arXiv:1806.05668v6 [math.FA] 1 Oct 2020
Operator E -norms and their use
M.E. Shirokov∗
Abstract
We consider a family of equivalent norms (called operator E -norms) on the
algebra B(H) of all bounded operators on a separable Hilbert space H induced
by a positive densely defined operator G on H. By choosing different generating
operator G one can obtain the operator E -norms producing different topologies,
in particular, the strong operator topology on bounded subsets of B(H).
We obtain a generalised version of the Kretschmann-Schlingemann-Werner
theorem, which shows continuity of the Stinespring representation of CP linear
maps w.r.t. the energy-constrained cb-norm (diamond norm) on the set of CP
linear maps and the operator E -norm on the set of Stinespring operators.
The operator E -norms induced by a positive operator
√ G are well defined for
linear operators relatively bounded w.r.t. the operator G and the linear space
of such operators equipped with any of these norms is a Banach space. We obtain
explicit
√ relations between the operator E -norms and the standard characteristics
simple upper
of G-bounded operators. The operator E -norms allow to obtain √
bounds and continuity bounds for some functions depending on G-bounded
operators used in applications.
Contents
1 Introduction
2
2 Preliminaries
3
3 Operator E -norms on B(H)
3.1 Equivalent definitions and equivalent norms . . . . . . . . . . . . . . .
3.2 Basic properties of the operator E -norms . . . . . . . . . . . . . . . . .
3.3 Properties of the E -norms related to tensor products . . . . . . . . . .
4
5
8
9
4 The E -version of the Kretschmann-Schlingemann-Werner theorem
10
5 Operator E -norms for unbounded operators
17
∗
Steklov Mathematical Institute, RAS, Moscow, email:msh@mi.ras.ru
1
1
Introduction
The algebra B(H) of all bounded linear operators on a separable Hilbert space H,
some its subalgebras and subspaces are basic objects in different fields of modern
mathematics and mathematical physics [4, 10, 13]. In particular, B(H) appears as
an algebra of observables in the theory of quantum systems while unital completely
positive maps between such algebras called quantum channels play the role of dynamical
maps in the Heisenberg picture [6, 25, 27].
The variety of different topologies on B(H), relations between them and their
”physical” sense are well known for anybody who is interested in functional analysis,
theory of operator algebras, mathematical and theoretical physics.
In this article we describe families of norms on B(H) producing different topologies on B(H), in particular, the strong operator topology on bounded subsets of B(H).
These norms depending on a positive densely defined operator G and a positive parameter E were introduced in [16] for quantitative analysis of continuity of the Stinespring
representation of a quantum channel with respect to the strong convergence of quantum
channels and the strong operator convergence of Stinespring isometries.1
Now we consider these norms (called the operator E -norms) in more general context
(assuming that G is an arbitrary positive operator). In Section 3 we consider equivalent
definitions and basic properties of the operator E -norms. We obtain explicit relations
between the operator E -norms and the equivalent norm on√B(H) also induced by a
positive operator G (which is commonly used in analysis of G-bounded operators).
The operator E -norms make it possible to obtain a generalization the KretschmannSchlingemann-Werner theorem. The original version of this theorem presented in [8]
shows continuity of the Stinespring representation of a completely positive (CP) linear
map with respect to the norm of complete boundedness (cb-norm in what follows)2
on the set of CP maps and the operator norm on the set of Stinespring operators.
Our aim was to obtain a version of this theorem for other (weaker) topologies on the
sets of CP maps and corresponding Stinespring operators, in particular, for the strong
convergence topology on the set of CP maps and the strong operator topology on the
set of Stinespring operators. By using the operator E -norms one can upgrade the
proof of the Kretschmann-Schlingemann-Werner theorem without essential changes.
The generalised version of this theorem and its corollaries are presented in Section 4.
In Section 5 the operator E -norms induced by a positive √
operator G are extended
to linear operators relatively bounded w.r.t. the operator G. We prove that the
linear space of such operators equipped with any
√of these norms is a Banach space. Its
subspace consisting of all operators with zero G-bound is the completion of B(H)
w.r.t. any of the operator E -norms. We obtain
√ explicit relations between the operator
E -norms and the standard characteristics of G-bounded operators.
1
2
Other applications of the operator E -norms are presented in the recent papers [18, 19, 20].
It is also called the diamond norm in the quantum information theory [1, 27].
2
The operator E -norms allow to√obtain simple upper estimates and continuity bounds
for some functions depending on G-bounded operators used in applications.
As a basic example we consider the operators associated with the Heisenberg Commutation Relation.
2
Preliminaries
Let H be a separable infinite-dimensional Hilbert space, B(H) – the algebra of all
bounded operators on H with the operator norm k·k and T(H) – the Banach space of
all trace-class operators on H with the trace norm k·k1 (the Schatten class of order 1)
[4, 13]. Let T+ (H) be the cone of positive operators in T(H). Trace-class operators
will be usually denoted by the Greek letters ρ, σ, ω, ... The closed convex subsets
T+,1 (H) = {ρ ∈ T+ (H) | Trρ ≤ 1} and S(H) = {ρ ∈ T+ (H) | Trρ = 1}
of the cone T+ (H) are complete separable metric spaces with the metric defined by
the trace norm. Operators in S(H) are called density operators or states, since any ρ
in S(H) determines a normal state A 7→ TrAρ on the algebra B(H) [4, 6]. Extreme
points of S(H) are 1-rank projectors called pure states.
Denote by IH the unit operator on a Hilbert space H and by IdH the identity
transformation of the Banach space T(H).
We will use the Dirac notations |ϕi, |ψihϕ|,... for vectors and operators of rank 1
on a Hilbert space (in this notations the action of an operator |ψihϕ| on a vector |χi
gives the vector hϕ|χi|ψi) [6].
