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1、Mixing and decoherence to nearest separable states in quantum measurementsa r X i v :0705.0733v 2 q u a n t -p h 26 M a y 2007Mixing and decoherence to nearest separable states in quantum measurementsAvijit Lahiri ?Dept of Physics,Vidyasagar Evening College,Kolkata 700006,INDIAWe illustrate through
2、numerical results a number of features of environment-induced decoher-ence under a broad class of apparatus-environment interactions in quantum measurements whereinthe reduced system-apparatus density matrix evolves towards the nearest separable state and,inaddition,there occurs a mixing in relevant
3、 groups of apparatus microstates (see below).The re-sulting ?nal state is unique and correctly embodies the measurement statistics even in the absenceof environment-induced superselection because of energy di?erences between these groups of states.The partial transpose remains non-positive throughou
4、t the process.The measurement problem has traditionally posed questions in quantum theory relating to interpretation at a basic level.More recently,however,investigations on the process of environment-induced decoherence seem to have provided some of the answers,at least as plausible explanations.Th
5、e present paper aims at illustrating in simple and general terms a number of features of this process whereby entangled states of a certain type (to be termed pure-mixed entanglement,see below)involving the measured system and the apparatus go through a Brownian-like evolution in the composite state
6、 space,tending towards the nearest separable state (in the sense explained in 1)with an attendant erosion of the entanglement and,additionally,there occurs a mixing in the apparatus states leading to a homogeneous probability distribution over relevant sets of microstates of the latter.The mixing ca
7、uses a partial erosion in classical correlations as well,but at the end the remaining classical correlation between the measured system and the apparatus correctly describes the possible results of measurement and their respective probabilities.Moreover,for the class of entangled states considered,n
8、on-positivity of the partial transpose seems to be retained throughout the decoherence process,thereby providing a convenient indicator for monitoring the entire process.More precisely,we consider three systems that interact with one another and that can be interpreted,in an appropriate context,as a
9、 measured system (S),a measuring apparatus (A),and an environment (E).The state spaces of the three systems have,in general,di?erent dimensions -a fact of crucial importance in the context of the quantum measurement problem.For the composite system SA,we consider a situation where pure states of S a
10、re entangled with mixed states of A,which we refer to as pure-mixed entanglement,and which arises naturally from pure-pure entanglement by environment-induced dephasing among sets of apparatus states (see below)during the process of decoherence.The former has been considered in 2in relation to the i
11、nformation gain in a quantum measurement.Our numerical results refer to a 2D system S with orthonormal states |s 1,|s 2(eigenstates of the observable to be measured),an apparatus A with two bunches of orthonormal states (say,|a 1,.,|a N 1,|b 1,.,|b N 2)forming subspaces of dimensions N 1,N 2,and an
12、environment E with a state space of dimension N e .The bunches of apparatus states are almost-degenerate microstates corresponding to macroscopic pointer states that may be superpositions or mixtures of these microstates (see,for instance,2,3).Mixed apparatus states are actually more relevant in the
13、 context of measurement 4,since the apparatus is distinct from the measured system in that it is a macroscopic system for which mixed states arise naturally due to environmental perturbations (see,e.g.,3).The S-A state we start from is given by=|c 1|2|s 1+|c 2|2|s 2where c 1,c 2are superposition coe
14、?cients of the system state (c 1|s 1+c 2|s 2)being measured,(A )a N 1i =1p i |a i and(A )b N 2i =1q i |b i are the mixed apparatus states entangled with |s 1,and |s 2respectively,and|(A )a N 1i =1p i |a i ,(3a)|(A )b N 2i =1q i |b i .(3b)are involved in the o?-diagonal terms representing the entangl
15、ed state.In these expressions,p i ,q i are two sets of weights specifying the degree of mixing in the two groups of apparatus states.A pure apparatus state corresponds to one of the relevant set of weights being unity,with all the other weights of the set being zero.The commonly discussed special ca
16、se of pure-pure entanglement corresponds to both the apparatus states being pure (and so,(A )a =|(A )a apparatus states,without loss of generality.Note that while |(A )a ,|(A )b have norms,in general,di?erent fromunity,(1)happens to be satisfy all the requirements of a density matrix.The pure-mixed
17、entangled state (1)has a structure similar to that of a pure-pure one of the form(P ?P )=|c 1|2|s 1The end result arising from the environment-induced decoherence operating on (4)and embodying correctly the measurement statistics (relating to the measurement variable under consideration,represented
18、by s 1|s 1(P ?P )=|c 1|2|s 1obtained by deleting the o?-diagonal terms,since one can assume that the latter are averaged out due to the deco-herence.In a similar vein,one expects that the disentangled state resulting from (1)would be(P ?M )=|c 1|2|s 1While the state (5)has the interesting property t
