian one and then back to the nonMarkovian one, as shown in Fig. 2(b). This behavior is individuated by the suppression and the successive reactivation of oscillations during the dynamics. It is worth noting that, although the qubit can experience nonMarkovian dynamics again for large enough J, the nonMarkovian dynamics curve is different from the original one for J = 0 in the sense that the oscillations arise before the coherence decays to zero. In general, the coupling of C_{1}C_{2} can enhance the quantum coherence also in the strong coupling regime between the qubit and the cavity C_{1}.
The oscillations of coherence, in clear contrast to the monotonic smooth decay in the Markovian regime, constitute a sufficient condition to signify the presence of memory effects in the system dynamics, being due to information backflow from the environment to the quantum system^{69}. The degree of a nonMarkovian process, the socalled nonMarkovianity, can be quantified by different suitable measures^{69,70,71,72}. We adopt here the nonMarkovianity measure which exploits the dynamics of the trace distance between two initially different states ρ_{1}(0) and ρ_{2}(0) of an open system to assess their distinguishability^{69}. A Markovian evolution can never increase the trace distance, hence nonmonotonicity of the latter would imply a nonMarkovian character of the system dynamics. Based on this concept, the nonMarkovianity can be quantified by a measure defined as^{69}
where is the rate of change of the trace distance, which is defined as , with . By virtue of , we plot in Fig. 3 the nonMarkovianity of the qubit dynamics for the conditions considered in Fig. 2(a,b). We see that if the qubit is initially weakly coupled to the cavity C_{1} (κ = 0.24 Γ_{1}) its dynamics can undergo a transition from Markovian to nonMarkovian regimes by increasing the coupling strengths J between the two cavities. On the other hand, for strong qubitcavity coupling (κ = 0.4 Γ_{1}), the nonMarkovian dynamics occurring for J = 0 turns into Markovian and then back to nonMarkovian by increasing J. We mention that such a behavior has been also observed in a different structured system where a qubit simultaneously interacts with two coupled lossy cavities^{73}.
Trapping qubit coherence in the longtime limit is a useful dynamical feature for itself that shall play a role for the preservation of quantum entanglement to be treated in the next section. We indeed find that the use of coupled cavities can achieve this result if the cavity C_{2} is perfect, that is Γ_{2} = 0 (no photon leakage). The plots in Fig. 2(c,d) demonstrate the coherence trapping in the longtime limit for both weak and strong coupling regimes between the qubit and the cavity C_{1} for different coupling strengths J between the two cavities. This behavior can be explained by noticing that there exists a bound (decoherencefree) state of the qubit and the cavity C_{2} of the form , with J and κ being the C_{1}C_{2} and qubitC_{1} coupling strengths. Being this state free from decay, once the reduced initial state of the qubit and the cavity C_{2} contains a nonzero component of this bound state , a longliving quantum coherence for the qubit can be obtained. For the initial state of the qubit and two cavities here considered and Γ_{2} = 0, the coherence defined in Eq. (8) gets the asymptotic value C , which increases with J for a given κ. We further point out that the cavity C_{1} acts as a catalyst of the entanglement for the hybrid qubitC_{2} system, in perfect analogy to the stationary entanglement exhibited by two qubits embedded in a common cavity^{24}. In the latter case, in fact, the cavity mediates the interaction between the two qubits and performs as an entanglement catalyst for them.
We now discuss the effect of nonresonant qubitC_{1} interaction (δ ≠ 0) on the dynamics of coherence. In Fig. 4(a–d), we display the density plots of the coherence as functions of detuning δ = ω_{0} − ω and rescaled time Γt for both weak and strong couplings. One observes that when δ departures from zero, the decay of coherence speeds up achieving the fastest decay around δ = J. It is interesting to highlight the role of the cavitycavity coupling parameter J as a benchmark for having the fastest decay during the dynamics under the nonresonant condition. For larger detuning tending to the dispersive regime , the decay of coherence is instead strongly slowed down^{48}. However, as shown in Fig. 5, stationary coherence is forbidden out of resonance when the cavity C_{2} is perfect. Since our main aim is the longtime preservation of quantum coherence and thus of entanglement, in the following we only focus on the condition of resonance between qubit and cavity frequencies.
