A random matrix theory of decoherence

Random matrix theory is used to represent generic loss of coherence of a fixed central system coupled to a quantum-chaotic environment, represented by a random matrix ensemble, via random interactions. We study the average density matrix arising from the ensemble induced, in contrast to previous studies where the average values of purity, concurrence and entropy were considered; we further discuss when one or the other approach is relevant. The two approaches agree in the limit of large environments. Analytic results for the average density matrix and its purity are presented in linear response approximation. The two-qubit system is analysed, mainly numerically, in more detail.


Introduction
Decoherence has become a subject of increasing interest, since it is not only related to the problems of measurement and emergence of classical behaviour [1,2,3,4] but has become one of the main stumbling blocks for quantum information solutions.Under most usual circumstances decay of coherence is exponential and this has been experimentally confirmed by experiment [5,6].Exponential decay of coherence typically holds if the Heisenberg time of the environment is to long to provide a relevant time scale, i.e. if decoherence is complete long before the Heisenberg time.As the global environment is always present and has, for all practical purposes, infinite Heisenberg time, our discussion seems spurious but this is not the case.Quantum information tasks require sufficiently low temperatures, high vacuum, good screening and careful selection of the states used etc, such that the global environment can be neglected to first approximation.We are left with a near environment consisting of only a few degrees of freedom which are involved in the control of the operations of the quantum gates.This near environment may have very specific properties, but is often governed by dynamics that might go for "quantum chaotic".In this paper we provide a generic description of decoherence which also covers such a situation.
This requires modelling of the environment, while avoiding a complete microscopic description.More specifically, we describe a generic environment with "quantum chaotic" dynamics by random matrix theory (RMT).The first step in this direction is the pioneering paper of Lutz and Weidenmueller [7,8] in which they propose a RMT description that reproduces the results of a Caldeira-Leggett model.Thus they show that RMT also works in the standard situation.We shall see that this is closely related to choosing a very dense spectrum in the environment, which sends the Heisenberg time to infinity, or, more generally, where decoherence is strong, long before the Heisenberg time of the environment is reached.While this is very reasonable for the far environment, the near environment can have level densities comparable to those of a central system, and this is where RMT models can produce new results.A model to study such effects was first proposed in [9,10] and further developed in [11,12].These models are closely related to one developed for fidelity decay [13], which was successfully tested in experiments [14,15,16].For decoherence, the models were compared to numerics with a spin chain environment [17].
In these models purity was calculated analytically in linear response approximation as averaged over the random matrices.For the purpose of quantum information, this was quite sufficient, as only high purities are needed.Yet, the beauty is, that these RMT models are sometimes exactly solvable by super-symmetric techniques.An attractive example are the solutions for the decay of the fidelity amplitude [18,19,20,21].Yet, these techniques nowadays are largely limited to the calculation of two-point functions.Purity is obviously of higher order, and thus we should look for an amplitude.Coherences, i.e. off-diagonal elements of the density matrix are good candidates, and their decay is often used to describe decoherence [6].This suggests to consider the average density matrix, something that has not been done so far.
Actually, in the entire formulation of the RMT models a basic question has not been addressed.If we consider the evolution of some initial state by an ensemble of Hamiltonians we also get an ensemble of density matrices for the central system.Normally, we would consider the average density matrix as the basic quantity from which observables such as purity or concurrence should be calculated.However, since these quantities are non-linear in the density matrix, their average (over the ensemble of density matrices) is different from their value computed from the average density matrix.At the end, it is the particular application which determines which kind of averaging must be chosen.We shall discuss this in section 5, but we will find in the previous sections, that for large environments the quantities we analyse, namely purity, entropy and concurrence converge to the same result for both averages.
We first formulate the random matrix model we use and derive a few basic relations, which will be needed in subsequent sections.Some obvious conditions the model must fulfil are presented before proceeding to calculate the average density matrix in linear response approximation.For the particular case of one qubit as a central system we outline how to obtain the exact answer with super-symmetric theory.Similar results for the average density matrix are presented in what we earlier called spectator configuration [22].This configuration has proven very useful to describe more complex central systems with high purity [23].We next concentrate on two-qubit central systems in the spectator configuration, where we display some properties of the density matrix ensemble including statistics of the eigenvalues and some further results on a curious relation to Werner states shown in previous papers [12,17].In section 5 we discuss in more details the meaning of an ensemble of density matrices.The discussion will include both situations where one should use the average density matrix to calculate non-linear functionals and others where this is not appropriate.This will lead us to conclusions and an outlook on the relevance of more realistic RMT environments.

