Content uploaded by Alberto Barchielli
Author content
All content in this area was uploaded by Alberto Barchielli on Nov 08, 2012
Content may be subject to copyright.
arXiv:quant-ph/0401114v1 20 Jan 2004
INSTRUMENTAL PROCESSES, ENTROPIES, INFORMATION
IN QUANTUM CONTINUAL MEASUREMENTS
A. BARCHIELLI
Politecnico di Milano, Dipartimento di Matematica,
Piazza Leonardo da Vinci 32, I-20133 Milano, Italy.
E-mail: Alberto.Barchielli@polimi.it
G. LUPIERI
Universit`a degli Studi di Milano, Dipartimento di Fisica,
Via Celoria 16, I-20133 Milano, Italy.
E-mail: Giancarlo.Lupieri@mi.infn.it
Dedicated to Alexander S. Holevo on his 60th birthday
In this paper we will give a short presentation of the quantum L´evy-Khinchin
formula and of the formulation of quantum continual measurements based on
stochastic differential equations, matters which we had the pleasure to work on
in collaboration with Prof. Holevo. Then we will begin the study of various en-
tropies and relative entropies, which seem to be promising quantities for measuring
the information content of the continual measurement under consideration and for
analysing its asymptotic behaviour.
1 A quantum L´evy-Khinchin formula
The theory of measurements continuous in time in quantum mechanics
(quantum continual measurements) started with the description of counting
exp eriments
1
and of situations in which an observable is measured imprecisely,
but with continuity in time;
2
both formulations are based on the notions of
instrument
1
and of positive operator valued measure. Soon after we suc-
ceeded in unifying the two approaches,
3
Holevo
4
realized that some quantum
analogue of infinite divisibility was involved and thus started a search of a
quantum L´evy-Khinchin formula;
5,6,7,8
a review is given in refs.
9,10
, while
a different approach is presented in refs.
11
.
Let H be a complex separable Hilbert space, T (H) be the trace-class on
H and S(H) be the set of statistical opera tors. We denote by L(H
1
; H
2
) the
space of linear bounded operators from H
1
into H
2
and set L(H) = L(H; H).
ha, τi = Tr{aτ }, τ ∈ T (H) , a ∈ L(H); kτ k
1
= Tr
√
τ
∗
τ
.
An instrument is a map-valued σ-additive measure N on some measurable
space (Y, B); the maps ar e from T (H) into itself, linear, completely po sitive
and normalized in the sense that Tr{N(Y)[τ]} = Tr{τ}.
The formulation of continual measurements given by Holevo
9
is based on
analogies with the L´evy processes and it is les s general, but more fruitful,
than the one initiated by our group
2
and based on the generalized stochastic
1
processes. In order to simplify the presentation, we will only consider the case
of one- dimensional processes. Let Y be the space of all real functions on the
positive time axis starting from zero, continuous from the right and with left
limits, and let B
b
a
, 0 ≤ a ≤ b, be the σ-alge bra generated by the increments
y(t) − y(s), a ≤ s ≤ t ≤ b. A time homogeneous instrumental process with
independent increments (i-process) is a family {N
b
a
; 0 ≤ a ≤ b}, where N
b
a
is
an instrument on (Y, B
b
a
) such that N
b+t
a+t
(E
t
) = N
b
a
(E) for arbitrary b ≥ a,
t ∈ R
+
, E ∈ B
b
a
, where E
t
= {y : T
t
y ∈ E},
T
t
y
(s) = y(s + t), and such that
N
b
a
(E) ◦N
c
b
(F ) = N
c
a
(E ∩ F ), 0 ≤ a ≤ b ≤ c, E ∈ B
b
a
, F ∈ B
c
b
. (1)
Every i-process is determined by its finite-dimensional distributions, which
have the structure
N
t
p
t
0
y(·) : y(t
1
) − y(t
0
) ∈ B
1
, . . . , y(t
p
) − y(t
p−1
) ∈ B
p
= N
t
p
−t
p−1
(B
p
) ◦ ···◦N
t
1
−t
0
(B
1
), (2)
where 0 ≤ t
0
< t
1
< ··· < t
p
, B
1
, . . . , B
p
∈ B(R), and
N
t
(B) = N
a+t
a
y(·) : y(a + t) − y(a) ∈ B
(3)
is independent of a by the time homogeneity. The instrument N
t
completely
determines the i-process and it is completely characterized by its Fourier
transform (characteristic function)
R
R
e
iky
N
t
(dy); Eq. (1) and the continuity
assumption
lim
t↓0
kN
t
(U
0
) − 1lk = 0 , for every neighbourhood U
0
of 0, (4)
imply that this characteristic function is of the form exp{tK(k)}, K(k) ∈
L
T (H)
. The quantum L´evy-Khinchin formula is the complete character-
ization of the gener ator K .
8
The structure of K can be written in different
equivalent ways and here we give an expression
12
which is particularly con-
venient for reformulating the theory of the continual measurements in terms
of stochastic differential equations, as illustra ted in the next section.
