Decoherence assisted spin squeezing generation in superposition of tripartite GHZ and W states
aa r X i v : . [ qu a n t - ph ] J un Decoherence assisted spin squeezing generation in superposition of tripartite GHZ andW states
Kapil K. Sharma ∗ and Swaroop Ganguly † Department of Electrical Engineering,Indian Institute of Technology Bombay, Mumbai 400076, India
E-mail: ∗ [email protected]: † [email protected] (Dated: June 6, 2018)In the present paper, we study spin squeezing under decoherence in the superposition of tripartitemaximally entangled GHZ and W states. Here we use amplitude damping, phase damping anddepolarisation channel. We have investigated the dynamics of spin squeezing with the interplay ofsuperposition and decoherence parameters with different directions of the mean spin vector. We havefound the mixture of GHZ and W states is robust against spin squeezing generation for amplitudedamping and phase damping channels for certain directions of the mean spin vector. However, thedepolarisation channel performs well for spin squeezing generation and generates permanent spinsqueezing in the superposition of GHZ and W states. I. INTRODUCTION
Spin squeezing (SS)[1–3] has strong connection withentanglement[4, 5] and play the important role in quan-tum information theory. SS characterizes the sensitiv-ity of quantum systems with respect to SU(2) rotations.It has its lucid applications in entanglement detection,precession enhancement in quantum meteorology, atomicclocks, Bose-Einstein Condensate states, gravitational in-terferometers etc.[6]. Recently there are many studies oninequalities of spin squeezing[7–10], which are progres-sively used to establish mathematical criteria to detectentanglement and negative pairwise correlations in quan-tum systems. Most importantly spin squeezing is a goodtool to study the quantum correlations in quantum en-sembles, where constitutes of the system cannot be ad-dressed individually and collective behavior of the systemplay the significant role, like in Bose-Einstein conden-sate. The framework of SS has also been used to detectthe signatures of quantum chaos[11, 12] and identifyingphase transitions in quantum systems[13, 14]. Primarytest bench for spin squeezing is the coherent spin states(CSS)[15, 16], which are states with minimal uncertaintyand have lucid properties. Kitagawa-Ueda has given amathematical criteria to study spin squeezing[1], whichis also experimentally verified. He has been studied spinsqueezing in coherent spin states (CSS) with one andtwo axis twisting Hamiltonians. Two-axis Hamiltonian isgiven as H = χ ( J x J y + J y J x ), which is non-linear Hamil-tonian and a good resource to generate spin squeezing inthe system. In fact, generating spin squeezing in varietiesof quantum systems by many techniques is a good area ofresearch. The scientific community needs the generationof spin squeezing mainly for two reasons as to increase thecoherence time in the system and to produce strong atom-atom interactions. There are many theoretical and exper-imental proposals to generate spin squeezing, these alsohave experimental manifestations in Bose-Einstein con-densate through particle collisions [17–20] and transfer ofsqueezing in atoms from squeezed light[21]. It is difficult to sustain squeezing in light for a long time, so scientificcommunity has an alternative to transfer this squeezingin atomic systems to store quantum information. Thistechnique has the great impact to design photonic quan-tum memories where photos are used as an informationcarrier. Another technique to generate the spin squeez-ing, which is progressively used is through nondemolitionquantum measurements. Apart from it, it is interesting tofind, that decoherence also plays the role in spin squeezinggeneration. It is obvious that quantum systems are tooevasive, such that these always interact with the environ-ment and hence decoherence effects take place. Studiesof decoherence on spin squeezing is an essential require-ment. Recently the spin squeezing under various deco-herence channels in CCS have been studied by X. Wanget al., they also found the phenomenon of spin squeezingsudden death (SSSD)[22]. With the motivations from X.Wang et al. study, we have shown the spin squeezingproduction