Random matrix models
We shall start with a central system that does not display any particular structure, and thus the entire system will be bi-partite consisting simply of this central system and an environment.Decoherence will be viewed as the entanglement with the environment after unitary evolution of the entire system.As we wish to describe a near environment with quantum chaotic features, we neglect the far environment entirely, as indeed we shall throughout this paper, assuming that decoherence resulting from the far environment occurs on a much longer time scale.
The Hilbert space is divided into two parts, that of the central system H c and that of the environment H e .The dimension of those Hilbert spaces is denoted by m = dim(H c ) and N = dim(H e ), respectively.Eventually, the case N → ∞ will be particularly important.For a factorized initial state of the form we solve the Heisenberg equation where the Hamiltonian is of the form such that H 0 = H c + H e .Note that typically, we shall not require the initial state to be pure.It may well be a product of two mixed states ̺ c and ̺ e .In the following step, we express the forward evolution of the whole system in terms of echo dynamics.
Echo dynamics means, that we consider a forward time evolution which includes the coupling between central system and environment and a backward evolution, which is driven by the internal dynamics of environment and central system alone.The latter will not modify the entanglement between the two.This is important, because then we may apply the linear response approximation to the echo-dynamics which results in a much larger range of validity.A systematic application of the interaction picture would do the same simplification.Hence, we write: where we have set = 1.In order to obtain the echo dynamics, we evolve the system backwards in time with the Hamiltonian H 0 : This may be interpreted as a forward evolution with the echo-operator M λ (t) [11].The evolution of the central system is then given by where u c = exp(−iH c t) and u e = exp(−iH e t).This expression for ̺ c (t) may be further simplified: where ̺c (t) = tr e [̺ M (t)] is the density matrix of the central system in the interaction picture.Note that for operators of the form 1 1 c ⊗ u e the cyclic permutations are allowed even though we are dealing with a partial trace.Equation ( 7) holds only as long as H 0 describes uncoupled dynamics.
Linear response approximation In the linear response approximation, the echo-operator M λ (t) reads where and Ṽ (t) = exp(iH 0 t) V exp(−iH 0 t).Starting from equation ( 8), we obtain the density matrix of the central system averaged over the coupling matrix V (this average will be denoted by angular brackets).Subsequently, we may also average over the environmental Hamiltonian H e .However, for the linear response result as such, this is not necessary.
In the interaction picture, we therefore obtain: where we have used the notation Note that I(t) is self-adjoint, while J(t) is not.According to Equation (7), this yields for the time evolution of the central system: The averaging procedure itself as well as the different possible statistical assumptions, will be discussed below.
Purity We can now obtain the purity of the average density matrix of the central system.In view of (7), we find for the purity of the averaged state ̺c (t): The purity may be more explicitly written in terms of the average state of the full system (in the interaction picture) as Note that in this expression, we cannot exchange the order of the partial traces, since ̺ M (t) is no longer a pure state in the product Hilbert space H of central system and environment.This is the reason why we cannot simply adopt the calculations in [22] to the present case.In linear response approximation we find with the help of ( 10) Since A J , A I and ̺ c are all Hermitian operators, these expectation values are real.Note the difference to the approach chosen in Refs.[9,17,22,23], where we computed the average purity rather then the purity of the average density matrix.This difference will be discussed as we proceed and taken up in detail in section 5.