The quantum L´evy-Khinchin formula for the generator K is: ∀τ ∈ T (H),
∀k ∈ R, ∀h, g ∈ H,
K(k)[τ] = L[τ ] + ikcτ −
1
2
r
2
k
2
τ + ikr (Rτ + τ R
∗
)
+
Z
R
∗
e
ikz
− 1
J[τ ](z) − ikzϕ
2
(z)τ
µ(dz) , (5)
2
where c ∈ R, r ∈ R, ϕ
2
(z) =
b
2
b
2
+ z
2
, b > 0,
L = L
0
+ L
1
+ L
2
, (6)
L
0
[τ] = −i[H, τ ] +
1
2
∞
X
j=1
L
j
τ, L
∗
j
+
L
j
, τL
∗
j
, (7)
L
1
[τ] =
1
2
([Rτ, R
∗
] + [R, τR
∗
]) , (8)
L
2
[τ] = −
1
2
J
∗
Jτ −
1
2
τJ
∗
J + Tr
L
2
ν
{Jτ J
∗
}, (9)
J [|hihg|] (z) =
∞
X
n=1
ν
dz × {n}
µ(dz)
|(Jh)(z, n) + hih(Jg)(z, n) + g|, (10)
R, H, L
j
∈ L(H ), H = H
∗
,
P
∞
j=1
L
∗
j
L
j
∈ L(H) (strong convergence), R
∗
=
R\{0}, ν is a σ-finite measure on R
∗
×N and µ(dz) =
P
∞
n=1
ν(dz ×{n}); we
assume that
Z
R
∗
ϕ
1
(z) µ(dz) ≡
∞
X
n=1
Z
R
∗
ϕ
1
(z) ν
dz × {n}
< +∞, (11)
with ϕ
1
(z) =
z
2
b
2
+ z
2
. Note that ϕ
1
(z) + ϕ
2
(z) = 1 and that µ is a L´evy
measure on R
∗
. Finally, J ∈ L
H; L
2
ν
(H)
, where L
2
ν
= L
2
(R
∗
× N, ν),
L
2
ν
(H) = L
2
(R
∗
× N, ν; H) ≃ L
2
ν
⊗ H. The fact that the operators H, R ,
L
j
, J ar e bounded is due to the assumption (4), which is therefore a strong
restriction from a physical point of view.
It is convenient to introduce also the characteristic functional of the whole
i-process as the solution of the equation: ∀a ∈ L(H), ∀τ ∈ T (H),
a, G
t
(k)[τ]
= ha, τi +
Z
t
0
a, K
k(s)
◦ G
s
(k)[τ]
ds , (12)
where k(t) is a real function, continuous from the left with right limits; let
us call it a test function. By taking k(t) = κ1l
[0,T )
(t), we get G
T
(k) =
exp{T K(κ)} and, similarly, by taking a more general step function for k we
get the Fourier transform of the finite-dimensional distributions (2), so that
G
t
completely characterizes the i-process.
The operators U(t) = exp{tL} = G
t
(0) = N
t
(R), t ≥ 0, form a completely
positive quantum dynamical semigr oup. We fix an initial state ̺ ∈ S(H) and
set η
t
= U(t)[̺]; η
t
is called the a priori state at time t beca use it represents
the state o f the system at time t, when no selection is done on the basis of the
results of the continual measure ment. The a priori states satisfy the master
equation
d
dt
η
t
= L[η
t
] , η
0
= ̺ . (13)
3
2 Stochastic differential equations
An alternative useful formulation of quantum continual measurements is based
on stochastic differential eq uations (SDE’s); it was initiated for the basic
cases by Belavkin
13
by using analogies w ith the classical filtering theory. The
general SDE’s corre sponding to the L´evy-Khinchin formula (5) were studied
in refs.
14
.
2.1 Output signal and reference probability
Let W be a one-dimensional standard continuous Wiener process and N (dz ×
dt) be a random Poisson measure on R
∗
×R
+
of intensity µ(dz)dt, independent
of W . The two process e s are realized in a complete standard probability
space (Ω, F, Q) with the filtration of σ-algebras {F
t
, t ≥ 0}, which is the
augmentation of the natural filtra tio n of W and N; we a ssume also F =
W
t≥0
F
t
. It is useful to introduce the compensated process
e
N(dz × dt) = N (dz × dt) − µ(dz)dt . (14)
In all the SDE’s such as E qs. (15), (17), (18), (19), (34), the presence of
integrals with respect either to the jump process N or to the compensated
processes
e
N or
˘
N (see (28)) is due to problems of convergence of the stochastic
integrals which arise when infinitely many small jumps are present (the case
R
R
∗
µ(dz) = +∞).
Now, by using W , N and all the ingredients entering the L´evy-Khinchin
formula (5), we are able to construct various random quantities which allow us
to ree xpress in a different form the i-process of the previous section. Firstly,
let us introduce the real process
Y (t) = ct + rW (t) +
Z
R
∗
×(0,t]
ϕ
1
(z)zN(dz ×ds) +
Z
R
∗
×(0,t]
ϕ
2
(z)z
e
N(dz ×ds) ,
(15)
which, under the reference probability Q, is a generic L´evy process; Y will
represent the output process of the continual measurement introduced in the
previous s e ction. In the following we shall need the quantity
Φ
t
(k) = exp
i
Z
t
0
k(s)dY (s)
(16)
and its stochastic differential
dΦ
t
(k) = Φ
t
(k)
Z
R
∗
e
ik(t)z
− 1 − ik(t)ϕ
2
(z)z
µ(dz) + ick(t)
−
1
2
r
2
k(t)
2
dt + irk(t)dW (t) +
Z
R
∗
e
ik(t)z
− 1
e
N(dz × dt)
. (17)
4
In Table 1 we summarize the rules of stochastic calculus fo r W and N , which
have been used in computing dΦ
t
(k) and which shall be used to compute all
the stochastic differentials in the rest of the paper.
