On the Choi-Jamiolkowski Correspondence in Infinite Dimensions

0.19 MB·

by  A. S. Holevo

On the Choi-Jamiolkowski Correspondence
1 in Infinite Dimensions
1 ∗
0
2

n A. S. Holevo
a
J Steklov Mathematical Institute

2
2

]
h Abstract
p
-
h We give a mathematical formulation for the Choi-Jamiolkowski
t (CJ) correspondence in the infinite-dimensional case close to one used
a
m in quantum information theory. We show that “unnormalized maxi-
[ mally entangled state” and the corresponding analog of the Choi ma-

trix can be defined rigorously as positive semidefinite forms on an
3
v appropriate dense subspace. The properties of these forms are dis-
6 cussed in Sec. 2. In Sec. 3 we prove a version of a result from
9
[14] characterizing the form corresponding to entanglement-breaking
1
0 channel by giving precise definitions of the separable CJ form and
4. the relevant integral. In Sec. 4,5 we obtain explicit expressions for CJ
0 formsandoperatorsdefiningBosonicGaussian channel. Inparticular,
0
a condition for existence of the bounded CJ operator is given.
1
:
v
i
X 1 Introduction
r
a
In this paper we give a mathematical formulation for the Choi-Jamiolkowski
(CJ) correspondence ([12], [3]) in the infinite-dimensional case in the form
closetooneusedinquantuminformationtheory(seee.g. [13]). Weshowthat
there is no need to use a limiting procedure (cf. [6]) to define “unnormalized
maximally entangled state” and the corresponding analog of the Choi matrix
[3]sincetheycanbedefined rigorouslyas, ingeneral, nonclosable formsonan
appropriate dense subspace. The properties of these forms are discussed in
∗
Support by the Institute Mittag-Leffler (Djursholm, Sweden) is gratefully acknowl-
edged.
1
Sec. 2. An important question is: when the CJ form is given by a bounded
operator. This is the case for entanglement-breaking channels: we prove
this in Sec. 3 along with a version of a result from [14] characterizing CJ
operators which correspond to such channels by giving precise definitions of
a separable operator and a relevant integral. In Sec. 4 we obtain explicit
expressions for CJ forms and operators defining a general Bosonic Gaussian
channel. In Sec. 5 we give a decomposition of CJ form into product of the
four principal types and a necessary and sufficient condition for existence of
the bounded CJ operator.
2 Positive semidefinite forms
In what follows , ,... denote separable Hilbert spaces; T( ) denotes the
H K H
Banach space of trace-class operators in , and S( ) – the convex subset
H H
of all density operators. We shall also call them states for brevity, having in
mind that a density operator ρ uniquely determines a normal state on the
algebra B( ) of all bounded operators in . Equipped with the trace-norm
H H
distance, S( ) is a complete separable metric space. It is known [5], [4]
H
that a sequence of quantum states ρ converging to a state ρ in the weak
n
{ }
operator topology converges to it in the trace norm. Moreover, it suffices
that lim ψ ρ ψ = ψ ρ ψ for ψ in a dense linear subspace of .
n n
h | | i h | | i H
Definition 1 A channel is a linear map Φ:T( ) T( ) with the prop-
A B
H 7→ H
erties:
1) Φ(S( )) S( ); this implies that Φ is bounded map [4] and hence
A B
H ⊆ H
it is uniquely determined by the infinite matrix [Φ( i j )], where i is a
| ih | {| i}
fixed orthonormal basis in .
A
H
2) The block matrix [Φ( i j )] is positive semidefinite in the sense that
| ih |
for any finite collection of vectors ψ
i B
{| i} ⊆ H
ψ Φ( i j ) ψ 0. (1)
i j
h | | ih | | i ≥
Xij
The Choi-Jamiolkowski (CJ) form of the channel, associated with the
basis i , is defined by the relation (2) below.
{| i}
In weconsider thedensedomainwhichisinvariant under“local”
B A
H ⊗H
bounded operators X X :
B A
⊗
= lin ψ ψ : ψ ,ψ .