Statistical (random matrix) assumptions
We divide the statistical assumptions employed into two parts.A basic part which is invoked when we compute the density matrix of the central system, and an optional part, which makes stronger assumptions on the dynamics of the environment in order to obtain more explicit results.
• We assume that V is taken from the Gaussian unitary ensemble.This allows to work in the eigenbasis of H 0 , where H 0 is diagonal.The transformation into this basis is necessarily a unitary transformation, such that the ensemble for V remains unchanged.In addition, due to the separability of H 0 = H e + H c , the eigenbasis may be chosen as the product basis |ij of eigenstates of H c and H e , respectively.
• The choice of H c is quite arbitrary as long as it remains fixed.In contrast, we eventually assume H e to be randomly chosen from an appropriate ensemble, say one of the classical ensembles.In that case, the two-point form factor b 2 (t) [24] is the only characteristic quantity of the spectrum.The two-point form factor may well be left unspecified.However, as specific examples we may consider Poisson spectra (b 2 (t) = 0), as well as GOE-or GUE-spectra [24].

Average density matrix of the central state
Here, we calculate the behaviour of the central system, when averaging over the randommatrix coupling to the environment.The state of the central system is then described by the density matrix ̺ c (t) .In view of Eqs. ( 10) and ( 12), we start by calculating the terms A J and A I . where We use the implicit summation convention for indices appearing more than once.The diagonal matrices C c and C e are defined in analogy to Ref. [22] with the only difference that we use the subscript c instead of 1 to refer to the central system: The separable result in ( 17) is due to the choice of the GUE as random matrix ensemble for the coupling.In the GOE case, we would obtain an additional term, which breaks the separability. where , again in full analogy to Ref. [22].For the second term A I , we obtain where This expression is a weighted version of C c (t), with weights given by the diagonal elements of ̺ c .It may be understood as a mapping of the density matrix ̺ c onto a diagonal matrix.It will be denoted by We will encounter a generalization of C c (t) again in the following section.Note that We may write: Note that the double integral A I is Hermitian though the integrand is not.We thus have obtained the relevant quantities for computing the average density matrix in terms of correlation functions.In order to obtain an explicit expression, we can subsequently average over the environmental Hamiltonian also.

Master equation
In order to arrive at the desired master equation, we will make two approximations.First, we assume that the time scale for decoherence is much shorter than the Heisenberg time τ H of the environment.We then approximate the spectral correlation function in the environment with a simple delta function: In the case of the random matrix environment C e (t) → Ce (t), the latter also contains a smooth part, which becomes important at times of the order of τ H and beyond; for the precise form of Ce (t) see [22].Second, we assume that the state ̺(t) of the full system (central system and environment) remains a product state throughout the whole evolution.One common argument is that ̺ e does not change in the course of time because the environment is much bigger than the central system [25].Together, these approximations are consistent with a Markov approximation.We then use the linear response result to advance the density matrix ̺ c (t) for a time step, which should be small compared to the decoherence time but large compared to the mixing time in the environment.We thereby obtain a closed differential equation for ̺ c (t) , the desired master equation.
As before, we restrict the discussion to the GUE coupling.Note though that a GOE coupling can be treated along the same lines and would yield precisely the same result.In this respect, on the level of a master equation approach, the evolution of the central system does not distinguish between systems, where the coupling breaks the time-reversal symmetry and others that do not.This is a general feature of the Fermi golden rule approximation.With (25) and due to C c (0) = m 1 1 c , we find: where we have used the fact that C c [̺ c ](0) = 1 1 c .With these two results and (10), we obtain for the derivative of ̺c (t) at t = 0: Since we are working in the vicinity of t = 0, we may replace ̺ c with ̺c (0) on the right hand side and thereby close the above relation: We therefore obtain as a closed evolution equation for the density matrix of the central system: which constitutes the desired master equation [25].

Purity of the average density matrix
We calculate the purity on the basis of (15).That implies the calculation of two terms, tr( A J ̺ c ) and tr( A I ̺ c ).For the first term we obtain from (19): Similarly, we obtain: where If ̺ c is a pure state, the quantity S c (τ −τ ′ ) agrees with that defined in Ref. [22].Overall, we obtain For a large environment (N → ∞) and for a pure state ̺ c , this result agrees with the average purity calculated in Ref. [22].For finite N, their difference is calculated in Appendix A. Note that neither result requires any averaging over the dynamics in the environment.The averaging over the coupling is sufficient.