Table 1. The Ito table and an example of application with only the jump part.
dt dW (t) N(dz × dt)
dt 0 0 0
dW (t) 0 dt 0
N(dz
′
× dt) 0 0 δ(z − z
′
) N(dz × dt)
f
X +
R
R
∗
C(z)N (dz × dt)
−f(X) =
R
R
∗
f
X + C(z)
− f (X)
N(dz ×dt)
2.2 A linear SDE and the instruments
Let us consider now the linear SDE for σ
t
∈ T (H), σ
t
≥ 0: ∀a ∈ L(H),
ha, σ
t
i = ha, ̺i +
Z
t
0
ha, L[σ
s
]ids +
Z
t
0
ha, Rσ
s
+ σ
s
R
∗
idW (s)
+
Z
R
∗
×(0,t]
ha, J[σ
s
](z) − σ
s
i
e
N(dz × ds). (18)
We call the σ
t
non normalized a posteriori states (nnap states); the reason
will be clarified in the following. The coe fficient of the jump term should
be wr itten as J[σ
s
−
](z) − σ
s
−
, with the following meaning: when there is a
jump of N , i.e. when N(dz × ds) = 1, the nnap state before the jump σ
s
−
is
transformed into the state after the jump σ
s
+
= J[σ
s
−
](z); however, we prefer
to simplify the notation and not to write the superscripts “minus”. Similar
considerations apply to all the other SDE’s.
By using Table 1 to differentiate Φ
t
(k)ha, σ
t
i, we get
d
Φ
t
(k)ha, σ
t
i
= Φ
t
(k)
a, K
k(t)
[σ
t
]
dt +
a, irk(t)σ
t
+ Rσ
t
+ σ
t
R
∗
dW (t) +
Z
R
∗
D
a, e
ik(t)z
J[σ
t
](z) − σ
t
E
e
N(dz × dt)
(19)
and by taking the expectation we see that the terms with dW and
e
N disa ppear
and that the resulting equation is the same as Eq. (12), which defines G.
Therefore, we have
ha, G
t
(k)[̺]i = E
Q
[Φ
t
(k)ha, σ
t
i] , (20)
an equation s howing that Y (t) and σ
t
completely determine the characteristic
functional of the continual measurement and, so, the whole i-pr ocess. In
5
particular, by taking k = 0 we obtain that the expectatio n va lue of the nnap
states gives the a priori s tates: E
Q
[ha, σ
t
i] = ha, η
t
i.
2.3 The physical probability and the a posteriori states
Let us now study the norm of the nnap states: kσ
t
k
1
= h1l, σ
t
i = Tr{σ
t
}. By
taking the trace of Eq. (18) we get
d kσ
t
k
1
= kσ
t
k
1
m(t)dW (t) +
Z
R
∗
I
t
(z) − 1
e
N(dz × dt)
, (21)
where
m(t) = hR + R
∗
, ρ
t
i, I
t
(z) = h1l, J[ρ
t
](z)i = hJ(z), ρ
t
i, (22)
hg|J(z)hi =
∞
X
n=1
ν
dz × {n}
µ(dz)
h(Jg)(z, n) + g|(Jh)(z, n) + hi, ∀g, h ∈ H,
(23)
ρ
t
=
(
kσ
t
k
−1
1
σ
t
if kσ
t
k
1
> 0
̺ otherwise
(24)
The op e rators ρ
t
belong to S(H) and will be called a posteriori states, as
explained below. Note the common initial state: η
0
= σ
0
= ρ
0
= ̺. It is
possible to show that kσ
t
(ω)k
1
is a martingale and that it can be us e d as a
local density with respect to Q to define a new probability P
̺
on (Ω, F), the
physical probability, by
P
̺
(dω)
F
t
= kσ
t
(ω)k
1
Q(dω)
F
t
, o r
P
̺
(dω)
Q(dω)
F
t
= kσ
t
(ω)k
1
. (25)
By taking a = 1l in (20) and by using the new physical pr obability we can
write
h1l, G
t
(k)[̺]i = E
P
̺
[Φ
t
(k)] . (26)
This equation shows that the Fourier transform of all the probabilities involved
in the continual measurement is given by the characteristic functional of the
process Y (t) under the probability P
̺
. It is this fact which substantiates the
interpretation of P
̺
as the physical probability and of Y (t) as the output
process.
It is pos sible to prove that under the physical probability P
̺
˘
W (t) = W (t) −
Z
t
0
m(s) ds (27)
6
is a standard Wiener process and N(dz × dt) is a point process of stochastic
intensity I
t
(z)µ(dz)dt; we set
˘
N(dz × dt) = N (dz × dt) − I
t
(z)µ(dz)dt . (28)
The typical properties of the trajectories of the output signal can be visualized
in a particularly s imple manner when
R
R
∗
ϕ
2
(z)zµ(dz) < +∞; in this case we
can write
Y (t) = Y
cbv
(t) + r
˘
W (t) +
Z
R
∗
×(0,t]
zN (dz × ds) (29)
where
R
R
∗
×(0,t]
zN (dz × ds) is the jump part, with jumps of amplitude z and
intensity I
s
(z)µ(dz)ds, r
˘
W (t) is a c ontinuous pa rt proportional to a Wiener
process and
Y
cbv
(t) = t
c −
Z
R
∗
ϕ
2
(z)zµ(dz)
+
Z
t
0
m(s)ds (30)
is a continuous part with bounded variation.