Twoqubit entanglement preservation
So far, we have studied the manipulation of coherence dynamics of a qubit via an adjustment of coupling strength between two cavities. We now extend this architecture to explore the possibility to harness and preserve the entanglement of two independent qubits, labeled as A and B. We thus consider A (B) interacts locally with cavity C_{1A} (C_{1B}) which is in turn coupled to cavity C_{2A}(C_{2B}) with coupling strength J_{A} (J_{B}), as illustrated in Fig. 6. That is, we have two independent dynamics with each one consisting of a qubit j (j = A, B) and two coupled cavities C_{1j} − C_{2j}. The total Hamiltonian is then given by the sum of the two independent Hamiltonians, namely, , where each H_{j} is the singlequbit Hamiltonian of Eq. (1). Denoting with Γ_{1j} (Γ_{2j}) the decay rate of cavity C_{1j} (C_{2j}), we shall assume Γ_{1A} = Γ_{1B} = Γ as the unit of the other parameters.
As known for the case of independent subsystems, the complete dynamics of the twoqubit system can be obtained by knowing that of each qubit interacting with its own environment^{41,42}. By means of the singlequbit evolution, we can construct the evolved density matrix of the two atoms, whose elements in the standard computational basis are
where ρ_{lm}(0) are the density matrix elements of the twoqubit initial state and , are the timedependent functions of Eq. (7).
We consider the qubits initially in an entangled state of the form . As is known, this type of entangled states with suffers from entanglement sudden death when each atom locally interacts with a dissipative environment^{7,8,9}. As far as nonMarkovian environments are concerned, partial revivals of entanglement can occur^{38,41,42,43,44,74,75,76,77,78,79,80,81,82,83,84} typically after asymptotically decaying to zero or after a finite dark period of complete disappearance. It would be useful in practical applications that the nonMarkovian oscillations can occur when the entanglement still retain a relatively large value. With our cavitybased architecture, on the one hand we show that the Markovian dynamics of entanglement in the weak coupling regime between the atoms and the corresponding cavities (i.e., C_{1A} and C_{1B}) can be turned into nonMarkovian one by increasing the coupling strengths between the cavities C_{1A}C_{2A} and (or) C_{1B}C_{2B}; on the other hand, we find that the appearance of entanglement revivals can be shifted to earlier times. We employ the concurrence^{85} to quantify the entanglement (see Methods), which for the twoqubit evolved state of Eq. (10) reads CAB . Notice that the concurrence of the Belllike initial state is CAB . In Fig. 7(a) we plot the dynamics of concurrence CAB in the weak coupling regime between the two qubits with their corresponding cavities with ( has been assumed). For twoqubit initial states with , , the entanglement experiences sudden death without coupled cavities . By incorporating the additional cavities with relatively small coupling strength, e.g., J_{A} = 0.5 Γ and J_{B} = Γ, the concurrence still undergoes a Markovian decay but the time of entanglement disappearance is prolonged. Increasing the coupling strengths J_{A}, J_{B} of the relevant cavities drives the entanglement dynamics from Markovian regime to nonMarkovian one. Moreover, the entanglement revivals after decay happen shortly after the evolution when the entanglement still has a large value. In general, the concurrences are enhanced pronouncedly with J_{A} and J_{B}. A comprehensive picture of the dynamics of concurrence as a function of coupling strength J is shown in Fig. 7(c) where we have assumed J_{A} = J_{B} = J. In Fig. 7(b) we plot the dynamics of in the strong coupling regime between qubit j and its cavity C_{1j} with for which the twoqubit dynamics is already nonMarkovian in absence of cavity coupling, namely the entanglement can revive after dark periods. Remarkably, the figure shows that when the coupling J_{j} between C_{1j} and C_{2j} is activated and gradually increased in each location, multiple transitions from nonMarkovian to Markovian dynamics surface. We point out that the entanglement dynamics within the nonMarkovian regime exhibit different qualitative behaviors with respect to the first time when entanglement oscillates. For instance, for , the nonMarkovian entanglement oscillations (revivals) happen after its disappearance, while when and the entanglement oscillates before its sudden death. These dynamical features are clearly displayed in Fig. 7(d).