We will pay a special attention to the class of unbounded densely defined positive
operators on H having discrete spectrum of finite multiplicity. In Dirac’s notations any
such operator G can be represented as follows
G=
+∞
X
k=0
Ek |τk ihτk |
(1)
on the domain D(G) = {ϕ ∈ H | k=0 Ek2 |hτk |ϕi|2 < +∞}, where {τk }+∞
k=0 is the
orthonormal basis of eigenvectors of G corresponding to the nondecreasing sequence
{Ek }+∞
k=0 of eigenvalues tending to +∞. We will use the following (cf.[29])
Definition 1. An operator G having representation (1) is called discrete.
The set S(H) is compact if and only if dim H < +∞. We will use the following
Lemma 1. [5] If G is a discrete unbounded operator on H then the set of states ρ
in S(H) satisfying the inequality TrGρ ≤ E is compact for any E ≥ inf kϕk=1 hϕ|G|ϕi.
We will also use the following simple lemma.
Lemma 2. [28] If f is a concave nonnegative function on [0, +∞) then for any
positive x < y and any z ≥ 0 the inequality xf (z/x) ≤ yf (z/y) holds.
P+∞
3
Operator E -norms on B(H)
3
Let G be a positive semidefinite operator on H with a dense domain D(G) such that
inf {kGϕk | ϕ ∈ D(G), kϕk = 1} = 0.
(2)
We will assume that for any positive operator ρ in T(H) the value of TrGρ (finite or infinite) is defined as supn TrPn Gρ, where Pn is the spectral projector of G corresponding
to the interval [0, n].
For given E > 0 consider the function on B(H) defined as
.
kAkG
E =
sup
ϕ∈H1 ,hϕ|G|ϕi≤E
kAϕk,
(3)
√
where√H1 is the unit sphere in H and it is assumed that hϕ|G|ϕi = k Gϕk2 if ϕ lies
in D( G) and hϕ|G|ϕi = +∞ otherwise. This function can be also defined as
p
.
TrAρA∗ ,
(4)
kAkG
=
sup
E
ρ∈S(H):TrGρ≤E
where the supremum is over all states ρ in S(H) satisfying the inequality TrGρ ≤ E.3
The coincidence of the r.h.s. of (3) and (4) for any A ∈ B(H) is shown in [26].
It is easy to see that the function A 7→ kAkG
E is a norm on B(H). Definition (3)
G
shows the sense of the norm k · kE (as a constrained version of the operator norm
k · k) while definition (4) is more convenient for studying its analytical properties. In
particular, by using definition (4) the following proposition is proved in [16].4
Proposition 1. For any operator A ∈ B(H) the following properties hold:
a) kAkG
E tends to kAk as E → +∞;
p
b) the function E 7→ kAkG
is concave and nondecreasing on R+ for p ∈ (0, 2];
E
√
√
√
for
any
unit
vector
ϕ
in
D(
G),
where
K
=
max{1,
k
Gϕk/
E}.
c) kAϕk ≤ Kϕ kAkG
ϕ
E
We will call the norms k · kG
E the operator E-norms on B(H). Property b) in
Proposition 1 shows that
p
G
kAkG
E2 /E1 kAkG
for any E2 > E1 > 0.
(5)
E1 ≤ kAkE2 ≤
E1
Hence for given operator G all the norms k·kG
E , E > 0, are equivalent on B(H).
Remark 1. The definition of the operator E -norm is obviously generalized to
operators between different Hilbert spaces H and K. It is easy to see that all the above
3
In the previous versions of this posting the coincidence of the r.h.s. of (3) and (4) was conjectured,
but it was proved only under the assumption that the operator G is discrete (Definition 1).
4
In [16] condition (2) was not assumed. We use this assumption here, since it simplifies analysis of
G+λI
the norms k·kG
= kAkG
E without reduction of generality (note that kAkE
E−λ for all A and λ > 0).
4
and below results concerning properties of the operator E -norms remain valid (with
obvious modifications) for this generalization. √
Since the set D( G) is dense in H, property c) in Proposition 1 shows that the
topology generated by any of the norms k·kG
E on bounded subsets of B(H) is not weaker
than the strong operator topology. On the other hand, it is not stronger than the norm
topology on B(H). The following proposition characterizes these extreme cases.
Proposition 2. A) The norm k·kG
E , E > 0, is equivalent to the operator norm k·k
on B(H) if and only if the operator G is bounded.
B) The norm k · kG
E , E > 0, generates the strong operator topology on bounded
subsets of B(H) if and only if G is an unbounded discrete operator (Definition 1).
Proof. A) If G is a bounded operator then k·kG
E = k·k for any E ≥ kGk.
If G is a unbounded operator and Pn is the spectral projector of G corresponding
to the interval [n, +∞) then kPn k = 1 for all n. By noting that TrPn ρ ≤ E/n for any
state ρ such that TrGρ ≤ E, it is easy to see that kPn kG
E → 0 as n → +∞.
B) The ”if” part of this assertion is proved in [16].
Assume there is a spectral projector of the operator G corresponding to a finite
interval [0, E0 ] with infinite-dimensional range H0 . Since hϕ|G|ϕi ≤ E0 for any unit
vector ϕ in H0 , we have kAkG
E = kAk for any A ∈ B(H0 ) and E > E0 . So, any of the
norms k·kG
,
E
>
0,
generates
the norm topology on B(H0 ) in this case .
E
Different types of operator convergence can be obtained by using the norm k · kG
E
induced by different operators G.
Example. Let H = H1 ⊕ H2 and G = G1 ⊕ G2 , where Gk is a positive densely
defined operator on a separable Hilbert space Hk satisfying condition (2), k = 1, 2. By
using definition (4) and the triangle inequality it is easy to show that
q 2
2
G1
G2
2
1
≤ kAkG
(6)
+ (1 − p) kAP2 kG
p kAP1 kG
E ≤ kAP1 kE + kAP2 kE
E
E
k
for any p ∈ [0, 1], where Pk is the projector on the subspace Hk and kAPk kG
E , k = 1, 2,
are defined in accordance with Remark 1.
