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Simple quantum systems as a source of coherent information
B.A.Grishanin, V.N.Zadkov

International Laser Center and Department of Physics, M.V.Lomonosov Moscow State University
119899, Moscow, Russia

ABSTRACT
The coherent information concept is used to analyze a variety of simple quantum systems. Coherent information

was calculated for the information decay in a two-level atom in the presence of an external resonant field, for the
information exchange between two coupled two-level atoms, and for the information transfer from a two-level atom to another atom and to a photon field. The coherent information is shown to be equal to zero for all full-measurement procedures, but it completely retains its original value for quantum duplication. Transmission of information from one open subsystem to another one in the entire closed system is analyzed to learn quantum inforination about the

forbidden atomic transition via a dipole active transition of the same atom. It is argued that coherent information can be used effectively to quantify the information channels in physical systems where quantum coherence plays an important role.

1.

INTRODUCTION

The concept of noisy quantum channel may be used in many information-carrying applications, such as quantum

communication, quantum cryptography, and quantum computers.1 Shannon's theory of information25 is a purely classical one and cannot be applied to quantum mechanical systems. Therefore, much recent work has been done on quantum analogues of the Shannon theory.611 The coherent information introduced in7'9 is suggested to be analogous to the concept of mutual information in classical information theory. It is defined by
IC

=

8out

Se

(1)

S0, is the entropy of the information channel output and 8e 5 the entropy exchange6'9 taken from the channel reservoir. If S0t -- 8e > 0, then, expressed in qubits, I describes a binary logarithm of the Hilbert space dimension, all states of which are transmitted with the probability p = 1 in the limit of infinitely large ergodic ensembles.
where

Otherwise, we set I = 0.
The validity of the coherent information concept was proved in,9'10 and it was used successfully for quantifying the resources needed to perform physical tasks. Coherent information is expected to be as universal as its classical analogue, Shannon information, and it characterizes a quantum information channel regardless of the nature of both quantum information and quantum noise. In contrast to Shannon information in classical physics, however, coherent information is expected to play a more essential role in quantum physics. The capacity of information channels in classical physics can be estimated, in most cases, even without relying on any information theory, at least within an order of magnitude. This, however, is not feasible in quantum physics and the coherent information concept, or a similar concept, must be used to quantify the information capacity of the channel. An analysis of the quantum information potentially available in physical systems is especially important for planning experiments in new fields of physics, such as quantum computations, quantum communications, and quantum cryptography,1'11 where the coherent information of the quantum channel determines its potential efficiency. In this paper, we apply the coherent information concept to an analysis of the quantum information exchange between two systems, which in general case may have essentially different Hilbert spaces. For this purpose, we must specify the noisy quantum information channel and its corresponding superoperator S, which transforms the initial state of the first system into the final state of another system. A classification scheme for possible quantum channels connecting two quantum systems is shown in Fig. 1.12 In addition to the two-time channels shown in the figure, we consider also their one-time analogues. Two-time quantum channels are widely used in quantum communications and measurements, whereas one-time quantum channels are appropriate for quantum computing and quantum teleportation.
International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, Sergei N. Bagaev, Victor N. Zadkov, Sergei M. Arakelian, Editors, Proceedings of SPIE Vol. 4429 (2001) © 2001 SPIE · 0277-786X/01/$15.00

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quanthmchannelg

time

Figure 1. Classification of possible quantum channels connecting two quantum systems. 1 -+ 1, information is transmitted from the initial state of the system to its final state (a); 1 --+ 2, information is transmitted from subsystem 1 of the system (1+2) to subsystem 2 of the system (b) ; 1 -- (1 + 2) , information is transmitted from subsystem 1 of the system (1+2) to the whole system (1+2) (c).
The paper is organized as follows. In section 2, we explain key definitions and review superoperator representation technique, which is used throughout the paper. In the following sections we consider a variety of quantum channels that correspond to the classification scheme shown in Fig. 1. Section 3 discusses the coherent information transfer between quantum states of a two-level atom (TLA) in a resonant laser field at two time instants (Fig. la) .The same type of quantum channel (1 --+ 1) can be considered for a system that contains two (or more) subsystems. This case is analyzed in section 4, using a spinless model of the hydrogen atom as an example. Coherent information transfer between two different quantum systems is considered in section 5. The analysis includes coherent information transfer between (i) two unitary coupled TLAs (Fig. ib), (ii) two TLAs coupled via the measuring procedure (Fig. ib), (iii) an arbitrary system and its duplication (Fig. ic), (iv) a TLA in the free space photon field (Fig. ib), and (v) two TLAs via the free space photon field (Fig. ib). Finally, section 6 concludes with a summary of our results.

2. KEY DEFINITIONS AND CALCULATION TECHNIQUE 2. 1. Notations and superoperator representation technique
This subsection explains key notations and briefly reviews the symbolic superoperator representation technique,13 which is especially convenient for mathematical treatment of coherent information transmission through a noisy quantum channel. The most general symbolic representation of a superoperator is defined by the expression

S = i: Ski (k ® Il),

(2)

where the substitution symbol 0 must be replaced by the transforming operator variable and (kI is an arbitrarily

chosen vector basis in Hilbert space H, to which the transformed operators are applied. In order to correctly apply this transformation to a density matrix, operators kl must obey the positivity condition for the block operator S = (k1)14 and orthonormalization condition

T8klk1,

(3)

which provides normalization for all normalized operators 3 with Tr 3 = 1.
Using symbolic representation (2), one can easily represent the production of superoperators S1, S2, which constitutes a symbolic representation of the superoperator algebra. For Ckl = 1k) (l it results to the identity superoperator, I, and for Ckl = 1k) (k 6k1--tO the quantum reduction superoperator R = I 1k) (kI @ 1k) (kI. The case of Ckj = represents the trace superoperator Trü, which is a linear functional in the density matrix space. The correspondence between the matrix representation S = (Smn) of the superoperator S in orthonormalized operator basis Ék and its symbolic representation (2) is given by

= 8(1k) (il) = Smn (il n Ik) em
mn

(4)

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and can be easily checked by substituting it in Eq. (2) and comparing with the standard definition of matrix elements

Sen = >m Smnm.