As expected according to the results obtained before on the singlequbit coherence, a steady concurrence arises in the longtime limit if the secondary cavities C_{2A}, C_{2B} do not lose photons, i.e., . Figure 8(a) shows the dynamics of concurrence for qubits coupled to their cavities with strengths , . We can readily see that, in absence of coupling with the secondary cavities (J_{A} = J_{B} = 0), the entanglement disappear at a finite time without any revival. Contrarily, if the local couplings C_{1j}C_{2j} are switched on and increased, the entanglement does not vanish at a finite time any more and reaches a steady value after undergoing nonMarkovian oscillations. Furthermore, the steady value of concurrence is proportional to the local cavity coupling strengths J_{A}, J_{B}. In Fig. 8(b), the concurrence dynamics for is plotted under which the twoqubit entanglement experiences nonMarkovian features, that is revivals after dark periods, already in absence of coupled cavities, as shown by the black solid curve for J_{A} = J_{B} = 0. Of course, in this case the entanglement eventually decays to zero. On the contrary, by adjusting suitable nonzero values of the local cavity couplings a considerable amount of entanglement can be trapped. As a peculiar qualitative dynamical feature, we highlight that the entanglement can revive and then be frozen after a finite dark period time of complete disappearance (e.g., see the inset of Fig. 8(b), for the shorttime dynamics with , and also ). We finally point out that the the amount of preserved entanglement depends on the choice of the initial state (i.e., on the initial amount of entanglement) of the two qubits. As displayed in Fig. 9, the less initial entanglement, the less entanglement is in general maintained in the ideal case of . However, since there is not a direct proportionality between the evolved concurrence CAB and its initial value CAB , the maximal values of concurrence do not exactly appear at (corresponding to maximal initial entanglement) at any time in the evolution, as instead one could expect. It can be then observed that nonzero entanglement trapping is achieved for α > 0.2.
Experimental paramaters
We conclude our study by discussing the experimental feasibility of the cavitybased architecture here proposed for the twoqubit assembly. Due to its cavity quantum electrodynamics characteristics, our engineered environment finds its natural realization in the wellestablished framework of circuit quantum electrodynamics (cQED) with transmon qubits and coplanar waveguide cavities^{64,86,87,88,89}. The entangled qubits can be initialized by using the standard technique of a transmissionline resonator as a quantum bus^{64,90}. Initial Belllike states as the one we have considered here can be currently prepared with very high fidelity^{90}. Considering uptodate experimental parameters^{86,87,88,89} applied to our global system of Fig. 6, the average photon decay rate for the cavity C_{1j} (j = A, B) containing the qubit is , while the average photon lifetime for the high quality factor cavity C_{2j} is ^{87}, which implies . The qubitcavity interaction intensity κ_{j} and the cavitycavity coupling strength J_{j} are usually of the same order of magnitude, with typical values . The typical cavity frequency is ^{64} while the qubit transition frequency can be arbitrarily adjusted in order to be resonant with the cavity frequency. The above experimental parameters put our system under the condition which guarantees the validity of the rotating wave approximation (RWA) for the qubitcavity interaction here considered in the Hamiltonian of Eq. (1).
In order to assess the extent of entanglement preservation expected under these experimental conditions, we can analyze the concurrence evolution under the same parameters of Fig. 8(a) for κ_{j}, J_{j}, which are already within the experimental values, but with instead of being zero (ideal case), where . The natural estimated disappearance time of entanglement in absence of coupling between the cavities is , as seen from Fig. 8(a). When considering the experimental achievable decay rates for the cavities C_{2j}, we find that the entanglement is expected to be preserved until times t^{*} orders of magnitude longer than , as shown in Table 1. In the case of higher quality factors for the cavities C_{2j}, such that the photon decay rate is of the order of , the entanglement can last even until the order of the seconds. These results provide a clear evidence of the practical powerful of our simple twoqubit architecture in significantly extending quantum entanglement lifetime for the implementation of given entanglementbased quantum tasks and algorithms^{14,90,91,92}.
It is worth to mention that nowadays cQED technologies are also able to create a qubitcavity coupling strength comparable to the cavity frequency, thus entering the socalled ultrastrong coupling regime^{93}. In that case the RWA is to be relaxed and the counterrotating terms in the qubitcavity interaction have to be taken into account. According to known results for the single qubit evolution beyond the RWA^{94}, it appears that the main effect of the counterrotating terms in the Rabi Hamiltonian is the photon creation from vacuum under dephasing noise, which in turns induces a bitflip error in the qubit evolution. This photon creation would be instead suppressed in the presence of dissipative (damping) mechanisms^{94}. Since our cavitybased architecture is subject to amplitude damping noise, the qualitative longtime dynamics of quantum coherence and thus of entanglement are expected not to be significantly modified with respect to the case when RWA is retained. These argumentations stimulate a detailed study of the performance of our proposed architecture under the ultrastrong coupling regime out of RWA, to be addressed elsewhere.