Assume that G1 is a discrete unbounded operator (Def.1) and G2 is a bounded
operator. Then it follows from (6) and Proposition 2 that
n
o
n
o n
o
k·kG
lim
A
=
A
⇔
s.o.lim
A
P
=
A
P
∧
k·klim
A
P
=
A
P
n
0
n
1
0
1
n
2
0
2
E
n→∞
n→∞
n→∞
for a bounded sequence {An } ⊂ B(H), where s.o.- lim denotes the limit w.r.t the strong
operator topology. So, in this case the norm k·kG
E generates a ”hybrid” topology on
bounded subsets of B(H) – some kind of the Cartesian product of the strong operator
and the norm topologies.
3.1
Equivalent definitions and equivalent norms
Recall that T+,1 (H) denotes the positive part of the unit ball in T(H). Denote by H≤1
the unit ball in H.
5
Proposition 3. A) For any A ∈ B(H) and E > 0 the following expressions hold
p
TrAρA∗ ,
∀A ∈ B(H),
(7)
kAkG
sup
kAϕk =
sup
E =
ϕ∈H≤1 :hϕ|G|ϕi≤E
ρ∈T+,1 (H):TrGρ≤E
i.e. the suprema in definitions (3) and (4) can be taken, respectively, over all vectors in
H≤1 satisfying the condition hϕ|G|ϕi ≤ E and over all operators in T+,1(H) satisfying
the condition TrGρ ≤ E.
B) If the operator G is unbounded then for any A ∈ B(H) and E > 0 the conditions
hϕ|G|ϕi ≤ E in (3) and TrGρ ≤ E in (4) and can be replaced, respectively, by the
conditions hϕ|G|ϕi = E and TrGρ = E.
C) If G is a discrete unbounded operator (Def.1) then for any A ∈ B(H) and E > 0 the
suprema in (3) and (4) are attainable. Moreover, if kAkG
E < kAk then the suprema in
(3) and (4) are attained, respectively, at unit vector ϕ0 in H such that hϕ0 |G|ϕ0 i = E
and at a state ρ0 in S(H) such that TrGρ0 = E.
Remark 2. It is easy to see that assertion A of Proposition 3 is not valid if the
operator G doesn’t satisfy condition (2).
Proof of Proposition 3. A) It suffices to show that the last expression in (7) does
not exceed kAkG
. Let ρ be an operator in T+,1(H) such that TrGρ ≤ E and r = Trρ.
. −1 E
Then ρ̂ = r ρ is a state such that TrGρ̂ ≤ E/r. So, by using concavity of the function
2
E → kAkG
and Lemma 2 in Section 2 we obtain
E
2 G 2
TrAρA∗ = rTrAρ̂A∗ ≤ r kAkG
≤
kAk
.
E/r
E
B)5 Show first that the inequality TrGρ ≤ E can be replaced by the equality
TrGρ = E in (4). Assume that there exist E > 0 and A ∈ B(H) such that
p
sup
TrAρA∗ ≤ kAkG
(8)
E −ε
ρ∈S(H):TrGρ=E
√
for some ε > 0. Let ρε be a state such that TrAρε A∗ > kAkG
E − ε/2 and TrGρε < E.
For each natural n > E there exist a state σn and a number pn ∈ (0, 1) such that
TrGσn ∈ (n, +∞) and TrG̺n = E, where ̺n = (1 − pn )ρε + pn σn . It is clear that
pn → 0 as n → +∞. Hence TrA̺n A∗ tends to TrAρε A∗ contradicting (8).
2
For any ε > 0 let ρε be a state such that TrAρε A∗ > [kAkG
E − ε] and TrGρε = E.
By Corollary 1 in [26] there is a probability measure µ on S(H) supported by pure
states such that
Z
ρε = σµ(dσ) and TrHσ = E for µ-almost all σ.
(9)
Since that function σ 7→ TrAσA∗ is affine and continuous, we have
Z
2
TrAσA∗ µ(dσ) = TrAρε A∗ > [kAkG
E − ε] .
5
If kAkG
E < kAk then this assertion can be derived from properties a) and b) in Proposition 1.
6
2
It shows existence of a pure state σε such that TrAσε A∗ > [kAkG
E − ε] and TrHσε = E.
C) In this case the set of pure states ρ satisfying the condition TrGρ ≤ E is compact
by Lemma 1 in Section 2. Hence, by the coincidence of the r.h.s. of (3) and (4), the
supremum in (4) is attained at some pure state ρ0 . By Proposition 1 the condition
G
G
′
kAkG
E < kAk shows that kAkE ′ < kAkE for any E < E. Hence TrGρ0 = E. Consider the following norm on B(H) depending on a positive parameter E:
n
o
√
G
2
k|Ak|E = sup kAϕk ϕ ∈ D( G), kϕk + hϕ|G|ϕi/E ≤ 1 .
(10)
This norm
√ naturally appears in analysis of operators relatively bounded w.r.t. the
operator G (see Section 5). We will obtain relations between the norms k · kG
E and
G
k| · k|E assuming that G is an arbitrary positive operator satisfying condition (2).
By using definitions (3) and (10) it is easy to show that
p
G
G
1/2kAkG
(11)
E ≤ k|Ak|E ≤ kAkE
for any E > 0. This inequality and inequality (5) imply that
p
G
k|Ak|G
2E2 /E1 k|Ak|G
for any E2 > E1 > 0
E1 ≤ k|Ak|E2 ≤
E1
(12)
and any A ∈ B(H). The above inequalities show that all the norms in the families
G
{k · kG
E }E>0 and {k| · k|E }E>0 are equivalent to each other.
G
In fact, all the norms k · kG
E and k| · k|E are equivalent
√ on the space of all linear
operators on H relatively bounded w.r.t. the operator G, and each of these norms
G
makes this space a Banach space. Moreover, the functions E 7→ kAkG
E and E 7→ k|Ak|E
are completely determined by each other (Remark 6 in Section 5). The main advantages
G
of the operator E -norm k · kG
E in comparison with the norm k| · k|E are the following:
p
• the concavity of the function E 7→ kAkG
for any p ∈ (0, 2];6
E
• the appearance of the norm k · kG
E in the generalized Kretschmann-SchlingemannWerner theorem (Section 4);
G
• the simple estimation of kΦ(A)kG
E via kAkE , where Φ : B(H) → B(H) is any
2-positive linear map satisfying the conditions of Proposition 4E (Section 3.2).
p
Remark 3. To show that the function E 7→ k|Ak|G
is not concave in general
E
for any p ∈ (0, 2] it suffices to consider two-dimensional Hilbert space H = C2 and the
operators
√
1 0
2 0
and A =
G=
.
