Download ON THE GENERAL FORM OF QUANTUM STOCHASTIC

ON THE GENERAL FORM OF QUANTUM STOCHASTIC
EVOLUTION EQUATION.
V. P. BELAVKIN.
Abstract. A characterisation of the quantum stochastic bounded generators
of irreversible quantum state evolutions is given. This suggests the general
form of quantum stochastic evolution equation with respect to the Poisson
(jumps), Wiener (di¤usion) or general Quantum Noise. The corresponding
irreversible Heisenberg evolution in terms of stochastic completely positive
(CP) cocycles is also characterized and the general form of the stochastic
completely dissipative (CD) operator equation is discovered.
1. Quantum Stochastic Filtering Equations
The quantum …ltering theory, which was outlined in [1, 2] and developed then
since [3], provides the derivations for new types of irreversible stochastic equations
for quantum states, giving the dynamical solution for the well-known quantum measurement problem. Some particular types of such equations have been considered
recently in the phenomenological theories of quantum permanent reduction [4, 5],
continuous measurement collapse [6, 7], spontaneous jumps [8, 9], di¤usions and
localizations [10, 11]. The main feature of such dynamics is that the reduced irreversible evolution can be described in terms of a linear dissipative stochastic wave
equation, the solution to which is normalized only in the mean square sense.
The simplest dynamics of this kind is described by the continuous …ltering wave
propagators Vt (!), de…ned on the space of all Brownian trajectories as an adapted
operator-valued stochastic process in the system Hilbert space H, satisfying the
stochastic di¤usion equation
(1.1)
dVt + KVt dt = LVt dQ;
V0 = I
in the Itô sense, which was derived from a unitary evolution in [13]. Here Q (t; !)
is the standard Wiener process, which is described by the independent increments
dQ (t) = Q (t + dt) Q (t), having the zero mean values hdQi = 0 and the multiplication property (dQ)2 = dt, K is an accretive operator, K + K y Ly L, and L
is a linear operator D ! H. Using the Itô formula
(1.2)
d Vty Vt = dVty Vt + Vty dVt + dVty dVt ;
Date : July 20, 1995.
1991 Mathematics Subject Classi…cation. Quantum Stochastics.
Key words and phrases. Quantum Jumps, State Di¤usion, Spontaneous Localization, Quantum Filtering, Stochastic Equations.
Published in: Stochastic Analysis and Applications 91–106, World Scienti…c, Singapore, 1996.
1
2
V. P. BELAVKIN.
and averaging h i over the trajectories of Q, one obtains dhVty Vt i 0 as a consequence of Ly L
K + K y . Note that the process Vt is necessarily unitary if the
…ltering condition K y + K = Ly L holds, and if Ly = L in the bounded case.
Another type of the …ltering wave propagator Vt (!) : 0 2 H 7! t (!) in H is
given by the stochastic jump equation
(1.3)
dVt + KVt dt = LVt dP;
V0 = I;
derived in [12] by the conditioning with respect to the spontaneous stationary reductions at the random time instants ! = ft1 ; t2 ; :::g. Here L = J I is the jump
operator, corresponding to the stationary discontinuous evolutions J : t 7! t+ at
t 2 !, and P (t; !) is the standard Poisson process, counting the number j! \ [0; t)j
compensated by its mean value t. It is described as the process with independent
increments dP (t) = P (t + dt) P (t), having the values f0; 1g at dt ! 0, with zero
2
mean hdPi = 0, and the multiplication property (dP) = dP + dt. Using the Itô
formula (1.2) with dVty dVt = Vty Ly LVt (dP+dt), one can obtain
d Vty Vt = Vty Ly L
K
K y Vt dt + Vty Ly + L + Ly L Vt dP:
Averaging h i over the trajectories of P, one can easily …nd that dhVty Vt i 0 under
the sub-…ltering condition Ly L K+K y . Such evolution is unitary if Ly L = K+K y
and if the jumps are isometric, J y J = I.