2.2. The calculation of coherent information
The entropy exchange in Eq. (1) for the coherent information is defined as
Se

S(ia), S(,) = --Trj1og2 ;,

(5)

where the joint input-output density matrix j3 is given, in accordance with,9'15 by

3a=S(Ipj)(pjI)®Ij)(jI.
Here

(6)

i) are the transformed eigenvectors of the input density matrix p) = = >p Ii) (ii, bar symbol stands for complex conjugation, and S is the channel input-output superoperator, so that the output density matrix Iout = SAD. Using superoperator representation (2) within the above defined eigen basis Ii), the density matrix (6) takes the form: (7) 13a (p.p.)1/4 ® (iI,

I)

the states of the output. Both the input and output marginal density matrices are = Troutj3. Finally, the coherent information (1) can be calculated, keeping in mind that S0, =
where

operators s

represent

given by the trace over the corresponding complementary system: i3out =

2.2.1. Two-time coherent information for two quantum systems
For the coherent information transfer between two quantum systems through the quantum channels shown in Figs ® P2, lb,c (1 --+ 2 or 1 --+ (1 + 2)), the initial joint density matrix must be taken in the product form p1+2 = where A = i3i and 12 are the initial marginal density matrices, the first one being an input. For the 1 --+ 2 quantum channel, the output is the state of the second system, since a transformation on these two systems is made and a certain amount of information is transmitted into the second system from the initial state of the first one.
The dynamical evolution ofthe joint (1+2) system is given by a superoperator 81+2 and the corresponding channel transformation superoperator, which converts I3out = S5jii, can be written as

where the trace is taken over the final state of the first system. The transformation is described in terms of Eq. (2) for the joint system as s (8) (nIst In) (icIp2 A) (kIÜ 1),

=i

k,clA n

where the product basis 1k) k) is used and indexes k, icstand for the first and second quantum systems, respectively. The operator coefficients 'kl in Eq. (2) now take the form:
Ski =

icA n

(nl

I)

Kid

P2 IA).

(9)

Superoperator S depends on both the dynamical transformation S12 and the initial state P2,and couples the initial state of the first system with the final state of the second system.

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2.2.2. One-time coherent information
One-time information quantities can be easily calculated if the corresponding joint density matrix is known. In the case of a single system, the corresponding channel is described by the identity superoperator I. For the joint input-output density matrix (6), we get a pure state = > Ii) Ipi) > (ciI (pi1 and then calculate the entropy 0 and the coherent information I = 8out = In the case of two systems, the input-output density exchange S = matrix is the joint density matrix 1+2, and the corresponding coherent information in system 2 on system 1 at time t is I (t) = S[32 (t)] -- S{,312 (t)] . In the case of unitary dynamics and a pure initial state of the second system, all initial eigenstates Ii) of the first system transform into the corresponding orthogonalset I'(t) of the (1+2) system, so that the joint entropy is time-independent and the coherent information yields I(t) = S[32(t)] -- S{3 (0)]. If the initial state of the first system is also a pure state, we get simply I (t) = S[,Ò2 (t)J . For the TLA case, this simply yields I = 1 qubit, if a maximally entangled state of two-atom qubits is achieved.

3. TLA IN A RESONANT LASER FIELD
In this section, we discuss the coherent information transfer between the quantum states of a TLA in a resonant laser field at two time instants (Fig. la). Such quantum channel with pure dephasing in the absence of an external field was considered in.9 In a more general case, coherent information, based on the joint input-output density matrix (6), can be readily calculated by using the matrix representation technique for the relaxation dynamics superoperator. An interesting question is how the coherent information depends on the applied resonance field. The field changes the relaxation rates of the TLA. These rates are presented with the real parts of the eigenvalues

Ak of the dynamical Liouvillian ¸ = 4 + LE of the TLA, where 4 and LE stand for the relaxation and field interaction Liouvillians. For simplicity, we will consider here relaxation caused only by pure dephasing, combined with the laser field interaction. The corresponding Liouvillian matrix in the basis of k = {I, ¸T3, &1 , cT2} reads

L=(g

F

where F is the pure dephasing rate, is the Rabi frequency, and & , of the matrix (10) can be readily calculated and are given by
Ak

'
2
--

(10)

, cr3

are the Pauli matrices. The eigenvalues

{0, --F, --(F + p2 _ 42)/2 --(F

/p2 4Ã2)/2}.