Towards exact expressions for the averaged Density matrix
For small dimensions of the central system exact expressions for the ensemble averaged entries of the reduced density matrix can in principle be obtained by methods similar to the ones used for the calculation of fidelity decay [20,21,26].For simplicity lets assume the central system to be a two-level system (i.e. one qubit), initially pure.We write where σ i , i = 1, 2, 3 are Pauli-matrices and σ 0 is the unit matrix.Then the ensemble average of ρ cn (t) takes the form We use an over-line to denote the ensemble average, because here, the average is not only over the coupling but also over the environmental Hamiltonian and even over initial states.Due to the latter, the above expression further simplifies to The quantities f nm (t) are formally average fidelity amplitudes or -equivalently-the ensemble averages of the trace of echo operators.The forward time evolution is given by the 2N × 2N matrix σ m H λ σ n and the backward time evolution is propagated by H λ .Fidelity amplitudes have been calculated exactly for several different random matrix ensembles using supersymmetry.The calculation of f nm (t) is therefore amenable to an exact treatment using the techniques of [21,27,28].We shall present the result in a future paper.

Decoherence within the spectator model
Dynamical model In the spectator model, the central system is divided into two parts where only one part is coupled to the environment (in our case this will be H 1 ).The Hilbert space of the environment is as before H e .The dimension of the different spaces are: m 1 = dim(H 1 ), m 2 = dim(H 2 ), and N = dim(H e ).The dimension of the total Hilbert space is N m, where m = m 1 m 2 .The Hamiltonian reads For an initial state ̺ c ⊗ ̺ e this means only H 1 and H e are coupled dynamically, whereas H 2 may affect the dynamics of the whole system only due to an initial entanglement, i.e. a non-separable ̺ c .

Linear response approximation
The spectator model may be treated along the same lines as the non-structured model considered in section 2. Thus we may use Eqs.(10) and (12) without change, such that the principal task is to compute A J and A I on the basis of ( 16) and ( 20), respectively.Hence, it is only the coupling matrix in the interaction picture together with its second moment, what we need to calculate anew.
To that end we introduce u 1 (t) = exp(−iH 1 t), u 2 (t) = exp(−iH 2 t) and denote with V 1,e the coupling matrix acting on the Hilbert space H 1 ⊗ H e .We therefore obtain: This result can be interpreted as the "old" random coupling matrix in the interaction picture acting in the Hilbert space H 1 ⊗ H e times the identity in H 2 .As a consequence: Ṽ1,e (τ ) Ṽ1,e (τ In the case of A I the calculation is a bit more involved: This result is quite similar to the expression in (21) for the same quantity in the nonstructured case, where we had defined the operator-mapping C c (t).One prominent feature of this mapping was the fact that it transformed any matrix into a diagonal one.
In the present expression this occurs only partially, namely in H 1 , while the density matrix remains in a sense untouched in H 2 .Thus we generalize C c (t), in what follows C 1 (t), to cases where its argument acts on a product space H c = H 1 ⊗ H 2 , in which case the operator mapping yields exactly what we obtained in Equation (43).
This allows us to write:

Master equation
Following the same lines as the derivation of the master equation in the non-structured case, we find in the Fermi golden rule limit: which may be continued in time, just as in the non-structured case, to give a closed differential equation for ̺c (t) : where ̺c (0) = ̺ c .
Werner state solution Recently, [17,22], it has been observed that the average purity and the average concurrence share a one-to-one correspondence during the decay of an initial Bell state.This correspondence is the same as for the one-parameter family of Werner states.Among other things, [17,22] consider the decoherence in a central twoqubit system coupled in the spectator configuration to a random matrix environment, and it is under these circumstances that the curious relationship can be observed.The master equation derived above resolves this puzzle, since, as we will show below, Werner states are solutions to that equation.The point is that the average purity (concurrence) agrees with the purity (concurrence) of the average density matrix in the limit of a large environment (N → ∞).In the case of purity, this has been shown analytically within the linear response approximation and numerically in section 4. In the case of concurrence this has been shown only numerically (in the same section 4).
Let us consider the two-qubit system of Sec. 4. We define a Werner state as a two-qubit mixed state which may be decomposed into a maximally entangled pure state and the identity matrix: where ψ is the maximally entangled initial state, and α(t) + β(t) = 1.Maximal entanglement is equivalent to the condition We will show that ̺ W (t) solves the master (48) for appropriately chosen coefficients α(t) and β(t).Note that (48) determines the time evolution of the two-qubits in the interaction picture.In order to obtain ̺ c (t) which describes the time evolution of the central system in the Schrödinger picture, we have to apply (12).If we have some single qubit dynamics, that may result in ̺ c (t) being no longer a Werner state.To complete the proof, we simply make the Ansatz: Substitution into the master equation (48) yields: Taking the trace on either side, we find: α + β = 0, in agreement with the normalisation condition above.However, projecting either sides on |ψ ψ|, we obtain: As announced, this determines α(t) and β(t) in such a way that the Werner state ̺ W (t) solves the master equation.

Purity
The linear response expression (15) for purity equally holds for the present spectator model.Thus where If ̺ c represents a pure state, the purity P (t) in linear response approximation depends on the initial state only via tr 2 ̺ c .This can be seen from: which is true for any density matrix ̺ c , and which is only true if ̺ c represents a pure state.

Two-qubit central systems
We shall proceed to analyse properties of the average density matrix for the particular case of a central system consisting of two qubits.Most of the analysis will be numerical, and thus we need a small central system for numerical expedience.A single qubit is of interest, but choosing this case would eliminate the option of considering entanglement within the central system.Anyway, the one qubit case will emerge as a special case, if the spectator is not entangled.
For the two qubit case, we consider not only the purity of the density matrix ρ but also its von Neumann entropy S(ρ) = − tr ρ log ρ.The latter quantity is preferred by many authors as a measure of decoherence.We shall quantify internal entanglement with concurrence, and study its properties in the spirit mentioned above.It is defined as C(ρ) = max{0, λ 1 − λ 2 − λ 3 − λ 4 } where λ i are the eigenvalues of the matrix ρ(σ y ⊗ σ y )ρ * (σ y ⊗ σ y ) in non-increasing order, ( * ) denotes complex conjugation in the computational basis, and σ y is a Pauli matrix [29].
Difference between the entropy (squares)/concurrence (triangles) of the average density matrix and their value for each density matrix, averaged over the ensemble.The colours indicate different times and the lines correspond to a fit with fixed slope of −1.This shows that the average density matrix captures the physics of the ensemble for the quantities considered, including purity in the large N limit.
In figure 1 we show the differences between the ensemble averages of C and S as well as their value for the average density matrix ρ, for different times and stepwise increasing the size of the environment.As the reader can see this difference approaches zero with increasing environment size as 1/N env .Similar results were obtained for purity, but are not shown in a figure.They are consistent with the results found in section 2. The upshot is then, that we see all these very relevant quantities do not depend on the form in which the average is taken.Whether this consequence of a concentration of measure to some generalized delta function may be conjectured, but the demonstration thereof will be then interesting subject of future research.
Concurrence and purity, for two non-interacting qubits coupled moderately to an environment, display a relation which corresponds well to that of a Werner state, while the members of the ensemble are not Werner states.It is thus reasonable to ask, whether the average density matrix corresponds to a Werner state.A 2 qubit Werner state, can be alternatively be defined as a density matrix with three degenerate eigenvalues, whose non-degenerate eigenstate is a maximally entangled state (i.e. a Bell state modulo local unitary transformations).We first examine under which circumstances we have a triple degeneracy.
Numerically, we construct a finite ensemble of N tot density matrices.We partition Standard deviation of the three closest eigenvalues of the average density matrix as a function of the size of the ensemble (see text for details).Different level splittings (differentiated by colours) and different times (differentiated by symbol type) are studied.When the coupled qubit has no internal Hamiltonian (blue symbols), the density matrix ρ has a triply degenerate eigenvalue in the large dimension limit.However when ∆ > 0, this triple degeneracy is lifted, as can be seen from an asymptotic non-zero value.
it in sets of equal size N par , and evaluate the average density matrix ρ Npar for each of the N tot /N par members of the partition (i = 1, • • • , N tot /N par ).The properties of ρ (i) Npar are studied.In particular we consider the standard deviation σ Werner of its three closest eigenvalues.The average σ Werner for each of the N tot /N par sets available.This quantity is plotted in figure 2 Indeed, if the coupled qubit has no internal dynamics, the average state, in the large dimension limit, is triply degenerate.In fact σ Werner → 0 as 1/ N par .If the qubit has an internal Hamiltonian, the degeneracy is lifted: a systematic splitting of the eigenvalues (and hence a deviation from Werner states) happens on a time scale set by the mean level spacing of the environment.
We also studied the concurrence of the eigenvector corresponding to the nondegenerate eigenvalue.It remains very close to one, independently of the concurrences of the other eigenvectors, which fluctuate without a clear pattern.We find the rather surprising result, that this property of the eigenfunction of the largest eigenvalue persists even in cases where the previous test shows, that we no longer deal with Werner states.Whether this property is characteristic of the average density matrix or whether it is common to many models will have to be analysed in future work.