This proves in both cases that the stochastic wave function t (!) = Vt (!) 0
is not normalized for each !, but it is normalized in the mean square sense to the
probability hjj t jj2 i
jj 0 jj2 = 1 for the quantum system not to be demolished
during its observation up to the time t. If jj t jj2 = 1, then the positive stochastic
b or counting P
b output
function jj t (!) jj2 is the probability density of a di¤usive Q
process up to the given t with respect to the standard Wiener Q or Poisson P input
processes.
Using the Itô formula for t (B) = Vty BVt , one can obtain the stochastic equations
(1.4)
d
t
(B) +
t
K y B + BK
Ly BL dt =
t
Ly B + BL dQ;
(1.5)
d
t
(B) +
t
K y B + BK
Ly BL dt =
t
J y BJ
B dP;
describing the stochastic evolution Yt = t (B) of a bounded system operator
y
B 2 L (H) as Yt (!) = Vt (!) BVt (!). The maps t : B 7! Yt are Hermitian
in the sense that Yty = Yt if B y = B, but in contrast to the usual Hamiltonian
y
dynamics, are not multiplicative in general, t B y C 6= t (B) t (C), even if they
are not averaged with respect to !. Moreover, they are usually not normalized,
Mt (!) := t (!; I) 6= I, although the stochastic positive operators Mt = Vty Vt
under the …ltering condition are usually normalized in the mean, hMt i = I, and
satisfy the martingale property t [Ms ] = Mt for all s > t, where t is the conditional expectation with respect to the history of the processes P or Q up to time
t.
Although the …ltering equations (1.3), (1.1) look very di¤erent, they can be
uni…ed in the form of quantum stochastic equation
(1.6)
dVt + KVt dt + K Vt d
= (J
I) Vt d + L+ Vt d
+
ON THE GENERAL FORM OF QUANTUM STOCHASTIC EVOLUTION EQUATION.
3
where + (t) is the creation process, corresponding to the annihilation
(t) on the
interval [0; t), and (t) is the number of quanta on this interval. These canonical
quantum stochastic processes, representing the quantum noise with respect to the
vacuum state j0i of the Fock space F over the single-quantum Hilbert space L2 (R+ )
of square-integrable functions of t 2 [0; 1), are formally given in [14] by the integrals
Z t
Z t
Z t
+ r
+
r
+
dr;
dr;
(t)
=
dr;
(t) =
(t) =
r
r
0
0
0
r
+
r
where
;
are the generalized quantum one-dimensional …elds in F, satisfying
the canonical commutation relations
r
+
s
;
= (s
r
r) I;
;
s
+
r ;
=0=
+
s
:
They can be de…ned by the independent increments with
(1.7)
h0jd
j0i = 0;
h0jd
+
j0i = 0;
h0jd j0i = 0
and the noncommutative multiplication table
(1.8)
d d =d ;
d
d =d
;
d d
+
=d
+
;
d
d
+
= dtI
with all other products being zero: d d
= d +d = d +d
= 0. The
standard Poisson process P as well as the Wiener process Q can be represented in
F by the linear combinations [16]
(1.9)
P (t) =
+
(t) + i
(t)
(t) ;
Q (t) =
+
(t) +
(t) ;
so the equation (1.6) corresponds to the stochastic di¤usion equation (1.1) if J = I,
L+ = L = K , and it corresponds to the stochastic jump equation (1.3) if
J = I + L, L+ = iL = K . The quantum stochastic equation for t (B) = Vty BVt
has the following general form
d
(1.10)
t
(B) +
+
t
t
K y B + BK
J y BL+
L BL+ dt =
K+ B +
t
L BJ
t
J y BJ
BK
d
B d
;
y
where L = Ly+ ; K+
= K , coinciding with either (1.4) or with (1.5) in the particular cases. The equation (1.10) is obtained from (1.6) by using the Itô formula (1.2)
L L+
with the multiplication table (1.8). The sub-…ltering condition K + K y
for the equation (1.6) de…nes in both cases the positive operator-valued process
Rt = t (I) as a sub-martingale with R0 = I, or a martingale in the case K + K y =
L L+ . In the particular case
J = S;
K
= L S;
L+ = SK+ ;
S y S = I;
corresponding to the Hudson–Evans ‡ow if S y = S 1 , the evolution is isometric,
and identity preserving, t (I) = I in the case of bounded K and L.