B A B A B B A A
H ×H { ⊗ ∈ H ∈ H }
2
Lemma 1 There is a unique sesquilinear positive semidefinite form Ω on
Φ
satisfying the relation
B A
H ×H
¯ ¯ ¯ ¯
Ω (ψ ψ ;ψ ψ ) = ψ Φ ψ ψ ψ = ψ Φ ( ψ ψ ) ψ ,
Φ B ⊗ A B′ ⊗ A′ h B| | Aih A′ | | B′ i h A′ | ∗ | B′ ih B| | Ai
(cid:0) (cid:1) (2)
where ψ¯ = +∞ i i ψ .
| i i=1 | ih | i
P
Proof. It is sufficient to show that for any ψ = ψj ψj
| BAi | Bi ⊗ | Ai ∈
j
P
B A
H ×H ψj Φ ψ¯j ψ¯k ψk 0,
h B| | Aih A| | Bi ≥
Xjk (cid:0) (cid:1)
and ψ = 0 implies that the above sum is equal to zero. Decomposing
BA
|ψAji|= i+i=∞1 cij|ii, we have |ψBAi = |ψ˜Bi i ⊗|ii where |ψ˜Bi i = cij|ψBj i
i j
P P P
andtheabove sum isequal to ψ˜i Φ( i j ) ψ˜j whence the result follows.
h B| | ih | | Bi
ij
(cid:3) P
Note that for any orthonormal basis e in
k B
{| i} H
Ω (e ψ ;e ψ ) = ψ ψ . (3)
Φ k ⊗ A k ⊗ A′ h A| A′ i
X
k
The expression on the left is a natural form generalization of partial trace
with respect to . The relation (3) shows that for Ω it is always given
B Φ
H
by the (form corresponding to) the identity operator I . Similarly defined
A
partial trace with respect to not always exists; however this is the case
A
H
when the expression Φ[I ] is well-defined in the sense of [10] and then it is
A
given by that operator. Positivity and the property (3) imply
1/2
Ω (ψ ψ ;ψ ψ ) [Ω (ψ ψ ;ψ ψ )Ω (ψ ψ ;ψ ψ )]
| Φ B ⊗ A B′ ⊗ A′ | ≤ Φ B ⊗ A B ⊗ A Φ B′ ⊗ A′ B′ ⊗ A′
ψ ψ ψ ψ . (4)
≤ k Bkk B′ kk Akk A′ k
Conversely, any sesquilinear positive semidefinite form Ω on
B A
H × H
satisfying the condition (3) uniquely defines a channel Φ such that Ω = Ω
Φ
(via the reversed relation (2)). The positivity of the form Ω can be used to
Φ
prove the Kraus decomposition for Φ similarly to the finite-dimensional case
[13]. Indeed, let H be the Hilbert space obtained by completion of
B A
H ×H
with respect to the inner product defined by the (factorized) form Ω . Then
Φ
3
anyorthonormal basis inHdefines acountable collection oflinear functionals
f + on such that
{| li}l=∞1 HB ×HA
+
∞
Ω (ψ ψ ;ψ ψ ) = f (ψ ψ )f (ψ ψ ).
Φ B ⊗ A B′ ⊗ A′ l B′ ⊗ A′ l B ⊗ A
X
l=1
Define the linear operators V : by the relation ψ V ψ =
l A B B l A
H → H h | | i
¯
f ψ ψ , then (3) implies that V are bounded operators satisfying
l B A l
⊗
(cid:0)+l=∞1 Vl∗Vl =(cid:1) IA and(2)impliestheKrausdecompositionΦ(ρ) = +l=∞1 VlρVl∗.
P If the form ΩΦ is closable [15], then it is defined by the uniqPue densely
defined selfadjoint positive operator, which we also denote Ω . If the domain
Φ
of the form Ω is the whole then the operator Ω is bounded. The
Φ B A Φ
H ⊗H
property (3) then reads
Tr Ω = I . (5)
B Φ A
In this case the defining relation (2) can be given a more familiar form
TrΩ (ρ X) = TrΦ ρ X = Trρ Φ (X), ρ T( ),X B( ),
Φ ⊤ ⊤ ∗
⊗ ∈ H ∈ H
(cid:0) (cid:1)
¯ ¯
where⊤ denotestranspositioninthebasis{|ii},sothat|ψAihψA′ | = (|ψA′ ihψA|)⊤.