These values are affected by the resonant laser field with respect to the unperturbed values 0, F, which also affects the resonant fluorescence spectrum of the TLA. At 11 > F/2 it results in so-called Mollow-triplet structure, centered at the transition frequency, which has been predicted theoretically16 and subsequently confirmed experimentally.17 From the information point of view, the resonant laser field might reduce the coherent information decay rate and, therefore, lead to the increase of information, although this information gain could intuitively be expected only from the laser-induced reduction of the relaxation rates of the relaxation superoperator 4. itself.1821 CaJculating the matrix of the evolution superoperator S = exp(Lt) and using its corresponding representation (2), the joint density matrix may be calculated analytically (6). Then (with the help of Eqs (5), (1)), the coherent information left in the TLA's state at time t may be calculated about its initial state. This state is chosen in the form of the maximum entropy density matrix 3o =1/2. The results of our calculations are presented in Fig. 2. They show the typical threshold-type dependence of the coherent information versus time, which is determined by the loss of coherence in the system. Also, the coherent information does not increase with an increase of the laser field intensity, as might be expected. The coherent information even decreases as the Rabi frequency increases. In addition, the results show a singularity in the first derivative of the coherent information dependence at time t = 0, which is a characteristic feature of the starting point of the decay of coherent quantum information. Initially, = iTjiTj+ with the input-output wave function the input-output density matrix (6) of the TLA is a pure state = Ji- Ij) i). Its eigenvalues Ak and the probabilities of the corresponding eigenstates are all equal to zero, except for the eigenstate corresponding to iJt Due to the singularity of the entropy function -- > Ak log Ak at Ak = 0 the derivative of the corresponding exchange entropy also shows a logarithmic singularity.

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1.0

Figure 2. The coherent information transmitted between the states of the TLA at two time instants, t = 0 and
t > 0, versus time Ft and the Rabi frequency Q/F (both are dimensionless). Another interesting feature of coherent information is its dependence on the initial (input) state jfij . might be chosen in the form of the eigenoperator possible,

If

it were

in =

Ikmin)j e1

of the Liouvillian, where I1min) is the eigenvector_corresponding to the minimum value IeAkI > 0 of the matrix L. Yet the vector I1min) 5 equal to {O, (IT + Jp2 4f2)/2Ã, 0, 1}, which corresponds to the linear space of operators
with zero trace due to the zero value of the first component. Therefore, the coherent information decay rate cannot be reduced by reducing the corresponding decay of atomic coherence.

4. COHERENT INFORMATION TRANSFER BETWEEN TWO SUBSYSTEMS OF THE SAME QUANTUM SYSTEM
In this section we investigate the quantum channel (1 -- 1, Fig. la) between two open subsystems A and B of a closed system A + B having a common Hilbert space sp (HA, HB), where HA and HB are the Hilbert subspaces of the subsystems A and B, respectively. In classical information theory, this situation corresponds to the transmission of part A C X of the values of an input random variable x E X. The situation where a receiver receives no message is also informative and means that x belongs to the supplement of A, x A. It can be described by the choice transformation C = PA + P0(1 --PA), where PA is the projection operator from X onto the subset A, PAX = x for x A and PAX = 0 (empty set) for x E A, Po is the projection from X onto an independent single-point set X0, and P0x = Xo. This transformation corresponds to the classical reduction channel, resulting in information loss only if A is not a single point. If A is a single point, we are able to get a maximum of one bit of information, for A can provide another point of the bit, so that for an input bit we have no loss of information. In quantum mechanics, the corresponding reduction channel is represented as the choice superoperator

C=P'A®PA+I0)(0ITr(1--PA)Ü(1--PA),

(11)

where state 0) is a quantum analogue of the classical single-point set, which is separate from all other states. Eq. (11) defines a positive and trace-preserving transformation, which can appropriately describe coherent information transfer between subsets of the entire system. The last term in Eq. (11) represents the total norm preservation, if all the states outside the output B-set are included. In our case, these states are included in the incoherent 0) (01 form, which in contrast to the classical one-bit analogue of a TLA yields no coherent information due to the complete destruction of the coherence.

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_tIL.

·

··....(·l=o m0\m-l m1
\....,·
.. .../··

1=1

n2

A

B

n1
Figure 3. A spinless model of the hydrogen atom. The information channel is made of the input forbidden
nim --+ n'l'rn' transition 100--200 and the output dipole active 100--210 transition.

Considering the coherent information transmitted from part A to part B of the system, which evolves in time, we deal with the channel superoperator

SAB CBSO(t)CA, S0(t) = U(t) 0 U1 (t)

(12)

with U(t) being the time evolution unitary operator. Here the input choice superoperator CA is shown just to define the total channel superoperator, regardless of the input density matrix. Otherwise, CA i5 already accounted in the input density matrix , defined as the operator in the corresponding subspace HA of the total Hilbert space H. Let us assume that the dynamical evolution of the system is determined and the Bohr frequencies wk and the corresponding eigenstates 1k) are found. Then, representing the projectors in terms of the corresponding input I/') and output IPm) wave functions, Eq. (12) gives the specified time evolution form

SAB(t) =
ll'EA

[11(t) + 10) (01

mB
mm'EB

(m Ii(t)) (j'(t)

im)]

iI 0
(13)

11'(t) =

: (Pm

bz(t)) (tp11(t) IPmI ) 1Pm) (Pm'I,

kbi(t)) = >::k

(k I,bi) 1k).

Let us consider the case of the orthogonal subsets of input/output wave functions, which is of special interest. Then, if there is only one common state fr/) in the sets /'i) IPm) and U(t0) = 1 holds for some to, we get SAB(to) = fri) (cb ® k) (01 +

0) (O :

(Pml

which means that the quantum system is reduced into a classical bit of the states 1c1) and 10) and no coherent information is stored in the subsystem B. Nevertheless, if the eigenstates 1k) of U(t) do not coincide with the input/output states kbi), I'm) the coherent information will increase with the time evolution. Hence, the information capacity of the channel is determined by quantum coupling of the input and output. To illustrate the coherent information transfer through the quantum channel considered in this section, let us analyze a typical intra-atomic channel between two two-level systems formed of two pairs of orthogonal states A = {ktio), i)} and B = {Rbo), lP2)} of the same atom. A spinless model of the hydrogen atom could serve as such is the ground s-state with n = 1, b1,2 are the s-state with 1 = 0, rn = 0 and p-state with 1 = 1, a system (Fig. 3): m = 0 of the first excited state with n = 2, respectively. In the absence of an external field, this quantum channel transmits no coherent information, as the 1 =0, m = 0 and 1 = 1, m = 0 states are uncoupled. In the presence of an external electric field applied along the Z-axis,