19、hat it is the closest separable state to (4),in the sense explained in 1,we indicate numerical evidence below to show that (6)is similarly the nearest separable state to (1).Thus,a neat description of the process of environment-induced decoherence would be that it is a quantum operation transforming
20、 the initial S-A entangled state to the nearest separable state.We see in our numerical results below that the decoherence process under quite general A-E interactions indeed shows a tendency towards such a state,but additionally involves another transformation in the apparatus states,namely one tow
21、ards maximum mixing in the two groups of states |a 1,.,|a N 1,and |b 1,.,|b N 2leading us to consider the separable state(P ?M )0=|c 1|2|s 1(A )a =1(A) b =1are,generically speaking,chaotic in the classical description.The quantum features of such interactions are known to be similar to those of ense
22、mbles ofrandom matrices.A large body of recent work has looked into the entangling power of chaotic interactions (see,e.g.,7,8,9),and a number of these also bring out random features in the density matrix ?uctuations in subsystems interacting with one another through such random matrices 10,11.The a
23、bove A-E interaction e?caciously entangles the apparaus states with the environment states and at the same time disentangles the system states from the apparatus states(see,e.g.,12,for an introduction to entanglement sharing in tripartite systems).The reduced S-A density matrix elements ?uctuate dur
24、ing the process,whereby S ?A undergoes a Brownian-like motion in the space of entangled states,tending to the nearest separable state (6)while at the same time deviating from the latter due to the mixing among the two groups of apparatus states alluded to above,?nally reaching the stae (7).The whole
25、 trajectory of the S-A state can be conveniently monitored through the partial transpose (with respect to,say,S)of S ?A ,de?ned asFIG.1:variation with time (arbitrary scale)of ?delity distance (see text)of the evolved reduced S-A state from the states (6)(dashed line)and (7)(solid line);N 1=3,N 2=4,
26、N e =22,=0.04,c 1=c 2=12;the weights p i ,q i in the initial state(1)are chosen randomly;the environment state is taken as a uniform mixture of basic states,the latter covering a small energy band in the middle of the energy gap between the two groups of apparatus states;this is an artefact for prev
27、enting mixing between the two groups;more generally,small values of prevent the mixing within the relevant time scale.where|,|are any two of the basic apparatus states mentioned above.It is known that the partial transpose is a positive operator for separable states while,for an entangled state,it m
28、ay or may not be positive.For the state(1),however,it happens to be a non-positive operator,with one negative eigenvalve.For instance,withN1=1,N2=2,for which the basic apparatus states are,say,|aand|b1,|b2,while(A)a and(A)b(refer to(2a),(2b)are given by|ab2|),the eigenvalues of the partial transpose
29、 of(1)are |c1c2|graphs,namely,the three basic principles governing the evolution of the reduced S-A state during decoherence in a quantum measurement are(a)a tendency towards the nearest separable state,(b)a tendency towards homoge-neous mixing between the groups of apparaus states corresponding to
30、distinct values of the pointer variable,and (c)Brownian-like?uctuations in the reduced density matrix.More detailed results will be reported elsewhere.One signi?cant fact to emerge is that the partial transpose remains non-positive throughout the process of decoherence. The above features,moreover,a
31、rise under the most general randomly generated S-A interactions,even in the absence of special environmental variables and states involved in any speci?c type of entanglement with the environment, viz.,one describing environmantal selection.The uniqueness of the?nal state resulting from the decohere
32、nce is a consequence of the energy di?erences between the groups of relevant apparatus states rather than of environmental selection.It has been argued in2that the information gain in a quantum measurement is actually the same as the classical correlation beteen the system and the apparatus states,a
33、nd by implication,the degree of quantum entanglement between the two is not relevant,the latter being precisely the information erased during the decoherence process. For a quantum system and an apparatus with a small and a large number of degrees of freedom respectively,the magnitude of quantum ent
34、anglement is actually small(see,for,instance14,results will also be presented in a separate communication)as compared with the distance from the initial entangled state to,say,faraway separable states.It is this smallness of the quantum information to be erased that is responsible for the process of
35、seeking out, in a Brownian-like evolution,of the appropriate disentangled state correctly embodying the measurement statistics. We have considered only projective measurements on pure system states in this paper.The results are seen to hold for measurements on mixed system states as well(to be repor
36、ted).It is not,however,clear whether corresponding statements can be made for more general POVM operations as well.An application of the features observed in this paper in devising a general numerical procedure for evaluating the magnitude of bipartitie entanglement will be described elsewhere.ACKNO
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