By rewriting Eq. (20) with the new probability, we have
ha, G
t
(k)[̺]i = E
P
̺
[Φ
t
(k)ha, ̺
t
i] . (31)
Because G is the Fourier transform of all the finite-dimensional distributions
and these distributions determine the who le i-process, this last equation is
equivalent to: ∀a ∈ L(H), ∀t ≥ 0, ∀E ∈ B
t
0
,
a, N
t
0
(E)[̺]
=
Z
{ω∈Ω:Y (·;ω)∈E}
ha, ρ
t
(ω)iP
̺
(dω) . (32)
This equation shows that ρ
t
is the state conditioned on the trajectory of the
output observed up to time t and ρ
t
has indeed the meaning of a posteriori
state at time t: the state we must attribute to the system under a selective
measurement up to t. By ta king k = 0 into Eq. (31) or E = Y into Eq. (32),
we get
ha, η
t
i = E
P
̺
[ha, ρ
t
i] , (33)
i.e. the a pos teriori states ρ
t
(ω) with the physical probability P
̺
(dω) realize
a demixture of the a priori state η
t
. Finally, by differentiating the definition
(24) of the a posteriori states, we get the SDE
ha, ρ
t
i = ha, ̺i +
Z
t
0
ha, Rρ
s
+ ρ
s
R
∗
− m(s)ρ
s
id
˘
W (s)
+
Z
R
∗
×(0,t]
ha, j(ρ
s
; z) − ρ
s
i
˘
N(dz × ds) +
Z
t
0
ha, L[ρ
s
]ids , (34)
j(τ; z) = (Tr {J[τ](z)})
−1
J[τ ](z) , τ ∈ S(H) ; (35)
Eq. (34) holds under the physical probability P
̺
.
7
3 Entropies and information
3.1 Quantum and classical entropies
In quantum measurement theory both quantum states and classica l probabil-
ities are involved and, so, quantum and classical entropies are relevant.
For x, y ∈ T (H), x ≥ 0, y ≥ 0, we introduce the functionals, with values
in [0, +∞],
15
S
q
(x) = −Tr{x ln x}, S
q
(x|y) = Tr{x ln x − x ln y}; (36)
if x, y ∈ S(H), S
q
(x) is the von Neumann entropy and S
q
(x|y) is the quantum
relative entropy. The von Neumann entropy can be infinite only if the Hilbert
space is infinite dimensional and it is zero only on the pur e states, while the
quantum relative entropy can be infinite even when the Hilbert space is finite
dimensional and it is zero only if the two states are eq ual.
A first q uantum e ntropy of interest is the a priori entropy S
q
η
t
, which
at time zero reduces to the entropy o f the initial state S
q
η
0
= S
q
(̺).
On the other hand, a classical entropy is the relative entropy (or Kullback-
Leibler informational divergence) of the physical probability P
̺
with respect
to the reference probability measure Q :
I
t
(P
̺
|Q) = E
P
̺
ln
P
̺
(dω)
Q(dω)
F
t
= E
Q
kσ
t
k
1
ln kσ
t
k
1
, (37)
Let us note that I
t
(P
̺
|Q) ≥ 0, I
0
(P
̺
|Q) = 0 and that I
t
(P
̺
|Q) is non de-
creasing, as one sees by computing its time derivative:
d
dt
I
t
(P
̺
|Q) = E
P
̺
1
2
m(t)
2
+
Z
R
∗
1−I
t
(z) + I
t
(z) ln I
t
(z)
µ(dz)
≥ 0 . (38)
If we consider two different initial states ̺
α
and ̺, with supp ρ
α
⊆ supp ρ,
we can introduce the quantum relative entropy S
q
(η
α
t
|η
t
) and the classical
P
̺
α
|P
̺
-relative entropy I
t
(P
̺
α
|P
̺
),
I
t
(P
̺
α
|P
̺
) = E
P
̺
α
ln
P
̺
α
(dω)
P
̺
(dω)
F
t
= E
Q
kσ
α
t
k
1
ln
kσ
α
t
k
1
kσ
t
k
1
. (39)
Here a nd in the following P
̺
α
, σ
α
t
, ρ
α
t
, η
α
t
, m
α
(t), I
α
t
(z) are defined by starting
from ̺
α
as P
̺
, σ
t
, ρ
t
, η
t
, m(t), I
t
(z) are defined by starting from ̺.
Let us stress the different behaviour in time of the two relative entropies;
this discussion will b e relevant later on. The quantum one starts from S
q
(̺
α
|̺)
at time zero and it is non increasing
S
q
η
α
t
η
t
= S
q
U(t − s) [η
α
s
]
U(t − s) [η
s
]
≤ S
q
η
α
s
η
s
, t > s ; (40)
this statement follows fr om the Uhlmann monotonicity theorem (ref.
15
Theor.