In the next sections we de…ne a multidimensional analog of the quantum stochastic equation (1.10) and will show that the suggested general structure of its
generator indeed follows just from the property of complete positivity of the map
t for all t > 0 and the normalization condition t (I) = Mt to a form-valued submartingale with respect to the natural …ltration of the quantum noise in the Fock
space F .
4
V. P. BELAVKIN.
2. The Generators of Quantum Filtering Cocycles.
The quantum …ltering dynamics over an operator algebra B B (H) is described
by a one parameter cocycles: = ( t )t>0 of linear completely positive stochastic
maps t (!) : B ! B. The cocycle condition
(2.1)
s
(!)
r
(! s ) =
r+s
(!) ;
8r; s > 0
means the stationarity, with respect to the shift ! s = f! (t + s)g of a given stochastic process ! = f! (t)g. Such maps are in general unbounded, but normalized,
0 with M0 = 1, or a
t (I) = Mt to an operator-valued martingale Mt = t [Ms ]
positive submartingale: Mt
[M
],
for
all
s
>
t,
.
t
s
Now we give a noncommutative generalization of the quantum stochastic CP
cocycles, which was suggested in [15] even for the nonlinear case. The stochastically di¤erentiable family with respect to a quantum stationary process, with
independent increments s (t) = (t + s)
(s) generated by a …nite dimensional
Itô algebra is described by the quantum stochastic equation
X
(2.2)
d t (Y ) = t
(Y ) d
:=
(Y )) d ;
Y 2B
t(
;
with the initial condition 0 (Y ) = Y , for all Y 2 B. Here
(t) with
2
+
m
f ; 1; :::; dg, 2 f+; 1; :::; dg are the standard time
(t) = tI, annihilation
(t),
m
m
creation +
n (t) = Nn (t) operator integrators with
n (t) and exchange-number
m; n 2 f1; :::; dg. The in…nitesimal increments d (t) = t (dt) are formally de…ned by the Hudson-Parthasarathy multiplication table [16] and the [ -property
[3],
(2.3)
d
d
d
=
;
[
= ;
where is the usual Kronecker delta restricted to the indices 2 f ; 1; :::; dg ;
2
[
y
f+; 1; :::; dg and
=
with respect to the re‡ection ( ) = +; (+) = of
: B ! B for the -cocycles t = t ,
the indices ( ; +) only. The linear maps
y y
, should obviously satisfy the [ -property [ = , where
where t (Y ) = t Y
y
[
=
,
(Y ) =
Y y . If the coe¢ cients b =
(Y ) are independent of
s
s
t, satis…es the cocycle property s
r = s+r , where t is the solution to (2.2)
with
(t) replaced by s (t). De…ne the (d + 2) (d + 2) matrix a = [a ] also
for = + and = , by
+
(Y ) = 0 =
(Y ) ;
8Y 2 B;
and then one can extend the summation in (2.2) so it is also over = +, and = .
By such an extension the multiplication table for d (a) = a d
can be written
as
(2.4)
y
d (a) d (a) = d
a[ a
in terms of the usual matrix product (ba) = b a and the involution a 7! a[ =
b; b[ = a can be obtained by the pseudo-Hermitian conjugation a[ = g a y g
respectively to the inde…nite Minkowski metric tensor g = [g ] and its inverse
g 1 = [g ], given by g =
I=g .
ON THE GENERAL FORM OF QUANTUM STOCHASTIC EVOLUTION EQUATION.
Let us prove that the "spatial" part
=(
6=+
6=
)
of
5
= + , called the quantum
6=+
stochastic germ for the representation : B 7! (B ) 6= , must be completely
stochastically dissipative for a CP cocycle in the following sense.