Discussion
In this work, we have analyzed the possibility to manipulate and maintain quantum coherence and entanglement of quantum systems by means of a simple yet effective cavitybased engineered environment. In particular, we have seen how an environmental architecture made of two coupled lossy cavities enables a switch between Markovian and nonMarkovian regimes for the dynamics of a qubit (artificial atom) embedded in one of the cavity. This feature possesses an intrinsic interest in the context of controlling memory effects of open quantum systems. Moreover, if the cavity without qubit has a small photon leakage with respect to the other one, qubit coherence can be efficiently maintained.
We mention that our cavitybased architecture for the single qubit can be viewed as the physical realization of a photonic band gap for the qubit^{95}, inhibiting its spontaneous emission. This property, then extended to the case of two independent qubits locally subject to such an engineered environment, has allowed us to show that quantum entanglement can be robustly shielded from decay, reaching a steadystate entanglement in the limit of perfect cavities. The emergence of this steadystate entanglement within our proposed architecture confirms the mechanism of entanglement preservation when the qubitenvironment interaction is dissipative: namely, the simultaneous existence of a bound state between the qubit and its local environment and of a nonMarkovian dynamics for the qubit^{40}. We remark that this condition is here shown to be efficiently approximated within current experimental parameters such as to maintain a substantial fraction of the entanglement initially shared between the qubits during the evolution. Moreover, we highlight that this goal is achieved even if the local reservoir (cavity) embedding the qubit is memoryless, thanks to the exploitation of an additional goodquality cavity suitably coupled to the first one. Specifically, we have found that, by suitably adjusting the control parameter constituted by this local cavity coupling, the entanglement between the separated qubits can be exploited for times orders of magnitude longer than the natural time of its disappearance in absence of the cavity coupling. These times are expected to be long enough to perform various quantum tasks^{14,90}.
Our longliving quantum entanglement scheme, besides its simplicity, is straightforwardly extendable to many qubits, thus fulfilling the scalability requirement for complex quantum information and computation protocols. The fact that the qubits are independent and noninteracting also allows for the desirable individual operations on each constituent of a quantum hardware. The results of this work provide new insights regarding the control of the fundamental nonMarkovian character of open quantum system dynamics and pave the way to further experimental developments towards the realization of devices able to preserve quantum resources.
Methods
Functions of the single qubit density matrix
Let us denote with L−1 the inverse Laplace transform of L(s). Then, the functions u_{t} and z_{t} appearing in Eq. (7) are expressed as
where
Entanglement quantification by concurrence
Entanglement for an arbitrary state ρ_{AB} of two qubits is quantified by concurrence^{3,85}
where χ_{i} are the eigenvalues in decreasing order of the matrix , with σ_{y} denoting the second Pauli matrix and corresponding to the complex conjugate of the twoqubit density matrix ρ_{AB} in the canonical computational basis .
Additional Information
How to cite this article: Man, Z.X. et al. Cavitybased architecture to preserve quantum coherence and entanglement. Sci. Rep. 5, 13843; doi: 10.1038/srep13843 (2015).
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Acknowledgements
In this work Z.X.M. and Y.J.X. are supported by the National Natural Science Foundation (China) under Grants Nos. 11574178, 11204156, 61178012 and 11247240 and the Promotive Research Fund for Excellent Young and MiddleAged Scientists of Shandong Province (China) under Project No. BS2013DX034. R.L.F. acknowledges support by the Brazilian funding agency CAPES [Pesquisador Visitante Especial Grant No. 108/2012].
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Z.X.M. and Y.J.X. performed the calculations. R.L.F. devised the initial idea and supervised the work. All the authors discussed the results and contributed to the preparation of the manuscript.
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Man, ZX., Xia, YJ. & Lo Franco, R. Cavitybased architecture to preserve quantum coherence and entanglement. Sci Rep 5, 13843 (2015). https://doi.org/10.1038/srep13843
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DOI: https://doi.org/10.1038/srep13843
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