5.3). The clas sical relative entropy starts from zero at time z e ro and it is non
8
decreasing, as one sees by computing its time derivative
d
dt
I
t
(P
̺
α
|P
̺
) = E
P
̺
α
1
2
m
α
(t) − m(t)
2
+
Z
R
∗
1 −
I
α
t
(z)
I
t
(z)
+
I
α
t
(z)
I
t
(z)
ln
I
α
t
(z)
I
t
(z)
I
t
(z)µ(dz)
≥ 0 . (41)
However, both relative entropies have the same bounds:
0 ≤ S
q
(η
α
t
|η
t
) ≤ S
q
(̺
α
|̺) , 0 ≤ I
t
(P
α
̺
|P
̺
) ≤ S
q
(̺
α
|̺) . (42)
The first statement is clear [se e Eq. (40)]. The second one too is a consequence
of the Uhlmann monotonicity theorem, as can be see n by considering the
“observation channel” Λ : L(H) → L
∞
(Ω, F
t
, Q) with predual Λ
∗
: ̺ → P
̺
∈
L
1
(Ω, F
t
, Q) (in ref.
15
see p. 138, Theor. 5.3 and the dis c ussions at pgs. 9
and 151).
3.2 Entropies and purity of the states
When one is studying the properties o f an instrument, a relevant question is
whether the a posteriori states are pure or not and, if not pure, how to measure
their “degree of mixing”. Ozawa
18
called quasi-complete an instrument which
sends e very initial pure state into pure a posteriori states. A first mea sure of
purity of the a posteriori states is the a posteriori entropy E
P
̺
S
q
ρ
t
, which
takes the initial value E
P
̺
S
q
ρ
0
= S
q
(̺). A related quantity, simpler to
study, is the a posteriori purity (or linear entropy)
p(t) = E
P
̺
[Tr {ρ
t
(1l − ρ
t
)}] , p(0) = Tr {̺ (1l − ̺)}. (43)
The a poster iori entropy and purity vanish if and only if the a poster iori s tates
are almost surely pure and one has p(t) ≤ E
P
̺
S
q
ρ
t
.
By the rules of stochastic calculus (Table 1) we get the time derivative of
the purity
d
dt
p(t) = ˙p
1
(t) − ˙p
2
(t) − ˙p
3
(t) , (44)
˙p
1
(t) = 2
∞
X
j=1
E
P
̺
h
Tr
n
ρ
t
L
∗
j
L
j
ρ
t
− ρ
1/2
t
L
∗
j
ρ
t
L
j
ρ
1/2
t
oi
, (45)
˙p
2
(t) = E
P
̺
h
Tr
n
ρ
1/2
t
(R + R
∗
− m(t)) ρ
t
(R + R
∗
− m(t)) ρ
1/2
t
oi
≥ 0, (46)
˙p
3
(t) =
Z
R
∗
E
P
̺
Tr
I
t
(z) j(ρ
t
; z)
2
− 2J[ρ
2
t
](z) + I
t
(z)ρ
2
t
µ(dz)
=
Z
R
∗
E
P
̺
h
I
t
(z)
−1
Tr
n
ρ
1/2
t
J(z)ρ
1/2
t
− I
t
(z)ρ
t
2
+ J[ρ
t
](z)
2
−
ρ
1/2
t
J(z)ρ
1/2
t
2
oi
µ(dz) .
(47)
9
Then, one can check the following points.
(a) If ρ
t
is almost surely a pure state, then one has ˙p
1
(t) ≥ 0, ˙p
2
(t) = 0,
˙p
3
(t) = −
R
R
∗
E
P
̺
Tr
j(ρ
t
; z) − j(ρ
t
; z)
2
I
t
(z)
µ(dz) ≤ 0.
(b) The a posteriori states are almost surely pure for all pure initial states (i.e.
the measurement is quasi complete) if and only if the following conditions
hold:
C1. L
0
[·] = −i[H, ·];
C2. j(τ ; z) is a pure s tate (µ-almost everywhere) for all pure states τ or,
equivalently, in (10)
Jh
(z, n) =
Jh
(z), ∀h ∈ H.
(c) Under the s ame conditions o ne has ˙p
1
(t) = 0, ˙p
3
(t) ≥ 0 for any initial
state; as ˙p
2
(t) ≥ 0 always, the purity decreases monotonically.
The prop e rties of the purity have a lso been used
17
to find sufficient con-
ditions (among which there is the quasi-completeness property) so that the
long time limit of the a posterio ri purity will vanish for e very initial state; note
that in a finite dimensional Hilbert space this is equivalent to the vanishing
of the limit of the a posteriori entropy.
Differentiating the a posterio ri entropy demands long computations in-
volving an integral representation of the logarithm (ref.
15
p. 51) and the
rules of stochastic calculus. We get
d
dt
E
P
̺
S
q
ρ
t
= E
P
̺
[D
1
(ρ
t
) − D
2
(ρ
t
) − D
3
(ρ
t
)] , (48)
where, ∀τ ∈ S(H),
D
1
(τ) =
X
j
Tr
L
∗
j
L
j
τ − L
j
τL
∗
j
ln τ
, (49)
D
2
(τ) =
Z
+∞
0
du Tr
uτ
(u + τ )
2
(R + R
∗
− Tr {(R + R
∗
) τ})
τ
u + τ
× (R + R
∗
− Tr {(R + R
∗
) τ}) +
τ
(u + τ )
2
[τ, R]
τ
u + τ
R
∗
−
τ
u + τ
, R
τ
u + τ
R
∗
,
(50)
D
3
(τ) =
Z
R
∗
µ(dz)
Tr {−J[τ ln τ](z)} − Tr {J[τ](z)}S
q
j(τ; z)
. (51)
From the time derivative of the a posterio ri entropy we have the following
results.