An example of nonclosable sesquilinear form is provided by the identity
channel Φ = Id, for which
Ω (ψ ψ ;ψ ψ ) = ψ ψ Ω Ω ψ ψ ,
Id B ⊗ A B′ ⊗ A′ h B ⊗ A| ih | B′ ⊗ A′ i
where Ω is the unbounded linear form on defined as
A A
h | H ×H
+
∞
Ω ψ ψ = i ψ i ψ ; ψ ,ψ ,
1 2 1 2 1 2 A
h | ⊗ i h | ih | i ∈ H
Xi=1
and Ω is the dual antilinear form which represents “unnormalized maxi-
| i
mally entangled state”, Ω = +∞ i i . The relation
| i i=1 | i⊗| i
P
Ω = (Φ Id )(Ω )
Φ A Id
⊗
holds in the weak sense i.e. as equality for the forms defined on .
B A
H ×H
Notice that (X X ) Ω = I X X Ω . If i is another basis
A ⊗ B | i ⊗ B A⊤ | i {| i′}
such that i = U i for a unitary(cid:0)U, then Ω (cid:1)= I UU Ω and
′ ′ ⊤
| i | i | i ⊗ | i
(cid:0) (cid:1)
Ω′Φ = I ⊗UU⊤ ΩΦ I ⊗UU⊤ ∗.
(cid:0) (cid:1) (cid:0) (cid:1)
4
In the case of bounded Ω this implies that Ω is the same for all choices
Φ Φ
k k
of the basis i . Notice also that transposition in the basis i is given
′
′ {| i} {| i}
by X⊤ = UU⊤ X⊤ UU⊤ ∗.
Fix a st(cid:0)ate σ(cid:1)in S((cid:0)HA) o(cid:1)f full rank, and let {|ii}+i=∞1 be the basis of eigen-
vectors of σ with the corresponding (positive) eigenvalues λ + . Consider
{ i}i=∞1
the purifying vector
+
∞
1/2
ψ = λ i i
| σi i | i⊗| i
Xi=1
in the space . Then the state
A A
H ⊗H
ρ (σ) = (Φ Id )( ψ ψ ) S( ) (6)
Φ A σ σ B A
⊗ | ih | ∈ H ⊗H
satisfying Tr ρ (σ) = σ uniquely determines the channel Φ via the relation
B Φ
1/2 1/2
Φ(|iihj|) = λ−i λ−j TrA(IB ⊗|jihi|)ρΦ(σ),
see [11]. The connection between ρ (σ) and Ω is
Φ Φ
ψ ψ ρ (σ) ψ ψ = Ω ψ σ1/2ψ ;ψ σ1/2ψ .
h B ⊗ A| Φ | B′ ⊗ A′ i Φ B ⊗ A B′ ⊗ A′
(cid:0) (cid:1)
Note that Ω is uniquely defined by its values on the dense domain =
Φ
D
σ1/2( ) due to the property (4).
B A
H × H
3 Separable operators and entanglement-breaking
channels
Let σ be a state in S( ) of full rank and Ω is a bounded positive operator
A
H
satisfying (5), then σ = I σ1/2 Ω I σ1/2 is a density operator in
BA B B
⊗ ⊗
such that Tr σ (cid:0)= σ. Let u(cid:1)s re(cid:0)mind that(cid:1)a state ρ S( )
B A B BA B A
H ⊗H ∈ H ⊗H
is called separable if it is in the convex closure (in the weak operator topology
and hence in the trace norm) of the set of all product states. Separable states
are precisely those which admit the representation
ρ = (ρ (x) ρ (x))µ(dx), (7)
B A
Z ⊗
X
where µ is a Borel probability measure on a complete separable metric space
, see [11].