Theorem 1. Suppose that the quantum stochastic equation (2.2) with 0 (B) = B
= ;
has a CP solution t ; t > 0. Then the germ-map = ( + ) =+; is conditionally completely positive
X
X
(Bk ) k = 0 )
h k j Bky Bl l i 0
k
k;l
= ;
Here 2 H H ; H = H Cd , and = ( ) =+; is the degenerate representation
(B) = B + , written both with in the matrix form as
(2.5)
where
that
=
m
+;
=
By =
(2.6)
;
=
m
+;
n
y
(B) =
By =
n
(B) ;
m
n
n;
=
Proof. Let us denote by D the H-span
=
n
nP
m
n
B
0
0
0
+
m
n
;
y
m
n
(B) ;
f
f
f
f
m
n
with
(B) = B
By =
n
m
2 H; f 2 Cd
m
n
such
y
(B)
L2 (R+ )
o
N
of coherent (exponential) functions f ( ) = t2 f (t), given for each …nite subset = ft1 ; :::; tn g
R+ by tensor products f n1 ;:::;nN ( ) = f n1 (t1 ) :::f nN (tN ),
n
where f ; n = 1; :::; d are square-integrable complex functions on R+ and f = 0
for almost all f = (f n ). The co-isometric shift Ts intertwining As (t) with A (t) =
f )( ) =
f ( + s). The complete posTs As (t) Tsy is de…ned on D by Ts (
itivity of the quantum stochastic adapted map t into the D-forms h j t (B) i,
for ; 2 D can be obviously written as
E
XXD f
h
y
(2.7)
f
;
X
Z;
h
0;
t
Z
X
X;Z f;h
where
h j
t
(f ; B; h ) i =
f
t
(B)
h
e
R1
t
f (s)y h (s)ds
;
f
B
6 0 for a …nite sequence of Bk 2 B, and for a …nite sequence of fl = fl1 ; :::; fld .
=
If the D-form t (B) satis…es the stochastic equation (2.2), the H-form t (f ; B; h )
satis…es the di¤erential equation
(2.8)
d
dt
+
t
y
(f ; B; h ) = f (t) h (t)
d
X
f m (t)
t
f ;
m
+
(B) ; h
m=1
+
d
X
m;n=1
t
(f ; B; h ) +
+
d
X
hn (t)
n=1
f m (t) hn (t)
t
(f ;
m
n
(B) ; h ) ;
t
t
f ;
f ;
+
n
(B) ; h
(B) ; h
6
V. P. BELAVKIN.
Pd
y
where f (t) h (t) = n=1 f n (t) hn (t). The positive de…niteness, (2.7), ensures
the conditional positivity
E
XX f
XXD f
h
y
0
f
;
X
Z;
h
(2.9)
B B =0)
Z
X
B
f
X;Z f;h
of the form t (f ; B; h ) = 1t ( t (f ; B; h ) B) for each t > 0 and of the limit 0
at t # 0, coinciding with the quadratic form
(2.10)
X
X
X
d
n
n
=
am m
am m (B) +
t (f ; B; h )
n (B) c + (B) ;
n (B) c +
dt
t=0
m;n
m
n
where a = f (0) ; c = h (0), and the ’s are de…ned in (2.5). Hence the form
XX
XX
m
y
n
h Xj
X y Z Z i :=
h m
Xj n X Z
Z
X;Z
+
X X
X;Z
n
h
Xj
;
X;Z m;n
y
n
Z
X Z
n
X
+
m
m
Xj
h
m
y
X Z
Z
+h
Xj
y
X Z j
!
Zi
P
P
P
with = f f ;
= f f af , where af = f (0), is positive if B B B = 0.
The components and
of these vectors are independent because for any 2 H
and
= 1 ; :::; d 2 H Cd there exists such a function a 7! a on Cd with a
P
P a
…nite support, that a a = ;
a = , namely, a = 0 for all a 2 Cd
a
P
d
a
n
except a = 0, for which
=
and a = en , the n-th basis element
n=1
a
d
n
in C , for which
= . This proves the complete positivity of the matrix form
, with respect to the matrix representation de…ned in (2.5) on the ket-vectors
= ( ).