(i) When τ is a pure state, D
1
(τ) = 0 if
P
j
Tr
τL
∗
j
(1l − τ ) L
j
= 0 and
D
1
(τ) = +∞ otherwise.
10
(ii) D
2
(τ) ≥ 0 for any state τ. When τ is a pure state D
2
(τ) = 0.
(iii) Under condition C2 one has D
3
(τ) ≥ 0 for any state τ.
(iv) When τ is a pure state, D
3
(τ) ≤ 0 in general and D
3
(τ) = 0 if condition
C2 holds.
Statements (i) and (iv) are easy to verify, while the pr oof of (iii) requires
arguments introduced in Section 3.3 and will be given there. In order to
study D
2
(τ) we need the spectral decomposition of τ : τ =
P
k
λ
k
P
k
, with
k 6= r ⇒ λ
k
6= λ
r
; by inserting this decomposition into Eq. (50) we get
D
2
(τ) =
1
2
X
k
λ
k
Tr
n
[P
k
(R + R
∗
− Tr {(R + R
∗
) τ}) P
k
]
2
o
+
1
2
X
k6=r
Tr {P
k
(R + R
∗
)P
r
(R + R
∗
)P
k
}
λ
k
λ
r
λ
k
− λ
r
ln
λ
k
λ
r
, (52)
which implies statement (ii).
3.3 Mutual entropies and amount of information
A basic concept in classical information theo ry is the mutual entropy (informa -
tion). For two nonindependent random variables it is the relative entropy of
their joint probability distribution with respect to the product of the marginal
distributions and it is a measure of how much information the two random
variables have in common. The idea of mutual entropy can be introduced also
in a quantum context, w hen tensor product structures are involved. Ohya used
the quantum mutual entropy in order to describe the amount of information
correctly transmitted through a quantum channel Λ
∗
from an input state ̺
to the output sta te Λ
∗
̺. T he star ting point is the definition of a “compound
state” which describes the correlation of ̺ and Λ
∗
̺; it depends on how one
decomposes the input state ̺ in elementary events (orthogonal pure states).
The mutual entropy of the state ̺ and the channel Λ
∗
is then defined as the
supremum over all such decompositions of the relative entropy of the com-
pound state with respect to the product state ̺ ⊗ Λ
∗
̺ (ref.
15
pp. 33–34,
139).
We want to generalize these ideas to our context, where we have not only
a quantum channel U(t), but also a classical output with probability law P
̺
;
let us note that σ
t
contains the a posteriori states and the probability law
and that it can be identified with a sta te on L(H) ⊗ L
∞
(Ω, F
t
, Q). Firstly,
we define a co mpound state Σ
t
describing the cor relation between the initial
state ̺ and the nnap sta te σ
t
. Let ̺ =
P
α
w
α
̺
α
be a decomposition of the
initial state into orthogonal pure states (an extremal Shatten decomposition);
if ̺ has degenerate eigenvalues, this decomposition is not unique. With the
11
notations of Section 3.1 we have
σ
t
=
X
α
w
α
σ
α
t
, ρ
t
=
X
α
w
α
kσ
α
t
k
1
kσ
t
k
1
ρ
α
t
, η
t
=
X
α
w
α
η
α
t
,
P
̺
=
X
α
w
α
P
̺
α
,
X
α
w
α
ρ
α
t
(ω)P
̺
α
(dω)
F
t
= ρ
t
(ω)P
̺
(dω)
F
t
.
(53)
The compound state Σ
t
will be a state on the von Neumann algebra A =
L(H) ⊗ L(H) ⊗ L
∞
(Ω, F
t
, Q) ≡ M
1
⊗ M
2
⊗ M
3
; a normal state Σ on A is
represented by a non neg ative random trac e-class operator
b
Σ on H ⊗ H such
that
R
Ω
Tr
H⊗H
n
b
Σ(ω)
o
Q(dω) = 1: Σ(A) =
R
Ω
Tr
H⊗H
n
b
Σ(ω)A(ω)
o
Q(dω),
A ∈ A. The relative entropy of the state Σ with respect to another state Π
with representative
b
Π is given by
S(Σ|Π) =
Z
Ω
Tr
H⊗H
n
b
Σ(ω)
ln
b
Σ(ω) − ln
b
Π(ω)
o
Q(dω) ; (54)
this formula is consistent with the general Araki-Uhlmann definition of relative
entropy in a von Neumann algebra (ref.
15
Chapt. 5 ).
We introduce the compound state Σ
t
on A by giving its representative
P
α
w
α
̺
α
⊗ σ
α
t
and we consider the different possible product states which
can be constructed with its marginal: Π
t
= Σ
t
M
1
⊗ Σ
t
M
2
⊗ Σ
t
M
1
with
representative kσ
t
k
1
̺ ⊗ η
t
, Π
1
t
= Σ
t
M
1
⊗ Σ
t
M
2
⊗M
3
with representative
̺ ⊗ σ
t
, Π
2
t
= Σ
t
M
2
⊗ Σ
t
M
1
⊗M
3
with representative
P
α
w
α
kσ
α
t
k
1
̺
α
⊗ η
t
,
Π
3
t
= Σ
t
M
1
⊗M
2
⊗ Σ
t
M
3
with representative kσ
t
k
1
P
α
w
α
̺
α
⊗ η
α
t
. The
different mutual entropies, i.e. the relative entropies of Σ
t
with res pect to
the different product states, are the object of interest. We can call S(Σ
t
|Π
t
)
the mutual input/output entropy; this is a new informational quantity, which
could be extended also to generic measurements represented by instruments.
