Genuine entanglement under squeezed generalized amplitude damping channels with memory

Mazhar Ali

Department of Electrical Engineering,Faculty of Engineering,Islamic University Madinah,Madinah 107,Saudi Arabia

Keywords: genuine entanglement,squeezed thermal baths,multipartite states

1.Introduction

Quantum correlations among several particles not only lead to counterintuitive predictions but also have a key role in future technologies.[1]There is growing interest among researchers to study quantum correlations in both theory and experiments.[2–6]Entanglement is a precious resource and is important for several applications.However, one of the big challenges is to preserve quantum correlations in quantum states.Quantum states interact with their environments and it is well-known that such interactions can degrade entanglement.[7–10]Many authors have studied the effect of the environment on entanglement for both bipartite and multipartite systems.[11–30]

Quantum channels describe the physical situations in which a given quantum state is transformed into another quantum state as a dynamical process.These processes are regarded as maps that preserve trace and are also completely positive,which means that a quantum state undergoing a quantum channel is again a valid quantum state.One such channel is the amplitude damping channel (ADC), which models spontaneous emission from excited atoms or energy dissipation at zero temperature.[31]This channel has been generalized to model dissipation at finite temperature[32]and is called the generalized amplitude damping(GAD)channel.Another generalization of this model takes squeezing into account,this model is called squeezed generalized amplitude damping(SGAD) channel.[33]The effect on qubit–qubit entanglement for this channel was studied,[34]where it was found that this channel cannot preserve entanglement.All of these channels are considered as memoryless and the effect of the channel can tions where this simplification is not true, andand channel is said to have memory.The effect of memory on spin chains,[36]entanglement-enhanced transmission of classical information in Pauli channels,[37]classical and quantum capacities of correlated ADC,[38]and others,[39,40]have been studied.Recently, qubit–qubit entanglement and quantum discord[41]under SGAD channel with memory has been studied.[42]be extended toNqubits simply asΦ=Φ.There are situa-

An array of beam-splitters can be used to model a squeezed reservoir.[43]Laser-cooled trapped ions can also mimic the dynamics of an atom with a squeezed vacuum bath.[44]Another method to mimic coupling is to use a squeezed bath via four-level atoms driven by weak laser fields.[45]Generalized ADCs are studied in experiments for decoherence and decay of atomic states.[46,47]

In this work, we study the effect of memory on genuine entanglement for three-qubit systems,which to our knowledge has not been studied before.We consider three qubits sent by the consecutive use of quantum channel with memory.We determine the asymptotic states for a most general initial state of three qubits.We find that squeezing parametermis not present in the asymptotic states.This observation is in agreement with the fact that a squeezed thermal bath can suppress quantum decoherence[34]but it is unable to preserve quantum entanglement.[48,49]We then analyze these asymptotic states for various initial states and find that in most cases they can be genuinely entangled depending on memory and the parameters of the initial states.We found that even if we start with bi-separable states, and then depending upon the thermal parametern(must not be very large)and the memory parameterµ(must be very large), asymptotic states can be genuinely entangled.Although the degree of genuine entanglement for these asymptotic states is quite small, it is an interesting feature of this dynamical process.

This paper is organized as follows.In Section 2, we briefly discuss squeezed generalized ADC for qubits and provide the general solution for an arbitrary state of three qubits.We briefly review the concept of genuine entanglement in Section 3 and present our main findings for two important families of quantum states.Finally,we conclude our work in Section 4.

2.Squeezed generalized amplitude damping channel for qubits

The quantum theory of damping takes a two-level atom(system) as a harmonic oscillator interacting with a reservoir(or bath), which can also be treated set of harmonic oscillators.The Hamiltonian in the interaction picture can be written as[50,51]

whereσ-=|b〉〈a| andσ+=|a〉〈b| are atomic lowering and raising operators.bk(b†k) are bath annihilation (creation) operators for each mode.νk=ckare density distributed frequencies,ωis atomic transition frequency andgkare coupling constants.After standard quantum optical approximations and tracing over the reservoir,we get a master equation for the system only.The squeezed generalized ADC is a noisy quantum channel in which a qubit interacts with a bath being initially in a squeezed thermal state with Markov and Born approximations.The master equation in the interaction picture is given as[50,51]

wherenis related to the number of thermal photons andmis the squeezing parameter.The complete positivity demands that forΩ ≥0,we must havem2≤n(n+1).We note that form=0,we have a qubit interacting with thermal reservoirs,and forn=m=0,we have(zero-temperature)amplitude damping process(vacuum reservoirs).