3. A Dilation Theorem for the Form-Generator.
The conditional positivity of the structural map with respect to the degenerate
representation written in the matrix form (2.6) obviously implies the positivity of
the dissipation form
X
XX
(3.1)
h X j (X; Z) Z i :=
h kj
(Bk ; Bl ) l i ;
X;Z
k;l
where
= = + and
corresponding to non-zero
is the dissipator matrix,
(X; Z) =
k
;
= Bk for any (…nite) sequence Bk 2 B, k = 1; 2; :::,
= ;
= ( ) =+;
B = B
B ; B 2 H; B 2 H . Here
y
X yZ
(X)
y
(Z)
(X)
y
(Z) + (X)
(I) (Z) ;
given by the elements
(3.2)
m
n
(X; Z)
=
m
n
X yZ + X yZ
m
n;
n
(X; Z)
=
n
X yZ
Xy
n
(Z) =
+
(X; Z)
=
+
X yZ
Xy
+
(Z)
n
+
+
y
(Z; X)
X y Z + X y DZ;
where D = + (I)
0 (D = 0 for the case of the martingale Mt ). This means
that the matrix-valued map
=[ m
n ], is completely positive, and as follows from
ON THE GENERAL FORM OF QUANTUM STOCHASTIC EVOLUTION EQUATION.
the next theorem, at least for the algebra B = B (H) the maps ,
following form
(3.3)
m
(B)
(B)
= 'm (B)
y
Km
B;
= ' (B)
y
K B
n
(B) = 'n (B)
BK;
m
,
n
7
have the
BKn
K + Ky
' (I)
6=+
where ' = (' ) 6= is a completely positive bounded map from B into the matrices
m
m
m
of operators with the elements 'm
n = n ; '+ = ' ; 'n = 'n ; '+ = ' : B ! B.
In order to make the formulation of the dilation theorem as concise as possible,
we need the notion of the [-representation of the algebra B in the operator algebra
A (E) of a pseudo-Hilbert space E = H H
H with respect to the inde…nite
metric
(3.4)
+
( j ) = 2Re
= ; ;+
+ 2
D
2
+k k +
for the triples = ( )
2 E, where
; + 2 H;
2 H ; H is a
2
pre-Hilbert space, and k kD = h j D i. The operators A 2 A (E) are given by
= ; ;+
3 3-block-matrices [A ] = ; ;+ , having the Pseudo-Hermitian adjoints jA[ =
(A j ), which are de…ned by the Hermitian adjoints Ay = A y as A[ = G 1 Ay G
respectively to the inde…nite metric tensor G = [G ] and its inverse G 1 = [G ],
given by
3
2
3
2
D 0 I
0 0 I
I
0 5
0 5;
G 1=4 0
(3.5)
G=4 0 I
I
0 0
I 0 D
with an arbitrary D, where I is the identity operator in H , being equal I =
m=1;:::;d
Cd = H .
[I m
n ]n=1;:::;d in the case of H = H
Theorem 2. The following are equivalent:
(i) The dissipation form
with + (I) = D, is
P (3.1), de…ned by the [-map
positive de…nite:
h
j
(X;
Z)
i
0.
X
Z
X;Z
(ii) There exists a pre-Hilbert space H , a unital y- representation j of B in
B (H ),
y
j B y B = j (B) j (B) ;
(3.6)
j (I) = I;
a (j; i)-derivation of B with i (B) = B,
y
k B y B = j (B) k (B) + k B y B;
(3.7)
y
having values in the operators H ! H , the adjoint map k (B) = k B y ,
with the property
k
B y B = B y k (B) + k
B y j (B)
of (i; j)-derivation in the operators H ! H, and a map l : B ! B having
the coboundary property
l B y B = B y l (B) + l B y B + k
(3.8)
B y k (B) ;
with the adjoint l (B) = l (B) + [D; B], such that
n
m
n
y
B y = k (B) Ln + B y Ln =
Lmy j
n
(B) = l (B) + DB,
y
(B) ;
and
(B) =
(B) Ln for some operators Ln : H ! H having the
adjoints Lny on H and Ln 2 B.