0 0
0
1
p
G
2E/(E + 1) if E > 1.
It is easy to see that k|Ak|G
E = 1 if E ∈ (0, 1] and k|Ak|E =
6
p
The function E 7→ k|Ak|G
is not concave in general for any p ∈ (0, 2] by Remark 3 below;
E
7
3.2
Basic properties of the operator E -norms
In the following proposition we collect properties of the operator E -norms used below.
Proposition 4. Let G be a positive densely defined operator on a Hilbert space H
satisfying condition (2) and E > 0.
p
G
∗ G
G
A) kAkG
kA∗ AkG
E = k|A|kE ≤
E for all A ∈ B(H) but kA kE 6= kAkE in general;
B) For arbitrary operators A and B in B(H) the following inequalities hold
G
G
m(A)kBkG
E ≤ kABkE ≤ kAkkBkE ,
√
where m(A) is the infimum of the spectrum of the operator |A| = A∗ A.
C) For arbitrary operators A and B in B(H) such that hAϕ|Bϕi = 0 for any ϕ ∈ H
the following inequalities hold
q
2
G
G
G 2
max kAkG
,
kBk
≤
kA
+
Bk
≤
[kAkG
E
E
E
E ] + [kBkE ] .
.
D) For any operator ρ in T+,1(H) with finite Eρ = TrGρ and arbitrary operators A
and B in B(H) the following inequalities hold
G
|TrAρB ∗ | ≤ kAρB ∗ k1 ≤ kAkG
Eρ kBkEρ .
E) For any 2-positive map Φ : B(H) → B(H) such that Φ(IH ) ≤ IH having the predual
.
map7 Φ∗ : T(H) → T(H) with finite YΦ (E) = sup{TrGΦ∗ (ρ) | ρ ∈ S(H), TrGρ ≤ E }
and arbitrary operator A in B(H) the following inequalities hold 8
p
p
kΦ(IH )k kAkG
kΦ(IH )kKΦ kAkG
KΦ = max{1, YΦ (E)/E}.
kΦ(A)kG
YΦ (E) ≤
E,
E ≤
Proposition 4 shows that the linear transformations A 7→ BA and A 7→ Φ(A) of
B(H), where B ∈ B(H) and Φ is a map with the properties pointed in part E, are
bounded operators w.r.t. the norm k·kG
E (in contrast to the transformation A 7→ AB).
G
G
G
p Proof. A) The equality kAkE = k|A|kE is obvious. The inequality kAkE ≤
kA∗ AkG
E follows from the operator Cauchy-Schwarz inequality
[TrA∗ Aρ]2 ≤ [Tr[A∗ A]2 ρ][Trρ].
To show that kA∗ kG
with kAkG
E may not coincide p
E take any operator G having form
G
(1). It is easy to see that k|τ0 ihτn |kE = E/En while k|τn ihτ0 |kG
E = 1 for all E > 0.
B) This assertion follows directly from definition (4) of the operator E -norm.
C) This assertion follows directly from definition (3) of the operator E -norm.
7
The map Φ∗ is defined by the relation TrΦ(A)ρ = TrAΦ∗ (ρ) for all A ∈ B(H) and ρ ∈ T(H).
Existence of Φ∗ is equivalent to normality of the map Φ, which means that Φ(supλ Aλ ) = supλ Φ(Aλ )
for any increasing net Aλ of positive operators in B(H) [4].
8
If G is a Hamiltonian of a quantum system described by the space H and Φ is a quantum channel
(in the Heisenberg picture) then YΦ (E)/E is the energy amplification factor of Φ.
8
D) The first inequality is obvious. Let U be the partial isometry from the polar
decomposition of AρB ∗ , i.e. AρB ∗ = U|AρB ∗ |. By using the operator Cauchy-Schwarz
inequality we obtain
kTrAρB ∗ k21 = [TrU ∗AρB ∗ ]2 ≤ [TrUU ∗AρA∗ ][TrBρB ∗ ] ≤ [TrAρA∗ ][TrBρB ∗ ],
where that last inequality is due to the fact that hUU ∗ ≤ IH . By
i Proposition 3A the
2
G
right hand side of this inequality does not exceed kAkG
Eρ kBkEρ .
E) By Kadison’s inequality and Proposition 3A we have
2
Tr[Φ(A)]∗ Φ(A)ρ ≤ kΦ(IH )kTrΦ(A∗ A)ρ = kΦ(IH )kTrA∗ AΦ∗ (ρ) ≤ kΦ(IH )k kAkG
YΦ (E)
for any A ∈ B(H) and any ρ ∈ S(H) such that TrGρ ≤ E (since the condition
Φ(IH ) ≤ IH guarantees that Φ∗ (ρ) ∈ T+,1(H)). This implies the first inequality. The
second inequality follows from (5). 3.3
Properties of the E -norms related to tensor products
If G1 and G2 are positive densely defined operators on Hilbert spaces H1 and H2
satisfying condition (2) then G12 = G1 ⊗ I2 + I1 ⊗ G2 is an operator on the Hilbert
space H12 = H1 ⊗ H2 with the same properties.910 The following proposition contains
several estimates for the operator E -norms of product operators used in Section 4.
Proposition 5. Let G1 , G2 and G12 be the operators described above.
A) For arbitrary operator A in B(H1 ) the following equalities hold
G1 ⊗ I 2
12
1
kA ⊗ I2 kG
= kA ⊗ I2 kE
= kAkG
E
E
B) For arbitrary operators A ∈ B(H1 ) and B ∈ B(H2 ) the following inequalities hold
q
G2
G12
G2
1
1
∗
sup kAkG
kBk
≤
kA
⊗
Bk
≤
sup
kA∗ AkG
(13)
x kB BkE−x ,
x
E−x
E
x∈(0,E)
x∈(0,E)
and
G1
G2
12
.
kA ⊗ BkG
E ≤ min kAkE kBk, kAkkBkE
(14)
Note that the lower and upper bounds in (13) and the r.h.s. of (14) tend to
12
kAkkBk = kA ⊗ Bk = limE→+∞ kA ⊗ BkG
as E → +∞.