in GHZ and W states and find a positive im-pact of decoherence on these states[23]. Spin squeezing inGHZ and W states also have been studied under particleloss and it has been found that GHZ state is very muchfragile in comparison to W states[24–26].These states areimportant states for executing applications in quantuminformation and study of spin squeezing in their super-position is also important[27, 28]. In continuation of ourstudy and taking the motivations from the limited studieson the effect of decoherence on spin squeezing, we proceedwith the spin squeezing generation in the superpositionof maximally entangled GHZ and W states in the presentletter. II. SPIN SQUEEZING, DECOHERENCECHANNELS AND SUPERPOSITION OF GHZAND W STATES
In this section we present the Kitagawa–Ueda (KU)spin squeezing criteria and decoherence channels underthose the decoherence dynamics of superposition of GHZand W states deal with. KU criteria is one of the mostsignificant criteria used to characterize the spin squeez-ing in symmetric states under particle exchange. Here wemention that GHZ and W state are symmetric and theirsuperposition is also symmetric, so the criteria is suit-able for the present study. Mathematically KU criteria isdefined as, [( △ J ϕ ) ] min ≤ J . (1)Where J is the spin quantum number, given as J = N and ( △ J ϕ ) is the variance of the total angular momen-tum along the perpendicular direction of the mean spinvector ( ~J mean ) in the collective system of spins. Meanspin vector play an important role to examine the spinsqueezing generation in the system, it is defined as, ~J mean = ( h J x i , h J y i , h J z i ) . (2)Where h J i =( x,y,z ) i are the expectation values of angularmomentum components along the ( x, y, z ) directions re-spectively. Rearranging the equation (1) we get ǫ = 4[( △ J ϕ ) ] min N ≤ . (3)Where N is the number of spins in the system and ( ǫ )is the spin squeezing parameter. For unsqueezed statesthe criteria satisfy the condition ( ǫ = 1), which meansthere is no quantum correlation present in the state. Forpure CSS uncorrelated state, the criteria carry the valueas ( ǫ = 1). If there are certain type of quantum correla-tions present in the state than ( ǫ ≤ | ψ i = √ α | GHZ i + p (1 − α ) | W i . (4)with the normalization condition |√ α | + | ( √ − α ) | = 1 . (5)Where, | GHZ i = 1 √ | i + | i (6) | W i = 1 √ | i + | i + | i (7)To begin with the decoherence dynamics, here we presentthe quantum decoherence channels, which play the im-portant role to study the spin squeezing characteris-tics in Eq.4. There are three important quantum chan-nels widely studied in quantum information communityas amplitude damping, phase damping and depolarisa-tion channel, the corresponding Kraus operators of thesechannels are given below, A. Amplitude damping channel
This channel is used to describe the energy loss inthe system. The Kraus operators of amplitude dampingchannel are given as, E = [[1 , T , [0 , √ e γt ] T ] (8) E = [[0 , T , [ p (1 − e − γt ) , T ] (9)Where [ . ] T represents the column of the matrix. B. Phase damping channel
This channel is the good model to represent the infor-mation loss in quantum system because of the relativephase produced in the system with system environmentinteraction. The channel do not involve the energy lossin the system as it is done in the case of amplitude damp-ing channel. The corresponding Kraus operators for thechannels are given as, E = [[ √ e − γt , T , [0 , √ e − γt ] T ] (10) E = [[ p (1 − e − γt ) , T , [0 , T ] (11) E = [[0 , T , [0 , p (1 − e − γt )] T ] (12)Where γt is the decay rate of decoherence. C. Depolarization channel
Depolarization channel is widely studied in polariza-tion encoding in quantum information, the correspondingKraus operators are given as, E = [[ √ e − γt , T , [0 , √ e γt ] T ] (13) E = [[0 , r
13 (1 − e − γt )] T , [ r
13 (1 − e − γt ) , T ] (14) E = [[0 , i r
13 (1 − e − γt )] T , [ − i r
13 (1 − e − γt ) , T ] (15) E = [[ r
13 (1 − e − γt ) , T , [0 , − i r
13 (1 − e − γt )] T ] (16)Where γt is the decoherence rate and the notation [ . ] T carry the usual meaning. III. RESULTS
In this section we present the results for spin squeez-ing dynamics for amplitude damping, phase damping anddepolarization channels.