8
V. P. BELAVKIN.
(iii) There exists a pseudo-Hilbert space, E, a unital [-representation | : B ! A (E),
and a linear operator L : H H ! E such that
L[ | (B) L =
(3.9)
(B) ;
8B 2 B:
(iv) The structural map
= + is conditionally completely positive with
respect to the matrix representation in (2.5).
Proof. The implication (i))(ii) generalizes the Evans-Lewis Theorem [17], and its
proof is similar to the proof of the dilation theorem in [18]. Let H be the pre-Hilbert
y
space of Kolmogorov decomposition (X; Z) = k (X) k (Z). It is de…ned as the
quotient space H = K=I of the H-span K = ( B )B2B , where B 2 H H is
not equal zero only for a …nite number of B 2 B, with respect to the kernel
8
9
<
=
X
I = ( B )B2B 2 Kj
h X j (X; Z) Z i = 0
:
;
X;Z
y
of the positive-de…nite form (3.1). The operators k (B) : H ! H H are de…ned
on the classes
of ( X )X2B 2 K as the adjoint
D
E X
y
k (B)
j
=
h X j (X; B) i
X
to the bounded operators k (B) : H H ! H , mapping the pairs =
into the equivalence classes
(B) = k (B) + k (B)
of ( Z (B) )Z2B , where
(B)
=
1
if
B
=
Z,
otherwise
(B)
=
0.
Let
us
de…ne
a linear operator j (B)
Z
Z
on H by
X
X
j (B)
(k (Z) + k (Z) ) =
(k (BZ)
k (B) Z + k (BZ) ) :
Z
Z
Obviously j (XB) = j (X) j (B), j (I) = I because k (I) = 0 and as follows from the
y
de…nition of the dissipation form, j (B) = j B y for all B 2 B. Thus j is a unital
y-representation, k is a (j; i)-cocycle, and k (B) = j (B) L , where L = k (I).
Moreover, as
B y B + B y (I) B
ByB
ByB
By
(B)
y
= B y (B) +
B y B + k (B) k (B) ;
y
= k (B) k (B) ;
y
= k (B) k (B) =
the property (3.8) is ful…lled, L j (B) L =
ByB
y
y
(B) B;
(B) with L = k (I) = L y , and
y
B y = k (B) L + B y L =
y
(B) ;
where L = (I) ; L+ = (I) = L y .
The proof of the implication (ii))(iii) can be also obtained as in [18] by the
explicit construction of E as H H
H with the inde…nite metric tensor G =
[G ] given above for ; = ; ; +, and D = (I). The unital [-representation
= ; ;+
| = [| ] = ; ;+ of B on E :
[
| X y Z = | (X) | (Z) ;
[
with | (B) = G
(3.10)
1
| (I) = I
y
| (B) G = | B y is given by the components
| = j;
|+ = k;
| =k ;
|+ = l;
| = i = |+
+
ON THE GENERAL FORM OF QUANTUM STOCHASTIC EVOLUTION EQUATION.
and all other | = 0. The linear operator L : H
can be de…ned by the components (L ; L ),
L = 0;
I 0
0 L
1[ 2
and L[ =
0
L+ = I;
L = 0;
D
L+
i k
0 L
@ 0 L A 4 0 j
1 0
0 0
Cd ,
H ! E, where H = H
L = Ln ;
L+ = 0;
L = (Ln ) ;
= Ly G, where L = L y ; L+ = L
1
30
l
0 L
k 5@ 0 L A =
1 0
i
9
y
. Then L[ |L =
l + Di
k L + iL
L k + L+ i
L jL
=
In order to
su¢ cient to show
P prove the implication (iii))(iv), it is P
P that the
vectors = B | (B) L B are positive, ( j )
0 if B (B) B = B B B =
P
+
0. But this follows immediately from the observation + =
B =
B | (B) L
P
2
B
=
0
such
that
the
inde…nite
metrics
(3.4)
is
positive,
(
j
)
=
k
k
0 in
B
B
this case.
P
The …nal implication (iv))(i) is obtained as the case I =
B6=I B B of
P
B
=
0.