E
2
Proof. A) It suffices to note that Tr[|A| ⊗ I2 ]ρ12 = Tr|A|2 ρ1 and that TrG12 ρ12 =
TrG1 ρ1 + TrG2 ρ2 for any state ρ12 ∈ S(H12 ), where ρ1 = TrH2 ρ12 and ρ2 = TrH1 ρ12
are the partial states of ρ12 .
9
If G1 and G2 are Hamiltonians of quantum systems 1 and 2 described by the spaces H1 and H2
then G12 is the Hamiltonian of the composite quantum system 12 [6].
10
Here and in what follows we write IX instead of IHX (where X = 1, 2, A, B, ..) to simplify notations.
9
B) For each xp∈ (0, E) and any ε > 0 there exist p
states ρ1 in S(H1 ) and ρ2 in
G1
2
2
S(H2 ) such that Tr|A| ρ1 > kAkx − ε, TrG1 ρ1 ≤ x, Tr|B|2 ρ2 > kBkG
E−x − ε and
p
TrG2 ρ2 ≤ E −x. Then TrG12 [ρ1 ⊗ρ2 ] ≤ x+E −x = E and Tr[|A|2 ⊗ |B|2 ][ρ1 ⊗ ρ2 ] ≥
G2
1
[kAkG
x − ε][kBkE−x − ε]. Since ε is arbitrary, this implies the left inequality in (13).
By the operator Cauchy-Schwarz inequality for any state ρ12 in S(H12 ) we have
p
p
Tr[|A|2 ⊗ |B|2 ]ρ12 ≤ Tr[|A|4 ⊗ I2 ]ρ12 Tr[I1 ⊗ |B|4 ]ρ12
=
p
Tr|A|4 ρ1
p
2 G2
1
Tr|B|4 ρ2 ≤ k|A|2 kG
TrG1 ρ1 k|B| kTrG2 ρ2 .
Since TrG12 ρ12 = TrG1 ρ1 + TrG2 ρ2 , this implies the right inequality in (13).
To prove inequality (14) it suffices to note that
A ⊗ B = [A ⊗ I2 ][I1 ⊗ B] = [I1 ⊗ B][A ⊗ I2 ]
and to apply Proposition 4B and part A of this proposition. 4
The E -version of the Kretschmann-SchlingemannWerner theorem
In this section we consider application of the operator E-norms to the theory of completely positive (CP) linear maps between Banach spaces of trace class operators on
separable Hilbert spaces (the Schatten classes of order 1). Since T(H)∗ = B(H), the
below results can be reformulated in terms of CP linear maps between algebras of all
bounded operators on separable Hilbert spaces. Nevertheless, the use of the ”predual
picture” is more natural for representation of our results. The theory of CP linear
maps between Banach spaces of trace class operators has important applications in
mathematical physics, in particular, in the theory of open quantum systems, where CP
trace-preserving linear maps called quantum channels play the role of dynamical maps
(in the Schrodinger picture), while CP trace-non-increasing linear maps called quantum
operations are essentially used in the theory of quantum measurements [6, 25, 27].
For a CP linear map Φ : T(HA ) → T(HB ) the Stinespring theorem (cf.[23]) implies
existence of a Hilbert space HE and an operator VΦ : HA → HB ⊗ HE such that
Φ(ρ) = TrE VΦ ρVΦ∗ ,
ρ ∈ T(HA ),
(15)
where TrE denotes the partial trace over HE . If Φ is trace-preserving (correspondingly,
trace-non-increasing) then VΦ is an isometry (correspondingly, contraction) [6, Ch.6].
The dual CP linear map Φ∗ : B(HB ) → B(HA ) has the corresponding representation
Φ∗ (B) = VΦ∗ [B ⊗ IE ]VΦ , B ∈ B(HB ).
(16)
10
The norm of complete boundedness (cb-norm in what follows) of a linear map
between the algebras B(HB ) and B(HA ) (cf. [10]) induces (by duality) the norm
.
kΦkcb =
sup
ρ∈T(HAR ),kρk1 ≤1
kΦ ⊗ IdR (ρ)k1
(17)
on the set of all linear maps between Banach spaces T(HA ) and T(HB ), where HR is
a separable Hilbert space and HAR = HA ⊗ HR . If Φ is a Hermitian preserving map
then the supremum in (17) can be taken over the set S(HAR ) [25, Ch.3].
The Kretschmann-Schlingemann-Werner theorem (the KSW-theorem in what follows) obtained in [8] states that
p
p
kΦ − Ψkcb
p
≤ inf kVΦ − VΨ k ≤ kΦ − Ψkcb ,
kΦkcb + kΨkcb VΦ ,VΨ
where the infimum is over all common Stinespring representations
Φ(ρ) = TrE VΦ ρVΦ∗
and Ψ(ρ) = TrE VΨ ρVΨ∗ .
(18)
In the proof of the KSW theorem it is shown that the quantity inf VΦ ,VΨ kVΦ − VΨ k
coincides with the Bures distance between the maps Φ and Ψ defined by the expression
β(Φ, Ψ) =
sup
ρ∈S(HAR )
β(Φ ⊗ IdR (ρ), Ψ ⊗ IdR (ρ)) ,
(19)
in which HR is a separable Hilbert space and β(·, ·) in the r.h.s. is the Bures distance
between operators in T+ (HBR ) defined as
q
p
(20)
β(ρ, σ) = kρk1 + kσk1 − 2 F (ρ, σ),
where
√ √
F (ρ, σ) = k ρ σk21
(21)
is the fidelity of the operators ρ and σ [6, 25, 27]. The Bures distance between CP
linear maps Φ and Ψ is connected to the operational fidelity of these maps introduced
in [2].