A. Dynamics with amplitude damping channel
Here we present the results obtained for spin squeezinggeneration with amplitude damping channel. We recallthat the direction of mean spin vector play an impor-tant role to examine the spin squeezing generation, itsdirection can be represented on the surface of unit Blochsphere with the combinations of angles ( θ, φ ). We haveplotted the spin squeezing parameter ( ǫ ) given in Eq.3with the mean spin vector directions { θ ∈ [0 , , φ ∈ [0 , } . In Eq.4, ( α = 0) corresponds to W states and( α = 1) corresponds to GHZ states. Here we present theresults in a table with the parameters ( θ, φ ) for whichthere is no SS generation found in superposition of GHZand W states with ( ∀ α ). θ φ θ = 0) than with ( ∀ φ, ∀ α ),no spin squeezing generation takes place. It is because,in Eq.3 the term ( △ J ϕ ) ] min ) = 0 , so it represents that,the minimum variance along the perpendicular directionof mean spin vector ( ~J mean ) falling in XY plane is zeroand decoherence do not induce the fluctuations in thesystem. So as long as the mean spin vector is along the zaxis, the superposition of GHZ and W states is robust tosupport spin squeezing generation. Similar conclusionscan be drawn from another combinations of parametervalues ( θ, φ ) presented in the table.Further results for different values of parameters ( θ, φ ),which support spin squeezing generation with the effectof decoherence ( γt ) have been shown in Fig.1. One im-portant result we have found that, for ( α = 0 . ǫ = 1) and the state | ψ i remain unsqueezedwith this value. B. Dynamics with phase damping channel
Here we present the results for spin squeezing genera-tion with phase damping channel under the effect of de-coherence. First, we present those values of parameters( θ, φ ) for which the spin squeezing generation does nottake place in the superposition state given in Eq.4. Thetable is given below, θ φ | ψ i remain un-squeezed with the parameter value ( α = 0 . C. Dynamics with depolarization channel
In continuation with the study, here we present theresults obtained for depolarization channel. This chan-nel has a totally different impact on spin squeezing gen-eration in comparison to amplitude damping and phasedamping channel. It generates the spin squeezing in thesystem even if the mean spin vector is along the z-axiswith ( θ = 0 , ≤ φ ≤ ∀ α ). It is an oppositecase to amplitude damping and phase damping channel.Looking into the graphical results shown in Fig.3, we findas the values of the parameters ( θ, φ ) vary, there are vari-ations in initial amplitude of spin squeezing, but as thedecoherence parameter γt advances, the spin squeezingdecay slowly, but never become zero. So the channel gen-erates permanent spin squeezing in the state | ψ i . It isvaluable to mention that, the decoherence channel con-tributes well for spin squeezing generation in the super-position of GHZ and W states. IV. CONCLUSION
In the present article, we have discussed the effect ofdecoherence on the superposition of GHZ and W statesunder amplitude damping, phase damping, and depolar-ization channels. For these channels, we have investigatedthe lucid signatures of spin squeezing generation. We alsohave investigated the sensitive directions of the mean spinvector for which the spin squeezing generation do not takeplace. The superposition parameter ( α = 0 .
9) has beenfound sensitive, which do not support spin squeezing gen-eration in both amplitude and phase damping channel.Amplitude damping and phase damping channels havean almost same impact on spin squeezing, while depo-larization channel has a totally different paradigm. Fordepolarization channel, as the decoherence rate increases,the degree of squeezing slowly decay but never be zero.So we have found the depolarization channel is a goodresource for generating permanent spin squeezing underdecoherence in the superposition of GHZ and W states.The present study can also be upgraded to study the spinsqueezing generation under particle loss along with deco-herence in the superposition of GHZ and W states. Sowe mention the results obtained in the paper and discus-sion may be useful for quantum information processingcommunity.
V. ACKNOWLEDGEMENT