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the considered two out of four initially degenerated states with n = 2 are split, due to the Stark shift into the new eigenstates Ii) = (hIi) + II)2))/v', 2) = (kbi) -- I/'2))/\/* The input 1 = 0 state oscillates with the Stark shift
frequency: ?/' (t)) = cos(w3t) Ibi) + sin(w3t) I/'2). Therefore, due to the applied electric field, the input state becomes coupled to the output state, which carries the coherent information. For our model, Eq. (13) presents the kj-operators in the form of a 3 x 3-matrix, where the third column and row introduce the phantom "vacuum" state 0):

/1 0 o\ ii=I 0 0 0 ,
21

I
I

\\0 0 0)

8121
822
1

/0 sinw8t 0\

\0

0 sinw3t

oo\ 00
0 o)

(0 0

\0

0

0 0

01, 0)

\o

0 0 0 sin2w3t o cos2w3t

Zero values of 812, 821 correspond to the absence of coherent information at t = 0 or to the absence of coupling. Choosing the input matrix in the maximum entropy form j3 = 1/2, we get the corresponding joint input-output matrix in the form

a=

!

0000
IC

¶

1

--

where x = sin w3t and the output density matrix pout is diagonal with the diagonal elements 1/2, x2/2, and (1 --x2)/2.

Calculating non-zero eigenvalues (1 x2)/2 of j3 and the entropies S0,t, S, we get the coherent information

=

[(1 + x2) log2(1 + x2) -- x2 log2 (x2)]/2.

This function is positive except for x = 0, where the coherent information is equal to zero, and its maximum is equal Thus coherent information on the state of the e.g., for the precession angle w8t = forbidden transition is available, in principle, from a dipole transition via Stark coupling. Its time-averaged value is (Ia) 0.46 qubit. This forbidden transition was discussed in22'23 as a potential source of information on spatial symmetry breaking caused by the weak neutral current.24'25 For example, if I, = 0, only the incoherent impact of the forbidden transition (by means of the ground state population no) remains and provides a classical-type of information on the interactions that cannot be observed directly. In this case, only one parameter--population--can be potentially measured, while exact knowledge of the phase of the transition demands I, = 1.

to 1 qubit at x =

5. COHERENT INFORMATION TRANSFER BETWEEN TWO QUANTUM SYSTEMS
In recent years, a few results have been published related to coherent information transfer in a system of two TLAs, including discussion of the problem from the entanglement measure viewpoint26 and the "eavesdropping problem" 27 A number of different experiments have been proposed to study controlled entanglement between two atoms.28'29 From the informational point of view, the coherent information transmitted in the system of two TLAs connected by a quantum channel depends both on the specific quantum channel transformation and the initial states of the TLAs. For the latter, it seems reasonable to assume that they can be represented by the product of the independent states
of each TLA: P1+2 = pin 0 P2·

In this section, we present a systematic treatment of the coherent information transfer between two different quantum systems. The analysis includes coherent information transfer between (i) two unitary coupled TLAs (subsection 5.1), (ii) two TLAs coupled via the measuring procedure (subsection 5.2), (iii) an arbitrary system and its duplicate (subsection 5.3), (iv) a TLA and the free space photon field (subsection 5.4), and (v) two TLAs coupled via the free space photon field (subsection 5.5).

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5.1. Two unitary coupled TLAs
Let us first examine a deterministic noiseless quantum channel connecting two TLAs (Fig. ib) . Such a channel can be described by the unitary two-TLA transformation, which is defined by the matrix elements Uk,k'j' with k, i, k', i' = 1, 2. Then, the channel transformation superoperator S describing the transformation -+ i3out = can be written in terms of the substitution symbol (see Eq. (2)), with operators skl = >I Skz, ii) (111, represented with the matrix elements of S (in accordance with Eqs (4), (9)), in the following form:

8k1,iw = ::
ma3

P2Um,,ka14nv,1j3.

(14)

The relation Tr 8k1 = > Skl, = k1 is valid here and ensures the correct normalization condition, whereas the
positivity of the block matrix

f\ I Skj)(

.ii 12
821

'\

22

ensures

the positivity of S.

For the no-entanglement transformation U = Ui ® U2, Eqs (2), (14) yield S = ITr 0, which means that the initial state i3 of the first TLA transfers into the final state, which is not entangled with the state % = U232U of
the second TLA.

We can simplify Eq. (14) by considering a pure state , so that together with an arbitrary choice of noentanglement transformation U, it seems reasonable to consider a special case of the pure state: P2a$ &J3aao. Keeping also in mind that 8k1,pv 5 linear on the density matrix P2 and the coherent information I is a convex function of 5,10 Eq. (14) simplifies to * (15) 8k1,jv = Ump,,kaoUmv,lao,
m

which means that the quantum channel is described only by the unitary transformation U. Here the summation is taken over only the states Irn) of the first TLA after the coupling transformation. The coherent information transmitted in systems of two unitary coupled TLAs with j5 = 1/2 and (p2)12 = /(i2)ii [1 -- (,2)ii] is shown in Fig. 4. A convex function of P2 shown, which has the maximum on the border, Pn (i32)11 = 0, 1. As in the case of a single TLA, the behavior of the coherent information preserves the typical threshold-type dependency on the coupling angle, which determines the degree of the coherent coupling of two TLAs with respect to the independent fluctuations of the second TLA.