Ensembles of density matrices and average density matrix
The very spirit of the treatment of decoherence and entanglement by RMT leads to ensembles of density matrices: each member of the RMT ensemble (and, in our approach, also each initial condition in the environment) induces a density matrix ρ i via unitary dynamics followed by partial tracing.For that ensemble of density matrices we may consider the average density matrix ρ = ρ i .A difference of background will lead at this point to a natural inclination to two different approaches: Physicists with a background in decoherence, quantum optics and quantum information will most likely tend to use this average density matrix to calculate any quantity desired [say C(ρ)], while others with a background in random matrix theory will be inclined to calculate the ensemble averages of the quantities of interest [say C(ρ i )].The difference between these two approaches disappears when one studies expected values of observables as tr Aρ = tr Aρ i or, as show in this paper, for some non-linear quantities in the limit of large dimension for the environment.Moreover, with quantitative entanglement witnesses [30], one can bound some of these non-linear quantities using (linear) observables.Yet the very concept of a relevant near environment, important for many of the new insights obtained form an RMT treatment, suggests to consider small environments as well.This leads us to discuss the appropriateness of either approach in typical applications.
One must recall, that the determination of any of these quantities is not achievable as a single measurement.Rather we need an ensemble of "identical" systems.The hinge is what we mean by "identical".If we have no control over the state or dynamics of the near environment, we must consider f (ρ) (f being an expectation value, or any of the non-linear quantities discussed).Two notes are important at this point: First, the problem discussed here appears for any type of averaging, not only for averaging over time evolutions.Second, for the GUE the ensemble a state average is implicit in the ensemble average, while this is not the case for a GOE [31], and thus has to be performed separately.However, if the near environment is known and its dynamics can be controlled, one should consider f (ρ).This can indeed be the case e.g. for an ion trap setup as the one discussed in the introduction, which might be used to experimentally verify our results.
The differentiation between both approaches is also relevant within quantum information tasks.Consider for instance Shor's algorithm.The success rate has to be calculated from the average density matrix ρ resulting from many realizations of the same experiment.For the teleportation of an unknown state, however, it is important to know the success rate at each attempt, and thus the quality of the process should result from an average quality for individual density matrices corresponding to each try.
At this point, to avoid confusion, we should recall that there is the option of obtaining a mixed density matrix without any entanglement by allowing probabilistic variations in the unitary time evolution of the central system itself in the absence of any significant coupling to the environment.Clearly we then have an ensemble of pure states, whose average naturally will not be pure.Yet errors due to variations of the unitary evolution are usually considered as a loss of fidelity.The purity of the mixed density matrix would then measure an average quantity closely related to fidelity.This explains our emphasis throughout this paper on the fact, that we need a fixed evolution for the central system.It also shows, that the separation of external and internal perturbations for the study of the stability of quantum information tasks is somewhat artificial in a practical sense, though it is very useful for theoretical studies.
The upshot of this discussion resides in the very fact, that for many quantum information tasks, we need repeated experiments.The decision, which average should be taken, is a subtle one.This emphasizes the importance of the result indicating, that the averages coincide in many cases.