The KSW theorem shows continuity of the map VΦ 7→ Φ and selective continuity of
the multi-valued map Φ 7→ VΦ with respect to the cb-norm topology on the set F(A, B)
of all CP linear maps Φ from T(HA ) to T(HB ) and the operator norm topology on the
set of Stinespring operators VΦ .
The cb-norm topology is widely used in the quantum theory, by it is too strong for
description of physical perturbations of infinite-dimensional quantum channels [17, 29].
Our aim is to obtain a version of the KSW theorem which would show continuity of the
map VΦ 7→ Φ and selective continuity of the multi-valued map Φ 7→ VΦ with respect
to weaker topologies on the sets of CP linear maps Φ and Stinespring operators VΦ . A
natural way to do this is to use the operator E -norms induced by some positive operator
11
G on HA (naturally generalized to operators between different separable Hilbert spaces,
see Remark 1) and the energy-constrained cb-norms
.
kΦkG
cb,E =
sup
ρ∈S(HAR ):TrGρA ≤E
kΦ ⊗ IdR (ρ)k1 ,
E > 0,
.
(where ρA = TrR ρ)
(22)
on the set of Hermitian-preserving linear maps from T(HA ) to T(HB ) introduced independently in [17] and [29] (the positive operator G is treated therein as a Hamiltonian
of a quantum system A).11 If G is a discrete unbounded operator (see Def.1) then
the topology generated by any of the norms (22) on bounded subsets of F(A, B) coincides with the strong convergence topology generated by the family of seminorms
Φ 7→ kΦ(ρ)k1 , ρ ∈ T(HA ) [17, Proposition 3].12
Following [15] introduce the energy-constrained Bures distance
βEG (Φ, Ψ) =
sup
ρ∈S(HAR ):TrGρA ≤E
β(Φ ⊗ IdR (ρ), Ψ ⊗ IdR (ρ)),
E > 0,
(23)
between CP linear maps Φ and Ψ from T(HA ) to T(HB ), where β(·, ·) in the r.h.s. is
the Bures distance between operators in T+ (HBR ) defined in (20) and HR is a separable
Hilbert space.
Remark 4. The infimum in (23) can be taken only over pure states ρ ∈ S(HAR ).
This follows from the freedom of choice of R, which implies possibility to purify any
mixed state in S(HAR ) by extending system R. We have only to note that the Bures
distance between operators in T+ (HXY ) defined in (20) does not increase under partial
trace: β(ρ, σ) ≥ β(ρX , σX ) for any ρ and σ in T+ (HXY ) [6, 25, 27].
The distance βEG (Φ, Ψ) turns out to be extremely useful in quantitative continuity
analysis of capacities of energy-constrained infinite-dimensional quantum channels [15,
Theorem 2]. By using the well known relations between the trace norm and the Bures
distance (20) one can show that for any E > 0 the distance βEG (Φ, Ψ) generates the
same topology on bounded subsets of F(A, B) as any of the energy-constrained cbnorms (22). The results of calculation of βEG (Φ, Ψ) for real quantum channels can be
found in [9].
Now we can formulate the E-version of KSW-theorem.
Theorem 1. Let G be a positive semidefinite densely defined operator on HA satG
isfying condition (2) and E > 0. Let k · kG
cb,E and k · kE be, respectively, the energyconstrained cb-norm and the operator E-norm induced by G. For any CP linear maps
Φ and Ψ from T(HA ) to T(HB ) the following inequalities hold
11
q
kΦ − ΨkG
qcb,E
q
≤
≤ inf kVΦ − VΨ kG
kΦ − ΨkG
E
cb,E ,
VΦ ,VΨ
G
G
kΦkcb,E + kΨkcb,E
(24)
Slightly different energy-constrained cb -norm is used in [11].
This topology is a restriction to the set F(A, B) of the strong operator topology on the set of all
linear maps from T(HA ) to T(HB ). The strong convergence of a sequence {Φn } ⊂ F(A, B) to a map
Φ0 means that limn→∞ Φn (ρ) = Φ0 (ρ) for all ρ ∈ T(HA ).
12
12
where the infimum is over all common Stinespring representation (18). The quantity
G
inf VΦ ,VΨ kVΦ − VΨ kG
E coincides with the energy-constrained Bures distance βE (Φ, Ψ) defined in (23). The infimum in (24) is attainable.
Proof. We will follow the proof of the KSW theorem (given in [8]) with necessary
modifications concerning the use of the energy-constrained cb-norms and the operator
E-norms (instead of the ordinary cb-norm and the operator norm).
To prove the first inequality in (24) assume that ρ is a state in S(HAR ) such that
TrGρA ≤ E. For a given common Stinespring representation (18) we have
k(Φ − Ψ) ⊗ IdR (ρ)k1 ≤ kVΦ ⊗ IR · ρ · VΦ∗ ⊗ IR − VΨ ⊗ IR · ρ · VΨ∗ ⊗ IR k1
≤ k(VΦ − VΨ ) ⊗ IR · ρ · VΦ∗ ⊗ IR k1 + kVΨ ⊗ IR · ρ · (VΦ∗ − VΨ∗ ) ⊗ IR k1
R
R
R
R
≤ k(VΦ − VΨ ) ⊗ IR kG⊗I
kVΦ ⊗ IR kG⊗I
+ k(VΦ − VΨ ) ⊗ IR kG⊗I
kVΨ ⊗ IR kG⊗I
E
E
E
E
G
G
G
≤ kVΦ − VΨ kG
E kVΦ kE + kVΦ − VΨ kE kVΨ kE .
The first and the second inequalities follow from the properties of the trace norm
(the non-increasing under partial trace and the triangle inequality), the third inequality follows from Proposition 4D, the last one – from Proposition 5A. By noting that
2
G
G 2
G
[kVΦ kG
in (24).
E ] = kΦkcb,E and [kVΨ kE ] = kΨkcb,E we obtain the first inequality
q
To prove the second inequality in (24) note that βEG (Φ, Ψ) ≤ kΦ − ΨkG
cb,E . This
p
follows from the inequality β(ρ, σ) ≤ kρ − σk1 valid
√ for any ρ and σ in T+ (H), which
√
is easily proved by using the inequality Tr( ρ − σ)2 ≤ kρ − σk1 (see the proof of
Lemma 9.2.3 in [6]). So, it suffices to show that
G
inf kVΦ − VΨ kG
E = βE (Φ, Ψ).