5.2. Two TLAs coupled via the measuring procedure
Here we will discuss a specific type of quantum channel connecting two TLAs,3° where the superoperator S is defined by the measuring procedure, which implements a different approach to the quantum information31 called measured information. We start with a channel formed of two identical two-level systems. In terms of wave function, the corresponding f ull measurement transformation of the first TLA state is defined as
ai=(Ãbi).
This
(16)

transformation provides full entanglement of some basis states fr/a), which do not depend on the initial state of the second TLA. The latter serves as a measuring device, yet fully preserves information on the basis states of the first system state /' = a q5). Eq. (16), being a deterministic transformation of the wave function, is neither linear nor unitary transformation with respect to Ão and, therefore, cannot represent a true deterministic transformation. The corresponding representation in terms of the two-TLA density matrices has the form:
P12

(Ãb (jI

P12

frj)

fr) I) I) (

(17)

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Figure 4. The coherent information transmitted between two unitary coupled TLAs versus population p11 of the diagonal initial density matrix of the second TLA and the coupling precession angle lIt.
This representation is linear on i12 and satisfies the standard conditions of physical feasibility,10'32 i.e completely positive and trace preserving. This matrix is in the form of > p fr/i) fr/j) (/j I ( I so that S(312 ) = S(2) . Due to the classical nature of the information represented here only with the classical indexes i and in accordance with the equations of section 2, the single-instant coherent information is zero. In the case of a two-time channel, the superoperator for the quantum channel connecting two TLAs can be readily derived from Eq. (2) with skj = fr/k) (IkI kl, (ki -- (kI, and k) --+ fr/k). After calculating the trace over the first TLA and replacing P12 with the substitution symbol 0, the equation takes the form:

M=>PkTrlEk®.
Here Pk 1k) (k
TLA and P2k
I

(18)

the orthogonal projectors representing the eigenstates of the "pointer" variable of the second 5 the orthogonal expansion of the unit (orthogonal map) formed of the same projectors. This orthogonal map determines here the quantum-to--classical reduction transformation TrlEk 0 = (kI ® Ick), which represents the procedure of getting classical information k from the first system. Applying the transformation and using Eq. (6) for the respective output and input-output density matrices, we get (18) to
are
k1k) (tl5k I

Iout =

Pk

frtk) (kI, Pa >Pk k) 71k) (7tk (kI,

(19)

where Pk (k Pin fr/k) = >:: pi

>i:: \/.7k

ment procedure. It is important to note (as it follows from Eq. (19)) that there is no coherent information in the system because vectors frk) are orthogonal and therefore the entropies of the density matrices (19) are obviously the same. Conversely, the measured information introduced in31 is not equal to zero in this case. We can easily generalize our result for a more general case of the quantum channel, when the second system has a different structure from the first and, therefore, they occupy different Hilbert spaces. This difference leads to the replacement of the basis states Icb) of the second system in our previous results with another orthogonal set = V q5), where V is an isometric transformation from the Hubert state H1 of the first system to the different Hilbert space H2 of the second system. After simple straightforward calculations, the final result is the same--there is no coherent information transmitted through the quantum channel. This result is a natural feature of coherent
information, in contrast to other information approaches (see, for example Ref.31).

@/k Ii )

I)

fr ) 2 are the eigenvalues of the reduced density matrix and rk) = are the normalized modified input states coupled with the output states Ic5k) after the measure-

(k

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It is interesting to discuss more general measuring-type transformations, for instance, the indirect (generalized) measurement procedure. This procedure was first applied to the problems of optimal quantum detection and measurement in33 and then, in a form of non-orthogonal expansion of unit ¸ (dA) ,in34 (E (dA) is equivalent to the positive operator-valued measure, POVM, used in the semiclassical version of quantum information and measurement theory'1'35'36). This indirect measuring transformation results from averaging a direct measuring transformation applied, not to the system of interest, but to its combination with an auxiliary independent system. The indirect-measurement superoperator in the general form can be written as

MPqTrqÜ,
where

(20)

Pq are the arbitrary orthogonal projectors and Eq 5 the general-type non-orthogonal expansion of the unit in H space (POVM). Note that eq ÃOq) (ÃOqI 5 a specific "pure" type of POVM, first used in quantum detection and estimation theory.33 The latter describes the full measurement in H 0 Ha for the singular choice of the initial auxiliary system density matrix p = The information transfer from the initial density matrix to the final output state is represented in Eq. (20) via the coupling provided by indexes q. Because the number Nq of q values can be greater than Dim H, it seems reasonable to suggest that some output coherent information is left about the input state. The corresponding output and input-output density matrices are given by

iout =

;:

qPq,

ja =

(il 'q li) IJq

® F) (31 ,

(21)

= Tr EqPin are the state probabilities given by the indirect measurement. In the case of full indirect measurement, it can be easily inferred theoretically or confirmed by numerical calculations for particular examples that no coherent information is available. The proof is based on the quantum analogue7 of the classical data processing theorem and the above discussed result on a full direct measurement. Therefore, in order to get non-zero coherent information, a class of incomplete (soft) measurements must be implemented, which are subject to more detailed quantum information analyzis.
where 15q

5.3. Quantum duplication procedure
In the previous subsection, we demonstrated that the classical-type measuring procedure defined by the transformation (17) completely destroys the coherent information transmitted through the quantum channel. Here we will consider a modified transformation for the quantum channel shown in Fig. ic, which preserves the coherent information:
P12
+ P12
WiI Tr2312 Icbj) Icb) frt) (IiI (/i1.

this equation off-diagonal matrix elements of the input density matrix ,Òi preserves the phase connections between different /. For the initial density matrix of a product type 3j110 ,Ò, in terms of the corresponding superoperator has the form:
In