B
B
4. The Structure of the Bounded Filtering Generators.
The structure (3.3) of the form-generator for CP cocycles over B = B (H) is
a consequence of the well known fact that the derivations k; k of the algebra
B (H) of all bounded operators on a Hilbert space H are spatial, k (B) = j (B) L
LB; k (B) = Ly j (B) BLy , and so
(4.1)
l (B) =
1 y
L k (B) + k (B) L + [B; D] + i [H; B] ;
2
where H y = H is a Hermitian operator in H. The germ-map whose components
are composed (as in (3.3)) into the sums of the components '
of a CP matrix
map ' : B ! B M Cd+1 and left and right multiplications, are obviously
conditionally completely positive with respect to the representation in (4). As
follows from the dilation theorem in this case, there exists a family L = L =
L+ ; Ln = Ln ; n = 1; :::; d of linear operators L : H ! H , having adjoints
Ly : H ! H such that ' (B) = Ly j (B) L .
The next theorem proves that these structural conditions which are su¢ cient for
complete positivity of the cocycles, given by the equation (2.2), are also necessary
if the germ-map is w*-continuous on an operator algebra B. Thus the equation
(2.2) for a completely positive quantum cocycle with bounded stochastic derivatives
has the following general form
d
t
(B) +
t
y
K B + BK
y
L j (B) L dt =
d
X
t
m
n
d
BKn d
n
Lym j (B) Ln
B
n
m
m;n=1
(4.2)
+
d
X
m=1
t
Lym j (B) L
y
Km
B d
+
m
+
d
X
t
Ly j (B) Ln
;
n=1
generalising the Lindblad form [17], for the norm-continuous semigroups of completely positive maps. The quantum stochastic submartingale Mt = t (I) is de…ned
10
V. P. BELAVKIN.
by the integral
Mt +
Z
t
s (D) ds = I +
0
(4.3)
+
Z
t
d
X
s
Z tX
d
s
n
m
Lyn Lm
d
m
n
0 m;n
Lym L
y
d
Km
+
m
+
0 m=1
Z tX
d
s
Ly Ln
Kn d
n
:
0 n=1
If the space K can be embedded into the direct sum H Cd = H ::: H of d
copies of the initial Hilbert space H such that j (B) = (B m
n ), this equation can
y
be resolved in the form t (B) = Ft BFt , where F = (Ft )t>0 is an (unbounded)
cocycle in the tensor product H F with Fock space F over the Hilbert space
Cd L2 (R+ ) of the quantum noise of dimensionality d. The cocycle F satis…es the
quantum stochastic equation
(4.4) dFt + KFt dt =
d
X
Lin
I
i
n
Ft d
i;n=1
n
i
+
d
X
Li Ft d
+
i
d
X
Kn Ft d
n
;
n=1
i=1
where Lin and Li are the operators in H, de…ning
'm
n (B)
(4.5)
=
d
X
i
Liy
m BLn ;
' (B) =
i=1
'm (B)
=
d
X
with
i=1
iy
i
Liy
m BL ;
'n (B) =
d
X
Liy BLin
i=1
y
i
Liy BLi
i=1
i=1
Pd
d
X
K + K y if submartingale) .
L L = K + K if Mt is a martingale (
Theorem 3. Let the germ-maps
of the quantum stochastic cocycle
over a
von-Neumann algebra B be w*-continuous and bounded:
(4.6)
! 21
d
X
2
= k k < 1;
k k = k (I)k < 1;
k k < 1;
k k=
k nk
n=1
where k k = sup fk (B)k : kBk < 1g ; k (I)k = sup fh ; (I) i jk k < 1 g
and t be a CP cocycle, satisfying equation (2.2) with 0 (B) = B and normalized
to a submartingale (martingale). Then they have the form (3.3) written as
(4.7)
with ' = '+ ;
map.