We can extend this model for multi-qubits either as uncorrelated noise in which we have to sum these three terms for each qubit separately and then solve the master equation in the interaction picture by takingΩA=ΩB=···=ΩN=Ω,nA=nB=···=nN=nandmA=mB=···=mN=m.This process is the simplest model for the memoryless quantum channel and the stochastic mapΦ(ρ)can be extended toΦ⊗N(ρ)forN-systems or uses of quantum channel.The result is that Kraus operators also have structureK=KA⊗KB⊗···KN.For a single qubit under such uncorrelated noise, the Kraus operators[52]can be written as

whereXj ̸=XA⊗XB⊗···XN.The density matrix for an arbitrary state of three qubits has a simple solution in this case

where

where 0≤µ ≤1 is degree of channel memory,which means that the noise is correlated with probabilityµ.We have the most general solution for the system and we can study the asymptotic states by takingt →∞.The examples that follow refer to such states asρ(∞).Although individual matrix elements have quite lengthy expressions,it is possible to study the evolution of entanglement numerically using MATLAB(as described in the following section).

3.Genuine entanglement under SGAD channels

Genuine entanglement can be detected and quantified via positive partial transpose (PPT) mixtures.[53–55]PPT mixtures are characterized by semidefinite programming(SDP) and this approach can be used to quantify genuine entanglement.[53]For bipartite systems, this procedure is equivalent to negativity,[56]so we can call it genuine negativity.For multi-qubits,it is bounded byE(ρ)≤1 with upper bound for certain pure states.[57]For mixed states,it is always less than 1.If this measure is positive, then the state is guaranteed to be genuinely entangled, otherwise we are not sure unless some other procedure indicates its entanglement properties.

It is well-known that for three qubits there are two inequivalent families of quantum states,namely,GHZ states andWstates,given as

where 0≤α ≤1.It is known that these states are genuinely entangled for 0.429≤α ≤1.[53]An interesting property of SGAD channel with memory is that for these states, the only non-zero matrix elements are only on the main diagonal and main off-diagonal of the density matrix, whereas all other elements remain zero(Xstructure).[58]A result of the detection of genuine entanglement states that the inequality

is satisfied by bi-separable states and the violation implies genuine entanglement.[59]This criterion is a necessary and sufficient condition for GHZ-diagonal states.[59]Application of this result toρ1(∞)gives us the condition that

This condition is always satisfied forn →∞because it simplifies toµ ≤(3-α)/2α.Hence,none of the asymptotic states withn →∞are genuinely entangled.However, it is possible that the condition is violated for certain values ofα,nandµ.Therefore, it is possible to have asymptotic states that are genuinely entangled.

In Fig.1,we plot genuine entanglementE(ρ)for asymptotic states withn=1.We observe that the stateρ1(∞) withα=1 is always genuinely entangled as long as memoryµ>0.For a mixture with white noise, if we takeα=0.95 (dashed line) orα= 0.9 (dotted–dashed line), then the asymptotic states are genuinely entangled for larger values of memoryµ.We also observe that most of the genuine entanglement is lost because asymptotic states have a much lower degree of genuine entanglement.In this case,our initial states are genuinely entangled and asymptotic states may be genuinely entangled,as seen in Fig.1.

Fig.1.Genuine entanglement for states at infinity plotted against memory parameterµ and n=1.We see that for α =1,asymptotic states are genuinely entangled for µ >0.If α <1,then some states are genuinely entangled if we increase parameterµ.