First of all, from Corollary 5.20 of ref.
15
, we obtain the chain rule
S(Σ
t
|Π
t
) = S(Σ
t
|Π
i
t
) + S(Π
i
t
|Π
t
) , i = 1, 2, 3 . (55)
Then, with some computations, we obtain the following re lations:
S(Π
1
t
|Π
t
) = E
P
̺
[S
q
(ρ
t
|η
t
)] = S
q
(η
t
) − E
P
̺
[S
q
(ρ
t
)] , (56)
S(Π
2
t
|Π
t
) =
X
α
w
α
I
t
(P
̺
α
|P
̺
) =
X
α
w
α
I
t
(P
̺
α
|Q) − I
t
(P
̺
|Q) , (57)
S(Π
3
t
|Π
t
) =
X
α
w
α
S
q
(η
α
t
|η
t
) = S
q
(η
t
) −
X
α
w
α
S
q
(η
α
t
) ; (58)
12
S(Σ
t
|Π
1
t
) = S(Π
2
t
|Π
t
) +
X
α
w
α
E
P
̺
α
[S
q
(ρ
α
t
|ρ
t
)]
= S(Π
2
t
|Π
t
) + E
P
̺
[S
q
(ρ
t
)] −
X
α
w
α
E
P
̺
α
[S
q
(ρ
α
t
)] ,
(59)
S(Σ
t
|Π
2
t
) =
X
α
w
α
E
P
̺
α
[S
q
(ρ
α
t
|η
t
)] = S
q
(η
t
) −
X
α
w
α
E
P
̺
α
[S
q
(ρ
α
t
)] , (60)
S(Σ
t
|Π
3
t
) = S(Π
2
t
|Π
t
) +
X
α
w
α
E
P
̺
α
[S
q
(ρ
α
t
|η
α
t
)]
= S(Π
2
t
|Π
t
) +
X
α
w
α
S
q
(η
α
t
) −
X
α
w
α
E
P
̺
α
[S
q
(ρ
α
t
)] ;
(61)
S(Σ
t
|Π
t
) = S(Π
2
t
|Π
t
) + S
q
(η
t
) −
X
α
w
α
E
P
̺
α
[S
q
(ρ
α
t
)] . (62)
The initial values are
S(Σ
0
|Π
0
) = S(Σ
0
|Π
1
0
) = S(Σ
0
|Π
2
0
) = S(Π
3
0
|Π
0
) = S
q
(̺) ,
S(Σ
0
|Π
3
0
) = S(Π
1
0
|Π
0
) = S(Π
2
0
|Π
0
) = 0 .
(63)
The quantity S(Π
1
t
|Π
t
) = E
P
̺
[S
q
(ρ
t
|η
t
)] is the a posteriori relative en-
tropy;
16
because Eq. (33) can be interpreted by saying that {P
̺
(dω), ρ
t
(ω)}
is a demixture of the a pr iori state η
t
, such a relative entropy is a measure of
how much such a demixture is fine. Let us observe tha t, for s ≤ t,
E
P
̺
[S
q
(ρ
t
|η
t
)] = E
P
̺
[S
q
(ρ
t
|U(t − s)[ρ
s
])] + E
P
̺
[S
q
(U(t − s )[ρ
s
]|η
t
)] . (64)
It follows that the variation in time of the a posteriori entropy is the sum of
two competing contributions of opposite sign:
∆ E
P
̺
[S
q
(ρ
t
|η
t
)] = E
P
̺
[S
q
(ρ
t+∆t
|U(∆t)[ρ
t
])]
+
E
P
̺
[S
q
(U(∆t)[ρ
t
]|U(∆t)[η
t
])] − E
P
̺
[S
q
(ρ
t
|η
t
)]
. (65)
The first term is clearly po sitive and represe nts an informa tion gain due to
the process of demixture induced by the measurement. The second term is
negative, once again as a cons equence of the Uhlmann monotonicity theorem,
and represents an informatio n loss due to the partial lack of memory of the
initial state induced by the dissipative part of the dynamics.
The quantity S(Π
2
t
|Π
t
) =
P
α
w
α
I
t
(P
̺
α
|P
̺
) has been introduced by
Ozawa
18
for a generic instr ument under the name of classical amount of
information. By the discussion in Section 3.1, eqs. (41) and (42), one obtains
that this quantity is non decreasing and bo unded:
0 ≤ S(Π
2
t
|Π
t
) =
X
α
w
α
I
t
(P
̺
α
|P
̺
) ≤
X
α
w
α
S
q
(̺
α
|̺) = S
q
(̺) . (66)
13
The supremum over all extremal Shatten decompositions of S(Π
3
t
|Π
t
) =
P
α
w
α
S
q
(η
α
t
|η
t
) is Ohya’s “mutual entropy of the input state ̺ and the chan-
nel U(t)”; by (40) S(Π
3
t
|Π
t
) is non increasing and by Theo r. 1.19 of ref.