X
5
A channel Φ is called entanglement-breaking if for arbitrary state ω
∈
S( ) the state (Φ Id )(ω) is separable. Channel Φ is entanglement-
A A A
H ⊗H ⊗
breaking if and only if there is a complete separable metric space , a Borel
X
S( ) -valued function x ρ (x) and a probability operator-valued Borel
B B
H 7→
measure (POVM) M (dx) on in such that
A A
X H
Φ(ρ) = ρ (x)µ (dx), (8)
B ρ
Z
X
where µ (E) = TrρM (E) for all Borel E . If ρ = ρ (σ), then ρ (x) in
ρ A Φ B
⊆ X
(8) is the same as in (7) while M (dx) is defined by the relation
A
ψ M (E) ψ = σ 1/2ψ ρ (x) σ 1/2ψ µ(dx); ψ ,ψ σ1/2( ),
h A′ | A | Ai Z h − A′ | A | − Ai A A′ ∈ HA
E
where the complex conjugate is in the basis i .
{| i}
Definition 2 A bounded positive operator Ω in satisfying (5) is
B A
H ⊗ H
called separable if it belongs to the closure, in the weak operator topology, of
the convex set of operators of the form ρ M , where M is a finite
α α α
⊗ { }
α
P
resolution of the identity in and ρ S( ). The weakly closed convex
A α B
H ∈ H
set of separable operators will be denoted C .
BA
Lemma 2 For Ω C the operator norm Ω 1.
BA
∈ k k ≤
Proof. It is sufficient to prove it for Ω = ρ M , since the weak
α α
⊗
α
P
operator limit does not increase the norm. Then
Ω = sup ψ Ω ψ sup ψ I M ψ ψ ψ = 1.
B α
k k h | | i ≤ h | ⊗ | i ≤ h | i
kψk=1 kψk=1 Xα
(cid:3)
It follows that in the definition 2 it is sufficient to consider sequences of
operators, weakly convergent on a dense subspace of .
B A
H ⊗H
Equipped the definition 2 and the construction of the integral (9) below
we can prove a rigorous version the corresponding result from [14].
6
Proposition 1 If the channel Φ is entanglement-breaking then its form is
given by a bounded operator Ω C .If Φ has representation (8) then
Φ BA
∈
Ω = ρ (x) M¯ (dx), (9)
Φ B A
Z ⊗
X
where the integral is defined in the proof.
Conversely, if a bounded operator Ω C then Ω = Ω , where the
BA Φ
∈
channel Φ is entanglement-breaking.
Proof. Theproofofthefirst statement requires sometheoryofintegration
with respect to a POVM.
Let bea complete separable metricspace withσ algebraofBorel sub-
X −
sets , let is a separable Hilbert space and M(E);B a POVM on
B H { ∈ B} X
withvaluesinB( ). Thenforanyψ thesetfunction ψ M(E) ψ ;B
H ∈ H {h | | i ∈ B}
is positive finite measure with total variation ψ 2; and for any ψ,ψ
′
k k ∈ H
the set function ψ M(E) ψ ;B is a complex measure of finite total
′
{h | | i ∈ B}
variation ψ ψ as follows from the inequality
′
k kk k
1
ψ M(E) ψ c ψ M(E) ψ +c 1 ψ M(E) ψ ; c > 0, (10)
′ − ′
|h | | i| ≤ 2 h | | i h | | i
(cid:2) (cid:3)
due to positivity of the operator M(E) for arbitrary Borel E .
⊆ X
A function x ρ(x) with values in S( ) will be called Borel function
→ H
if the scalar functions x ψ ρ(x) ψ are Borel for all ψ,ψ . Let
′ ′
→ h | | i ∈ H
µ be a σ finite measure on then the function x ρ(x) will be called
− X →
measurable if the functions x ψ ρ(x) ψ are µ measurable for all ψ,ψ
′ ′
→ h | | i − ∈
. This implies that for any A B( ) the scalar function x Trρ(x)A
H ∈ H →
is µ measurable. Since S( ) is separable with respect to the trace norm
− H
distance, this implies, by theorem 3.5.3 from [7], that x ρ(x) is strongly
→
measurable in the sense that there is a sequence of simple Borel functions
x ρ (x) such that ρ(x) ρ (x) 0 (modµ). If µ is a probability
→ n k − n k1 →
measure then the Bochner integral ρ(x)µ(dx) = lim ρ (x)µ(dx)
n n
exists and is an element of S( ). RX →∞RX
H
Now let Φ be entanglement-breaking channel with the representation (8).