In contrast, if we take initial state|GHZ2〉 mixed with white noise,then we would expect different results.As we saw in Eq.(14),|GHZ2〉 lives in decoherence free space.Therefore,its entanglement is not changing and forµ=1 andα=1,we must have the same state at infinity.Forα<1 andµ<1,we have to analyze the asymptotic states.In Fig.2, genuine entanglement is plotted for asymptotic states withn=1 for three choices of parameterα.We can see that forµ=1 andα=1, the states are stationary and genuine entanglement is fixed at maximum value of 1.However, as memoryµandαare decreased,the effects of uncorrelated noise degrade the genuine entanglement until for a specific value of memory the asymptotic states are no longer genuinely entangled.

The condition for bi-separable states forn →∞gives usµ ≤(6-α)/(13α), which is not true for the whole range ofα.It is violated forα<0.429 and the initial states are genuinely entangled for 0.429≤α ≤1.Since this result is for very largen, we expect that it is possible to start with a biseparable state and end up with a genuine entangled state,provided that memoryµis quite high and thatnis small.In Fig.3,we observe that all three bi-separable states can end up with genuine entangled states forµ ≥0.97.

Fig.2.Multipartite entanglement is plotted for asymptotic states against memoryµ.We take n=1 for this plot.

Fig.3.Genuine entanglement is plotted against memory µ.This plot is for n=0.1.We observe that all three initially bi-separable states may become genuinely entangled as a result of dynamical process.

Let us now take another type of genuine entangled state,namely,Wstate mixed with white noise,as

where 0≤β ≤1.We want to mention here that 1-β=αhas the same meanings and we could have usedαinstead ofβ.These states are genuinely entangled for 0≤β ≤0.521.[53]

Figure 4 depicts genuine entanglement for asymptotic states with initial states as genuinely entangled forβ=0,0.2 and bi-separable forβ=0.522.We observe that even though the initial state is bi-separable(forβ=0.522),the asymptotic state is genuinely entangled.We observe that all entangled asymptotic states have a very small degree of genuine entanglement.Nevertheless,it is an interesting feature of this channel that it can convert bi-separable states into genuine entangled states.

Fig.4.Genuine entanglement plotted against memory µ with n=1 for initially W states mixed with white noise.

4.Discussion and conclusion

We have studied genuine entanglement under the SGAD channel with memory for three qubits.We have obtained the most general solution for the system.Based on this solution,we are able to analyze the asymptotic states.We have found that in the case of|GHZ1〉states mixed with white noise and forn →∞, none of the asymptotic states are genuinely entangled.However, for small values ofnand depending on memoryµand initial states, it is possible to have genuinely entangled asymptotic states.We have also taken initial biseparable states and have found that under certain circumstances the asymptotic states may be genuinely entangled.We found this possibility not only in GHZ states but also forWstates mixed with white noise.It is an interesting feature of this dynamical process that it may bring bi-separable states to genuinely entangled states.However for this phenomenon to occur, we must have a very high degree of memoryµin the channel.We also observed that squeezing parametermis absent only among asymptotic states.It is well-known that expectation values of certain physical quantities at infinity are a function of average thermal photons〈n(ω)〉.[50]This has a simple interpretation.After a long time,the oscillator in contact with a heat bath gets thermalized, with the same average photon number as the thermal average at the oscillator’s frequency.The difference between two qubits under SGAD and three qubits under similar noise can be summarized as follows.It was found that for two-qubits singlet state with correlated noise, squeezing parametermdoes not affect the dynamics of entanglement and quantum discord.[42]In addition, some of the two-qubit asymptotic states are also not dependent on squeezing parameterm.It is possible that with singlet states as initial states, the asymptotic states withn →∞can be entangled even if initial states are separable.For three qubits,we get similar results with some differences.The first main difference is the fact that we can only get genuine entangled states at infinity ifnis not too large because we observed that none of the asymptotic states withn →∞are genuinely entangled.Similar to two qubits,for correlated noiseµ=1,the GHZ state living in decoherence free space is invariant under dynamics.For three qubits, we have two types of inequivalent genuine entangled states; whereas for two qubits, states are either entangled or separable.