=

are

taken into account, which

transformation from H to H 0 H,

Q = I frti) cbj) (cbjl (I

(c5I

(22)

This superoperator defines the coherent measuring transformation, in contrast to the incoherent transformation discussed in.31 The coherent measuring transformation converts into Ò2-independent state

= P2 =

(II Pin Ii) I) fri)

(iI

(23)

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which results in duplication of the input eigenstates / into the same states of the pointer variable k = >k k fr/'k) (tk I. Pure states of the input are transformed into the pure states of the joint (1+2)-system by doubling the pointer states:

This mapping is similar to the mapping given by Eq. (16). Of course, only the input states b equal to the chosen pointer basis states Ãb are duplicated without distortion because it is impossible to transmit non-orthogonal states using only orthogonal ones. The entropy of the output state with a density matrix (23) having the same matrix elements as is evidently the same as the input state, S0, = Sr S[3], due to the preservation of the coherence of all pure input states.
For the joint input-output states, the transformation (22) yields the corresponding density matrix (6) in HüH®H space:

ia

fr/k) frIk) (/i1 (iI ®\/ IXk) (xiI

(24)

where Pk, lXk) are the same as above, providing an expansion of the input density matrix in the form

=

>k 12k IXk) (Xk I. Taking into account that the first tensor product term in Eq. (24) is a set of transition projectors M , Pk1Pmn 5imPkn, we can apply easily proven algebraic rules valid for a scalar function f:

f(>.Pkl 0 Bkl) = >Pki 0 f(ul)kz,
where 1 = (IL) is the block matrix and Trf(Ekl kl ®Rkl) = Trf(i). Here I = (sJtIxk)(xii), and it is simply lix)) ((xII with IIx))k = /Xki, a vector in the H ® H space. All eigenvalues Ak of this matrix are equal to zero,
except one value corresponding to the eigenvector I Ix)). Calculation of the exchange entropy gives 8e 0, and, therefore, I = S. Consequently, the coherent duplication does not reduce the input information transmitted through the 1--+(1+2) channel, nor does it matter whether the = 0, or not. register k is compatible with the input density matrix, [k, If the channel is reduced to the one shown in Fig. lb and discussed in the previous subsection, by taking in Eq. ( 23) trace either over the first or the second system, we evidently come to the measurement procedure discussed in subsection 5.2. As a result, we can conclude that the coherent information is strictly associated with the joint system

but not with its subsystems. This natural property could be used in quantum error correction algorithms37 or for producing stable entangled states.38

5.4. TLA-to-vacuum field channel
In this subsection, we analyze the quantum channel between a TLA and a vacuum electromagnetic field (Fig. ib), which is an extension of the TLA in an external laser field, as considered in section 3. For this analysis, we will use a reduced model of the field, which is based on the reduction of the Hilbert space of the field in the Fock representation (Fig. 5). The problem, therefore, is reduced to that of the interaction of a two-level system with continuous multi-mode oscillator systems,39 a specific case of which is the interaction of an atom with the free photon field. However, to analyze the information in the system (atom+field), we do not need to consider the specific dependence of the wave function /'o (k, A) of the field photon on the wave vector (including polarization), because only its total probability and phase are significant. In the basis of the free atomic and field states for the vacuum's initial state ao = 0, we get from Eq. (15)

=

Ump,U,w,ij.

Greek letters are used to distinguish the photon field indexes, which in the general case include both the number of photons and their space or momentum coordinates. Matrix elements of this superoperator calculated via the
atom-to-field unitary evolution matrix Ump,kO coefficients (Table 1) are shown in Table 2.

Proc. SPIE Vol. 4429

73


1)a

%:O

.

:
0)
Fock states of the field

IO)a

Atom's

states

Figure 5. Structure of the joint Hubert space of the (atom+field) system. For the vacuum initial field state, both atomic states and only two Fock states of the field (10) and Ii)) are involved in the dynamics of the joint system (atom+field). The dynamics is entirely defined by just two states, I0)a I1)kA and 11)a 10), which are described by
/'0(k, A) and c1, respectively.

Table 1. Unitary (atom+field) to (atom+field) transformation Um2,ka for the vacuum initial photon field state, where indexes m,k stand for atomic quanta and jt,a--for the number of photons. Long dash symbol stays for the
elements not involved into the calculated terms Ski,,1,, (Table 2).

ka\ 00
01 10 11

00
1

01 0

10 11
0
Ci

0
0

0 b0(k,A)

Table 2. Atom-to-field transformation Ski,,1,,, which defines 1k) (11 --+ l) ("I superoperator transformation. Indexes k, 1 stand for atomic quanta and z, v--for the number of photons.
00 01 0 0 10
11

00
01 10
11

1

0

0 0
IC1 12

0(k,A)
0 0

bt(k,A)
0

0 0 0

b0(k, A),b(k', A')

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The choice of 0(k, A) as a basis for the photon field4° reduces the matrix of operator Skj,,, to the non-operator matrix transformation, which in terms of skl matrices has the form:

811=o o)'
821

I10
0

( o (1_e_t)h/2
O'\
'

.'

I

(1
where

--

_t)h/2 0 )

S12O (e_t 822
!\
0

0
0
1--

\
'\

)'

25

et

)

ci 2 exp(--'yt) describes the population decay of the totally populated initial excited state of the atom and f >i: 0(k, A)I2dk = 1 -- exp(--'yt) is the probability a photon will be detected. From Eq. (25), it follows that the structure of the photon field plays no role, and the transmitted information defined by the input-output density
matrix depends only on the photon emission probability by time t. The reduction of the photon field (only the photon numbers p, ii = 0, 1 were taken into account) leads to the conclusion that the photon states also are equivalent to those of a two-level system.