If x ρ (x) is a Borel function with values in S( ) and M (dx) a POVM
B B A
→ H
in we define the integral ρ (x) M¯ (dx) B( )as follows. For
A B A B A
H ⊗ ∈ H ⊗H
R
a simple function ρ (x) = Xρ 1 (x), where E is a finite decomposition
n α Eα { α}
α
P
7
of , we put Ω ρ (x) M¯ (dx) = ρ M¯ (E ). Apparently
n n A α A α
X ≡ ⊗ ⊗
R Pα
Ω C , hence Ω X 1. We also have
n BA n
∈ k k ≤
¯ ¯
ψ ψ Ω ψ ψ = ψ ρ (x) ψ ψ M (dx) ψ , (11)
h B ⊗ A| n| B′ ⊗ A′ i Z h B| n | B′ ih A′ | A | Ai
X
for all ψ ,ψ , ψ ,ψ . For two simple functions ρ (x),ρ (x) we
B B′ ∈ HB A A′ ∈ HA n ′m
have by the inequality (10)
ψ ψ (Ω Ω ) ψ ψ
|h B ⊗ A| n − ′m | B′ ⊗ A′ i|
1
ψ ψ ρ (x) ρ (x) c ψ¯ M (dx) ψ¯ +c 1 ψ¯ M (dx) ψ¯ .
≤ 2 k Bkk B′ kZ k n − ′m k1 h A| A | Ai − h A′ | A | A′ i
(cid:2) (cid:3)
X
Take ρ (x) such that ρ(x) ρ (x) 0 (modµ ) where σ is a state in
{ n } k − n k1 → σ
S( ) of full rank, then it follows that the forms ψ Ω ψ converge
HA h BA| n| B′ Ai
on a dense subspace of and are uniformly bounded. Hence there
B A
H ⊗H
exists uniquely defined bounded operator Ω, which is the limit of Ω in the
n
weak operator topology. We define ρ (x) M¯ (dx) = Ω, and it follows
B A
⊗
R
from the definition that Ω C . X
BA
∈
From the definition (2) of the form Ω we obtain
Φ
¯ ¯
Ω (ψ ψ ;ψ ψ = ψ ρ (x) ψ ψ M (dx) ψ ,
Φ B ⊗ A B′ ⊗ A′ i Z h B| B | B′ ih A′ | A | Ai
X
which is the limit of (11). Since, on the other hand, (11 ) converge to the
form defined by the operator Ω, we obtain that the form Ω is defined by
Φ
the operator Ω.
Conversely, let a bounded operator Ω C . Fix a state σ in S( ) of
BA A
∈ H
full rank. Using the fact that weak convergence of operators Ω = ρ
α
⊗
α
P
M C implies weak convergence of density operators σ = ρ
α BA BA α
∈ ⊗
α
P
σ1/2M σ1/2, we can prove that the state σ = I σ1/2 Ω I σ1/2 is
α BA B B
⊗ ⊗
separable. Basing on the decomposition (7) for σ(cid:0) and argu(cid:1)ing(cid:0)as in [11](cid:1)we
can construct POVM M (dx) and the family of states ρ (x) such that Ω is
A B
given by (9) and hence Ω = Ω for the corresponding entanglement-breaking
Φ
(cid:3)
channel Φ given by (8).