VΦ ,VΨ
(25)
G
Denote by αE
(Φ, Ψ) the l.h.s. of (25). Let CsG,E
S be∗ the subset of S(HA ) determined
by the inequality TrGρ ≤ E and N (Φ, Ψ) = VΦ VΨ , where the union is over all
common Stinespring representations (18). Then by using definition (4) we obtain
p
G
αE
(Φ, Ψ) = inf
sup
TrΦ(ρ) + TrΨ(ρ) − 2ℜTrNρ.
(26)
N ∈N (Φ,Ψ) ρ∈Cs
G,E
Following the proof of Theorem 1 in [8] show that N (Φ, Ψ) coincides with the set
.
M(Φ, Ψ) = {VΦ∗ (IB ⊗ C)VΨ | C ∈ B(HE ), kCk ≤ 1} ,
defined via some fixed common Stinespring representation (18). It will imply, in particular, that M(Φ, Ψ) does not depend on this representation.
To show that M(Φ, Ψ) ⊆ N (Φ, Ψ) it suffices to find for any contraction C ∈ B(HE )
a common Stinespring representation for Φ and Ψ with the operators ṼΦ and ṼΨ from
HA to HB ⊗ HẼ such that ṼΦ∗ ṼΨ = VΦ∗ (IB ⊗ C)VΨ .
13
1
2
1
2
Let HẼ = HE
⊕ HE
, where HE
and HE
are copies of HE . For given C define the
C
operators ṼΦ and ṼΨ from HA into HB ⊗ (HE1 ⊕ HE2 ) = HB ⊗ HE1 ⊕ HB ⊗ HE2 by
setting
p
ṼΦ |ϕi = VΦ |ϕi ⊕ |0i, ṼΨC |ϕi = (IB ⊗ C)VΨ |ϕi ⊕ IB ⊗ IE − C ∗ C VΨ |ϕi (27)
1
for any ϕ ∈ HA , where we assume that the operators VΦ and VΨ act from HA to HB ⊗HE
2
2
1
and HB ⊗ HE correspondingly, while the contraction C acts from HE to HE . It is easy
to see that the operators ṼΦ and ṼΨC form a common Stinespring representation for the
maps Φ and Ψ with the required property.
To prove that N (Φ, Ψ) ⊆ M(Φ, Ψ) take any common Stinespring representation
for the maps Φ and Ψ with the operators ṼΦ and ṼΨ from HA to HB ⊗ HẼ . By
Theorem 6.2.2 in [6] there exist partial isometries WΦ and WΨ from HE to HẼ such that
ṼΦ = (IB ⊗WΦ )VΦ and ṼΨ = (IB ⊗WΨ )VΨ . So, ṼΦ∗ ṼΨ = VΦ∗ (IB ⊗WΦ∗ WΨ )VΨ ∈ M(Φ, Ψ),
since kWΦ∗ WΨ k ≤ 1.
Since N (Φ, Ψ) = M(Φ, Ψ), the infimum in (26) can be taken over the set M(Φ, Ψ).
This implies
q
G
αE (Φ, Ψ) =
inf
sup
TrΦ(ρ) + TrΨ(ρ) − 2ℜTrVΦ∗ (IB ⊗ C)VΨ ρ
C∈B1 (HE ) ρ∈Cs
G,E
q
TrΦ(ρ) + TrΨ(ρ) − 2ℜTrVΦ∗ (IB ⊗ C)VΨ ρ
inf
= sup
(28)
s
C∈B
(H
)
1
E
ρ∈CG,E
r
TrΦ(ρ) + TrΨ(ρ) − 2 sup |TrVΦ∗ (IB ⊗ C)VΨ ρ|,
= sup
ρ∈CsG,E
C∈B1 (HE )
where the possibility to change the order of the optimization follows from Ky Fan’s
minimax theorem [22] and the σ-weak compactness of the unit ball B1 (HE ) of B(HE )
[4]. It is easy to see that
sup
C∈B1 (HE )
|TrVΦ∗ (IB ⊗ C)VΨ ρ| =
sup
C∈B1 (HE )
|hVΦ ⊗ IR ϕ|IBR ⊗ C|VΨ ⊗ IR ϕi|,
(29)
where ϕ is a purification of ρ, i.e. a vector in HA ⊗ HR such that TrR |ϕihϕ| = ρ.
Since for any common Stinespring representation (18) and any purification ϕ of a
state ρ the vectors VΦ ⊗IR |ϕi and VΨ ⊗IR |ϕi in HBER are purifications of the operators
Φ ⊗ IdR (|ϕihϕ|) and Ψ ⊗ IdR (|ϕihϕ|) in T(HBR ), by using the relation N (Φ, Ψ) =
M(Φ, Ψ) proved before and Uhlmann’s theorem [24, 27] it is easy to show that the
square of the r.h.s. of (29) coincides with the fidelity of these operators defined in (21).
Note also that TrΦ⊗IdR (σ) = TrΦ(σA ) and TrΨ⊗IdR (σ) = TrΨ(σA ) for any state σ in
G
S(HAR ). By Remark 4 these observations and (28) imply that αE
(Φ, Ψ) = βEG (Φ, Ψ),
i.e. that (25) holds.
The last assertion can be derived from the attainability of the infimum in the first
line in (28) which follows from the σ-weak compactness of the unit ball B1 (HE ). Theorem 1 shows continuity of the map VΦ 7→ Φ and selective continuity of the
multi-valued map Φ 7→ VΦ with respect to the energy-constrained cb-norm on the set
14
of CP linear maps Φ and the operator E -norm on the set of Stinespring operators VΦ .
Its basic assertion is the equality
βEG (Φ, Ψ) = inf kVΦ − VΨ kG
E.
(30)
VΦ ,VΨ
Some difficulty of applying Theorem 1 is related to the fact that the infimum in (30) is
over all common Stinespring representation (18). But by using the constructions from
the proof of this theorem one can obtain its versions which are more convenient for
applications, in particular, for analysis of converging sequences of CP linear maps.