Applying the transformation (25) to the input atom density matrix
,'.

_(Pii

P12

p1n--p12

)'
e_t)h/2

restricted to the real off-diagonal matrix elements, we get the output density matrix
Pout
--

( Pu

+ p22e_7t

P12 (1 --

P12 (1 --

e_t)h/2

P22(l --

and for P12

0 the respective input-output density matrix
P11

0 o 0

Pa--

--

0
0

P22e_Yt 0

[P11P221 -- e_yt)]h/2

0 [P11P22(1 -- 0 o o P22(l -- 0

For t --+ this expression yields a pure atom-photon state, which converts incoherent fluctuations of the atomic states, forming the incoherent ensemble, to equivalent coherent fluctuations of the photon states. The corresponding

eigenvalues are ) = {0, 0, l--P22 exp(--'yt), P22 exp(--'yt)}. Non-zero values are equal to the probabilities ofthe atomic states at time t. For the output (photon) density matrix i3out the eigenvalues are Aout {p22 [1 _ exp(')'t)], 1 -- p22[l --
exp(--'yt)J}, which are the probability that a photon will be emitted or not. These sets of eigenvalues determine the eigen probabilities of the joint input-output and marginal output matrices. The coherent information, defined by the difference of the corresponding entropies, then takes the form:
IC =
-- (1 -- P22 + xP22 log2(xp22) -- xp22) log2(1 -- xp22) -- (1

(1 -- x)p22

xp22) log2[1 -- (1 -- x)p22]+
log2(p22 --

(26)

where

x=

exp(--'yt).

This formula is valid for I > 0, otherwise, I =
[1 -- exp(--'yt)]

exp(--'yt) = 1/2, the time when the probability 1 -- P22 of the lower atomic state 1 -- P22 exp(--'-yt).

0. The corresponding critical point is of finding no photon is equal to the population

The results for calculating the coherent information are shown in Fig. 6 for two specific cases: P12 =0 (Fig. 6a) P12 < 1/2 (Fig. 6b). One can see from Fig. 6a that the coherent information is symmetrical with respect to the population Pu around the symmetry point P11 = 1/2. Increasing the excited state population P22 = 1 -- Pu and the corresponding photon emission yield does not increase the coherent information, because of the reduction of the source entropy, which determines the potential maximum value of the coherent information. For the same reason, the coherent information decreases when there is a non-zero coherent contribution to the initial maximum entropy atom state and completely vanishes for the pure coherent initial state (Fig. 6b).

and Pu = 1/2, 0

Proc. SPIE Vol. 4429

75


.--

0

o.o
P9:2

/
5

Figure 6. The coherent information transmitted in the atom-to-field quantum channel versus the dimensionless time yt and input atomic density matrix, which is taken either diagonal with the ground state matrix element p11
(a)

or as the sum of 1/2 and the real ("cosine-type") coherent contribution of the off-diagonal elements P1201 (b).

In accordance with section 2 and because of the purity of the initial field state, one-time information is equal to the difference of the entropies of the photon field only, represented by i3out, and the initial atomic state, represented For a pure initial state, expressed in the form of the excited atom state 2), and for 0 < t < x, we always by get non-zero information I = --x log2 x -- (1 -- x) log2(1 -- x) that yields 1 qubit for x = 1/2, when the excited state population is equal to the probability a photon will be emitted.

5.5. The transmission of coherent information between two atoms via a free space field
In this subsection, we will consider the quantum channel when information is transmitted from one atom to another

via the free space field (Fig. ib). Suppose that the second atom is initially in the ground state. In addition, we will restrict ourselves here to the long time scale approximation, in which the effects of the discrete nature of the retarding electromagnetic interaction are neglected.4144 Under such restrictions and approximations we have the Dicke problem,45 for which the well-known solution for the atomic state in the form of two decaying symmetric and antisymmetric Dicke states I) = (Ii) 2) + 2) 1))/v", a) = (Ii) 2) -- 2) I1))/V' and the stable vacuum state tO) = Ii) 1) can be written as:
c8(t) =
Ca(t)

exp[--('y8/2 + iA)t], Ca(O) exp[--('ya/2 _ iA)t],
c8(O)

(27)

Co (t) = Co (0) + [c8 (0)2 + Ca (0)2

C8(t)2

Ca

21/2 i(t)

Here c(t) is the amplitude of the stable vacuum component 1) 1), which has an incoherent contribution due to the spontaneous radiation transitions from the excited two-atomic states, e(t) is the homogeneously distributed random
phase, Ys,a and A are their decay rate and coupling shift, respectively, and Cs,a are the amplitudes of the Dicke states. In terms of the products of the individual atomic states Ii) i) for the corresponding initial amplitudes C12 (0) = 0, c22(0) = 0 the system's dynamics is described, according to the Dicke dynamics (27), by the following equations:

Cii(t) = Cii(0) + f(t)e(t)C2i(0),
C12(t) = fa(t)C12(0),

C21(t) = f3(t)c21(0), C22(t) = 0,

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Proc. SPIE Vol. 4429


f(t) =

{1

--

[exp(--y8t) + exp(--'yat)]/2}"2,

f8(t) = {exp[--('y3/2 + iA)t] + exp[--('ya/2 fa(t) {exp[--('y8/2 + iA)t] -- exp[--('ya/2

_ iA)tJ}/2.