Conclusions
We have presented a random matrix theory of decoherence and entanglement.The need for such a theory derives from two facts.On the one hand random matrix theory is known to provide a good generic description of properties of systems displaying, what is often known as quantum chaos or wave chaos.Indeed this statement is almost tautological as many authors nowadays omit any relation to classical chaos and use the relation to statistical properties of RMT as the definition of the latter.Fidelity decay is the other determining factor for the stability of quantum processes, in particular for those relating to quantum information.RMT has proven very successful in this field even in describing experiments [14,15,16], and comparison with numerics for dynamical spin chains have also given encouraging results [32,22,33].On the other hand the universal regime of exponential decay of coherence usually considered and derived in many ways (including RMT [7,8]) is basically founded in the typical situation of very long Heisenberg times in the environment.Intuitive arguments based on Fermi's Golden rule make this behaviour plausible almost independently of the model of the environment used, whenever decoherence occurs long before this time.Yet we argued, that situations with a near environment with fairly low level density and thus short Heisenberg times occur, and will become standard as quantum information systems with ever better isolation from the general environment are developed.It is under these circumstances, that RMT can provide the generic model to which the behaviour of specific systems should be compared.After describing the RMT model family, proposed to a large extent in earlier work, we proceed to analyse a point, which has set RMT models apart from other models of decoherence.The ensemble of evolution operators creates for the central system an ensemble of density matrices rather than a single density matrix.Consequently, properties such as entropy, purity and concurrence have so far been calculated as averages over that ensemble.However, we can also compute the average density matrix first, putting us on equal footing with other more conventional models.The various properties mentioned above are then determined from that single density matrix.This has also the great advantage to produce lower order quantities that have a fair chance to be calculated exactly using super-symmetric techniques.While we have not yet achieved this goal, we have calculated the average density matrix in linear response approximation.This as well as numerics allowed us to compare purity, von Neumann entropy as well as concurrence of the average density matrix to the average of these quantities over the ensemble of density matrices.The central finding is, that for large environments at constant Heisenberg time (or mean level distance) the difference between the two approaches converges to zero as the inverse of the dimension of the relevant Hilbert space of the environment.This indicates, that for decoherence we can often use the average density matrix, and thus the RMT models are really on the same footing with usual descriptions.Yet we have to note, that, when describing the entanglement between two smaller systems, deviations are important and the question which average the behaviour of a single system should be compared to depends on the particular experimental situation.Consistently, if we specialize to a two-qubit central system, we find that average concurrence and the concurrence of the average density matrix will also approach the same limit for large environments.We found the more surprising fact, that at least for small decoherence the eigenfunction of the dominant eigenvalue of the average density matrix remains to very good approximation a Bell state, if the initial state was a Bell state.
We have thus shown that purity and other quantities measuring entanglement yield the same result in the large environment limit, whether calculated from the average density matrix or as an average over the ensemble of density matrices.This was done analytically for purity in the linear response regime (section 2) and numerically, for purity, von Neumann entropy and concurrence beyond the linear response regime (section 4).
Taking into consideration the convexity of this quantities, this might imply that in the large dimension limit of the environment the measure for the density matrices becomes similar to a Dirac delta in the sense that all or a large class of convex functions could be calculated directly from the average density matrix.However other results [12] suggest that typical states are far from the average expected state.This apparent contradiction as well as the distribution of the density matrices as such shall be studied in a later paper.This work can also be readily extended considering more realistic RMT ensembles that RMT allows more realistic models then the classical ensembles.In particular the two-body random ensembles may play an important role particularly in their recent formulation for distinguishable spins [34].Also maps as generic models for gates are of possible interest, and we hope that we laid the foundation for more research in the domain of decoherence by "small" environments, where the Heisenberg time is shorter then or of the order of the decoherence time.