(4.8)
(B) = ' (B)
' m = 'm
+;
'=
'
'
(B) K
K y (B)
'n = 'n and 'm
n =
m
n,
'
'
K
K
;
and
K=
composing a bounded CP
K
K
with arbitrary K ; K , and K + K y ' (I). The equation (4.2) has the unique CP
solution , satisfying the condition s (I)
s [ t (I)] for all s < t ( s (I) = s [ t (I)]
if K + K y = ' (I)).
ON THE GENERAL FORM OF QUANTUM STOCHASTIC EVOLUTION EQUATION.
11
Proof. The structure (4.7) for the CP component
was obtained as a part of
the dilation theorem in the Stinespring form
(B) = Ly j (B) L = ' (B), where
L = L . In order to obtain the structure (4.7) for the bounded germ-maps
and , we can take into account the spatial structure k (B) = j (B) L LB of a
bounded (j; i)-derivation for a von-Neumann algebra B with respect to a normal
representation j of B and i (B) = B. Then
(B) = k (B) L + BL = Ly j (B) L
B Ly L
L
= Ly+ j (B) L
BK ;
y
where L+ = L; K = L L L . Hence (B) = ' (B) BK ; (B) = ' (B)
K B =
(B), where K = K y ; ' (B) = Ly j (B) L = ' (B), such that the
= ;
matrix-map ' (B) = (L j (B) L ) =+; with L = Ly ; L = Ly is CP. Taking into
account the form (4.1) of the coboundary l (B) = (B) DB which is due to the
spatial form iH + 21 D; B of the bounded derivation l (B) 21 Ly k (B) + k (B) L
on B, one can obtain the representation
1 y
L k (B) + k (B) L + DB + BD + i [H; B] = ' (B) BK K y B;
(B) =
2
where ' (B) = Ly j (B) L, K = iH + 12 Ly L D .
The existence and uniqueness of the solutions t (B) to the quantum stochastic
(B) B and the initial
equations (2.2) with the bounded generators
(B) =
conditions 0 (B) = B in an operator algebra B was proved in [20]. The positivity
of the solutions in the case of the equation (4.2), corresponding to the conditionally
positive germ-function (4.7), can be obtained by the iteration
Z t
(n+1)
(0)
y
(n)
(B)
=
V
BV
+
Vty (s) BVt (s) d ;
t
t
t
t (B) = B
s
0
of the quantum stochastic integral equation
Z t
y
BV
+
(4.9)
(B)
=
V
Vty (s) BVt (s)
t
t
s
t
d
;
0
with
(B) = ' (B) B . Here Vt = Vt (s) Vs with Vt (s) = Tsy Vt s Ts shifted
by the co-isometry Ts in D, is the vector cocycle, resolving the quantum stochastic
di¤erential equation
(4.10)
dVt + KVt dt +
d
X
Km V t d
m
=0
m=1
with the initial condition V0 = I in H. The equivalence of (4.2) and (4.9), (4.10) is
veri…ed by direct di¤erentiation of (4.9). In order to prove the complete positivity
of this solution, one should write down the corresponding iteration
Z t
(n+1)
y
y
(f
;
B;
h
)
=
V
BV
+
f (s) (n)
f ; ' Vty (s) BVt (s) ; h h (s) ds;
t
t
t
s
0
of the ordinary integral equation for the operator-valued kernels of coherent vectors,
de…ned in (2.7). Here g (s) = 1 g (s) such that
E X
XXD f
h
y
hXVt X jZVt Z i
t f ; X Z; h
Z =
X
X;Z f;h
+
Z t XXD
0 X;Z f;h
X;Z
f
X
(s) j
s
f ; ' X yZ ; h
h
Z
E
(s) ;
12
V. P. BELAVKIN.
where
B
=
P
g
g
B;
g
B
(s) =
P
g(s)
g
B
g (s). Then the CP property for
(n)
t ,
1)
immediately follows from the CP property of (n
; s < t and of '. The direct
s
(0)
iteration of this integral recursion with the initial CP condition t (B) = B gives
at the limit n ! 1 the minimal CP solution in the form of sum of n-tupol CP
integrals on the interval [0; t].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
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Mathematics Department, University of Nottingham,, NG7 2RD, UK.
E-mail address : vpb@@maths.nott.ac.uk