_ iA)t]}/2,

Applying these formulas to the input operators Ckl (O)c (0) k) (l of the first atom and then averaging the output over the final states of the first atom and the field fluctuations (the latter is represented here only with (t)), we get the symbolic channel superoperator transformation (1) (0) --+ ,(2)(t) = S(t)3(1)(O) and corresponding kl operators in the form:

8(t) =

1) (ii 0 Ii)

(ii +

+fa(t) 2) (21 0

I')

[f(t)2 + 1f8(t)12]

Ii) (21

2)
(21,

(ii

+Ifa(t)12 2) (21

0

2) (21

('I +f(t) 1) ('I 0 12)

,'

S21

-I
--

(1 O\ _(O f(t).\ s11_o o)' S12_O 0 )'
00\
fa(t) 0

(28

)

)

'

_ 822 --

(

f(t)2+1f3(t)12

0 Ifa(t)12

To further elucidate this problem, let us now discuss the case of two identical atoms having parallel dipole moments aligned perpendicular to the vector connecting the atoms. Here only two dimensionless parameters are essential: dimensionless time, 'yt, where 'y is the free atom's decay rate, and dimensionless distance, p =k0R, where R is the interatomic distance and k0 is the wave vector at the atomic frequency. Then, the dimensionless two-atomic

decay rates and the short distance dipole-dipole shift are given by29'38'44:
Ys,a/Y
respectively,

1

and A/'y = (3/4)/3,
sinp).

with g = (3/2)(p1 sin + Ã2 cos --

The coherent information may be calculated as previously described in subsection 5.4 by replacing exp(--'yt) with f(t)2 + 1f8(t)12 in Eq. (25). Then, the operators skj in Eq. (25) become similar to the corresponding operators in Eq. (28). The coherent information is given by the same Eq. (26) with x = f(t)2 + 1f8(t)12 which, however, now has (in contrast with a single-atom case considered in 5.4) new qualitative features arising from the specific oscillatory dependence of Ifs,a(t)12 on the interatomic distance p.

If there were no oscillations from the quasi-electrostatic dipole-dipole coupling, i.e. as in the case of A =0, the coherent information would always be equal to zero, because the threshold x < 0.5 would not be achieved. Parameter (1 -- x) corresponds to the population of the excited state of the second atom for the initial state 12) of the first atom, and for the optimal value P22 = 1/2 of its initial population (from the information point of view), we have 1 --x < 1/4 and x 3/4. Oscillations in Ifa(t)12 lead to the interference between the two decaying Dicke components, so that

the maximum of the population n2 =
information becomes a non-zero value.

1--

x goes to the larger values, maximally up to n2 = 1, and the coherent

Functions n2(, 'yt) and I(Ão, 'yt), calculated with Eq. (26) are shown in Fig. 7. For the considered geometry, they serve as the universal measures for a system of two atoms independent of their frequency or dipole moments.
As can be seen from Fig. 7a, the population decreases rapidly versus time because of the decay of the short-lived Dicke component. Both the population and the coherent information (Fig. 7b) show strong oscillations at smaller interatomic distances p. At cp --+ 0 the long-lived Dicke state yields an essential population even at infinitely long times, but it does not yield any coherent information after the total decay of the other short-lived Dicke state.

6. CONCLUSIONS
In this paper, we have shown that the coherent information concept can be used effectively to quantify the interaction

between two real quantum systems, which in general case may have essentially different Hilbert spaces, and to elucidate the role of quantum coherence specific for the joint system. For a TLA in a resonant laser field, coherent information in the system does not increase as the intensity of the external field increases, unless the external field modifies the relaxation parameters.
77

Proc. SPIE Vol. 4429


(a)

(b)
1.0 0.8

.
0
ò2

0.6
0.4

oE

0.2

0.2

DistaCe (

0.6

0.5

Iistafl

W

Figure '7. Excited state population of the second atom (a) and the coherent information (b) in a system of two atoms interacting via the free space field versus time yt and the interatomic distance cp = woR/c (both are dimensionless). = 1/2. The input density matrix is diagonal with the ground state matrix element

As an example of information transmission between the subsystems of a whole system, the hydrogen atom was considered. The coherent information in the atom was shown to transfer from the forbidden atomic transition to the dipole active transition in an external electric field, due to coupling through Stark splitting. For two unitary coupled TLAs, the maximum value I = 1 qubit of the coherent information was shown to be achieved for a complete unitary entanglement of two TLAs and I = 0, for any kind of measuring procedure discussed in subsection 5.2. For the information exchange between a TLA and a free-space vacuum photon field via spontaneous emission, the coherent information was shown to reach a non-zero value at the threshold point of the decay exponent exp(--'yt) equal to 1/2, when the probability of finding no photon is equal to the population of the lower atomic state. At its maximum, the coherent information can reach the value of I = 1 qubit. For the information transfer between two atoms via vacuum field, when the atoms are located at a distance of the order of their transition wavelength, the coherent information was shown to be a non-zero value, only because of the coherent oscillations of the Dicke states, which originate from the dipole-to-dipole short distance electrostatic-like 1/R3 interaction. In contrast, the semiclassical information received from the quantum detectTon procedure results from the population correlations.38

ACKNOWLEDGMENTS
This work was partially supported by the programs "Fundamental Metrology" , "Physics of Quantum and Wave Phenomena" , and "Nanotechnology" of the Russian Ministry of Science and Technology. The help of C. M. Elliott in preparing the manuscript is much appreciated.

REFERENCES
1. C. P. Williams and S. H. Clearwater, Explorations in Quantum Computing (Telos/Springer-Verlag, New York,
1998).

2. C. E. Shannon, Bell System Tech. J. 2'T, 379 (1948).

3. C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press,
Urbana, 1949). 4. R. G. Gallagher, Information Theory and Reliable Communication (John Wiley and Sons, New York, 1968).

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Proc. SPIE Vol. 4429


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