15
it
is bounded by
0 ≤ S(Π
3
t
|Π
t
) =
X
α
w
α
S
q
(η
α
t
|η
t
) ≤ min { S
q
(̺), S
q
(η
t
)} . (67)
For general instruments Ozawa
18
introduced an entropy defect, which
he called the amount of information; it measures how much the a posteriori
states are purer than the initial state (or less pure, when this quantity is
negative). In the case of continual measurements it is defined by
16
I
t
(̺) = S
q
(̺) − E
P
̺
S
q
ρ
t
. (68)
If an equilibrium state exists, η
eq
∈ S(H) and L[η
eq
] = 0, by (56) we have
S
q
(η
eq
) ≥ I
t
(η
eq
) = E
P
η
eq
[S
q
(ρ
t
|η
eq
)] ≥ 0. For a quasi-complete continual
measurement one has
S
q
(̺) ≥ I
t
(̺) ≥ S(Π
2
t
|Π
t
) ≥ 0 , I
t
(̺) ≥ I
s
(̺) , t ≥ s . (69)
The first statement was proved by Oz awa
18
for a generic quasi-complete in-
strument, while the second one fo llows from the first one by using conditional
exp ectations.
16
We have I
t
(̺) − I
s
(̺) = E
P
̺
S
q
(ρ
s
) − E
P
̺
[S
q
(ρ
t
)|F
s
]
; but
S
q
(ρ
s
)−E
P
ρ
[S
q
(ρ
t
)|F
s
] is the amount of information at time t when the initial
time is s and the initial state is ρ
s
and, so, it is non-negative for a quasi-
complete measurement. From the monotonicity of I
t
(̺) one obtains that the
time derivative of E
P
̺
S
q
ρ
t
is negative and this ho lds in particular a t time
zero for any choice of the initial state and also for R = 0. T his proves the
statement (iii) of Section 3.2.
Acknowledgments
Work supported in part by the European Community’s Human Potential
Programme under contract HPRN-CT-2002-00279, QP-Applications, and
by Istituto Nazionale di Fisica Nucleare, Sezione di Milano.
References
1. E.B. Davies, Quant um Theory of Open Systems (Academic Press, Lon-
don, 1976).
2. A. Barchielli, L. Lanz, G.M. Prosperi, Nuovo Cimento 72B, 79–121
(1982); Found. Phys. 13, 779–8 12 (1983).
3. A. Barchielli, G. Lupieri, J. Math. Phys. 26, 2222–22 30 (1985).
4. A.S. Ho le vo, Theor. Probab. Appl. 31, 493–497 (1986); Theor. Probab.
Appl. 32, 131–136 (1987); in Proceedings of the International Congress
of Mathematicians 1986, 1011–1020 (1987).
14
5. A.S. Holevo, in Quantum Probability and Applications III, eds. L. Ac-
cardi, W. von Waldenfels, Lect. Notes Math. 1303, 128–147 (1987);
in Quantum Probability and Applications IV, eds. L. Accardi, W. von
Waldenfels, Lect. Notes Math. 1396, 229–255 (1989).
6. A. Barchielli, G. Lupieri, in Quantum Probability and Applications IV,
eds. L. Accardi, W. von Waldenfels, Le c t. Notes Math. 1396, 107–127
(1989); in Probability Theory and Mathematical Statistics Vol. I, 78–90,
eds. B. Grigelionis et al. (Mokslas, Vilnius, and VSP, Utrecht, 1990).
7. L.V. Denisov, A.S. Holevo, in Probability Theory and Mathematical
Statistics, Vol. I, 261–270, eds. B . Grigelionis et al. (Mokslas, Vilnius
and VSP, Utrecht, 1990).
8. A.S. Holevo, Theor. Probab. Appl. 38, 211–216 (1993).
9. A.S. Holevo, in L´evy Processes, 225-239, eds. O. E. Barndorff-Nielsen, T.
Mikosch, S. Resnick (Birkhauser, Boston, 2001).
10. A.S. Holevo, Statistical Structure of Quantum Theory, Lect. Notes Phys.
m 67 (Springer, Berlin, 2001).
11. A. Barchielli, Probab. Theory Rel. Fields 82, 1–8 (1989); A. Barchielli,
G. Lupieri, Probab. Theory Rel. Fields 88, 1 67–194 (1991); A. Barchielli,
A.S. Holevo, G. Lupieri, J. Theor. Probab. 6, 231–265 (19 93).
12. A. Barchielli, A.M. Paganoni, Nagoya Math. J. 141, 29 –43 (199 6).
13. V.P. Belavkin, in A. Blaqui`ere (Ed.), Modelling and Control of Systems,
Lecture Notes in Co ntrol and Information Sciences 121 (Springer, Berlin,
1988) pp. 245–265; Phys. Lett. A 140 (1989) 355–358; J. Phys. A: Math.
Gen. 22 (1989) L1109–L1114.
14. A. Barchielli, A.S. Holevo, Stoch. Process. Appl. 58, 29 3–317 (1995); A.
Barchielli, A.M. Paganoni, F. Zucca, Stoch. Process. Appl. 73, 69–86
(1998).
15. M. Ohya, D. Petz, Quantum Entropy and Its Use (Springe r, Ber lin, 1993).
16. A. Barchielli, in Quantum Communication, Computing, and Measure-
ment 3, eds. P. Tombesi, O. Hirota, 49 –57 (Kluwer, New York, 2001).
17. A. Barchielli, A.M. Paganoni, Infinite Dimensional Anal. Quantum
Probab. Rel. Topics 6, 223–243 (2003).
18. M. Ozawa, J. Math. Phys. 27, 759–763 (1986).
15