8
4 Bosonic Gaussian channels
Let (Z,∆)beacoordinatesymplectic space, dimZ = 2s,with thesymplectic
form
0 1
∆(z,z′) = zt∆z′; ∆ = diag −
(cid:20) 1 0 (cid:21)
j=1,...,s
where t denotes transposition in Z. Let be the space of irreducible repre-
H
sentation of the Canonical Commutation Relations (CCR)
i
W(z)W(z ) = exp ∆(z,z ) W(z +z ). (12)
′ ′ ′
(cid:18)2 (cid:19)
HereW(z) = exp(iR z); z Z,istheWeylsystemandR = [q ,p ;...,q ,p ]
1 1 s s
· ∈
is the row-vector of the canonical variables in , see [8] for more detail. We
H
will continue to denote transposition in associated to a basis i ; from
⊤
H {| i}
(12) it follows that W(z) = exp(iR z); z Z, is the Weyl system for
⊤ ⊤
· ∈
the symplectic space (Z, ∆). The canonical transposition associated with
−
the Fock basis in is given by R = [q , p ;...,q , p ]; all the others are
⊤ 1 1 s s
H − −
obtained from this by unitary conjugations. In what follows transposition is
arbitrary if not stated otherwise.
Let be the representation space of CCR with a coordinate symplectic
A
H
space (Z ,∆ ) and the Weyl system W ( ), with a similar description for
A A A
·
. Let Φ : T( ) T( ) be a channel. Then the following relation
B A B
H H 7→ H
holds
1
ΩΦ = (2π)s Z WB(−z)⊗Φ∗(WB(z))⊤d2sz, (13)
ZB
in the sense of forms defined on . Here d2sz is the element of sym-
B A
H ×H
plectic volume in Z and is a transposition in .
B ⊤ A
H
Indeed,
¯ ¯ ¯ ¯
ψ Φ (W (z)) ψ = TrΦ( ψ ψ )W (z), (14)
′ ∗ B ′ B
h | | i | ih |
¯ ¯
where Φ( ψ ψ ) is trace-class operator and hence (14) is square-integrable
′
| ih |
as a function of z Z . Similarly, the function ψ W( z) ψ is also square-
B ′
∈ h | − | i
integrable. By the inversion formula for the noncommutative Fourier trans-
form (cf. ch. V of [8]), applied to the right-hand side of (14),
¯ ¯
Ω (ψ ψ ;ψ ψ ) = ψ Φ( ψ ψ ) ψ
Φ B ⊗ A B′ ⊗ A′ h B| | Aih A′ | | B′ i
1
= ψ W ( z) ψ ψ¯ Φ (W (z)) ψ¯ d2sz,
(2π)s Z h B| B − | B′ ih A′ | ∗ B | Ai
9
so that finally we get the formula (13).
Now let Φ be a (centered) Gaussian channel [1],
1
Φ (W (z)) = W (Kz)exp ztµz , z Z ,
∗ B A B
(cid:18)−2 (cid:19) ∈
where
i
µ ∆ ; ∆ = ∆ Kt∆ K. (15)
K K B A
≥ ±2 −
Then (13) implies
1 1
Ω = W ( z) W (Kz) exp ztµz d2sz. (16)
Φ (2π)s Z B − ⊗ A ⊤ (cid:18)−2 (cid:19)
The unitary operators W (z) = W (z) W ( Kz) = exp(iR z),
BA B A ⊤ BA
⊗ − ·
where
R = R I I R K,
BA B ⊗ − ⊗ A⊤
satisfy the Weyl-Segal CCR with (possibly degenerate) symplectic form de-
termined by the matrix ∆ . This representation in the space is
K B A
H ⊗ H
reducible (even if ∆ is nondegenerate, see the footnote below). Finally
K
1 1
Ω = exp( iR z)exp ztµz d2sz. (17)
Φ (2π)s Z − BA · (cid:18)−2 (cid:19)
ZB
If µ is nondegenerate, then the form Ω is given by bounded operator
Φ
with
1
Ω . (18)
Φ
k k ≤ √detµ
This follows from (17) taking into account the fact that norm of the Weyl
operators is equal to 1.
In what follows we shall consider the four basic cases. Later it will be
convenient to use reverse enumeration, so we start with the last, the most
degenerate case.
Case 4. If µ = 0 then ∆ = 0 and R is the vector operator with
K BA
commuting selfadjoint components. Thus
1
Ω = exp( iR z)d2sz = δ(R ), (19)
Φ (2π)s Z − BA · BA
10