Theorem 2. Let G be a positive semidefinite densely defined operator on HA satisfying condition (2), βEG and k·kG
E be, respectively, the energy-constrained Bures distance
and the operator E-norm induced by G. Let Φ be a CP linear map from T(HA ) to
T(HB ).
A) There is a Stinespring representation of Φ with the operator VΦ′ : HA → HB ⊗ HE ′
such that
(31)
βEG (Φ, Ψ) = inf kVΦ′ − VΨ kG
E,
VΨ
for any CP linear map Ψ : T(HA ) → T(HB ), where the infimum is over all Stinespring
representations of Ψ with the same environment space HE ′ . The infimum in (31) is
attainable.
B) If G is an unbounded discrete operator (Def.1) and VΦ : HA → HB ⊗ HE is the
operator from a given Stinespring representation of Φ such that dim HE = +∞ then
G
βEG (Φ, Ψ) ≤ inf kVΦ − VΨ kG
E ≤ 2βE (Φ, Ψ),
VΨ
for any CP linear map Ψ : T(HA ) → T(HB ), where the infimum is over all Stinespring
representations of Ψ with the same environment space HE .
Proof. If VΦ : HA → HB ⊗ HE is the operator from a Stinespring representation of
Φ such that dim HE = +∞ then, since any separable Hilbert space can be isometrically
embedded into HE , we may assume that any CP linear map Ψ : T(HA ) → T(HB ) has
a Stinespring representation with the same environment space HE . Denote by VΨ the
Stinespring operator of Ψ in this representation. Let ṼΦ and ṼΨC be the operators from
1
2
1
2
HA into HB ⊗ (HE
⊕ HE
) = (HB ⊗ HE
) ⊕ (HB ⊗ HE
) defined by formulae (27), where
1
2
HE and HE are copies of HE and C is a contraction in B(HE ). The arguments from
the proof of Theorem 1 show that βEG (Ψ, Φ) = kṼΨC0 − ṼΦ kG
E for some C0 ∈ B(HE )
depending on Φ and Ψ. So, to obtain assertion A it suffices to take ṼΦ in the role of
VΦ′ .
To prove assertion B we will use the above operators ṼΦ and ṼΨC0 as follows. Assume
first that the operator C0 is nondegenerate, i.e. ker C0 = {0}. Let U be the isometry
G
from the polar decomposition of C0 , i.e. C0 = U|C0 |. Since kṼΨC0 − ṼΦ kG
E = βE (Ψ, Φ),
it follows from Proposition 4C that
p
G
G
G
2
IB ⊗ IE − |C0 | VΨ ≤ βEG (Ψ, Φ) (32)
k(IB ⊗ C0 )VΨ − VΦ kE ≤ βE (Ψ, Φ) and
E
15
Hence the triangle inequality and Proposition 4B imply that
G
k(IB ⊗ U)VΨ − VΦ kG
E ≤ k(IB ⊗ C0 )VΨ − VΦ kE
(33)
G
G
+k(IB ⊗ C0 )VΨ − (IB ⊗ U)VΨ kG
E ≤ βE (Ψ, Φ) + kIB ⊗ (IE − |C0 |)VΨ kE .
Since C0 is a contraction, by using Proposition 4B and the second inequality in (32)
we obtain
p
2
G
G
kIB ⊗(IE −|C0 |)VΨ kG
≤
kI
⊗(I
−|C
|
)V
k
≤
kI
⊗
IE − |C0 |2 VΨ kG
B
E
0
Ψ
B
E
E
E ≤ βE (Ψ, Φ)
G
Thus, it follows from (33) that k(IB ⊗ U)VΨ − VΦ kG
E ≤ 2βE (Ψ, Φ). Since U is an
isometry, (IB ⊗ U)VΨ is a Stinespring operator for Ψ.
Since G is a discrete unbounded operator on HA , the set CsG,E of states ρ in S(HA )
such that TrGρ ≤ E is compact by Lemma 1. Using this and taking into account the
continuity of the expression under the square root in the first line in (28) as a function
on the Cartesian product of the set CsG,E and the set B1 (HE ) equipped with the weak
operator topology, it is easy to show that the first infimum in (28) can be taken over
the dense subset of B1 (HE ) consisting of non-degenerate operators. This allows to
omit the assumption ker C0 = {0}. If {Vn } is a sequence of operators from HA to HB ⊗ HE converging to an operator
V0 : HA → HB ⊗ HE w.r.t. the norm k·kG
E then the first inequality in (24) implies that
the sequence of CP maps Φn (ρ) = TrE Vn ρVn∗ converges to the map Φ0 (ρ) = TrE V0 ρV0∗
w.r.t. the norm k·kG
cb,E and for each n the following inequalities hold
q
hq
i
G
G
G
G
G
G
kΦn −Φ0 kG
≤
β
(Φ
,
Φ
)
+
kΦ
k
kΦ
k
n
0
n cb,E
0 cb,E ≤ kVn −V0 kE kVn kE + kV0 kE .
cb,E
E
Theorem 2 allows to describe all sequences of CP linear maps converging w.r.t. the
energy-constrained cb-norm.
Corollary 1. Let {Φn } be a sequence of CP linear maps from T(HA ) to T(HB )
converging to a CP linear map Φ0 with respect to the norm k·kG
cb,E .
A) There exist a separable Hilbert space HE ′ and a sequence {Vn } of operators from
HA into HB ⊗ HE ′ converging to an operator V0 with respect to the norm k·kG
E such
that Φn (ρ) = TrE ′ Vn ρVn∗ for all n ≥ 0 and
q
G
kVn − V0 kG
=
β
(Φ
,
Φ
)
≤
kΦn − Φ0 kG
n
0
E
E
cb,E .
B) If G is an unbounded discrete operator (Def.1) and V0 : HA → HB ⊗ HE is the
operator from a given Stinespring representation of the map Φ0 such that dim HE =
+∞, then for any ε > 0 there exists a sequence {Vn } of operators from HA into
HB ⊗HE converging to the operator V0 with respect to the norm k·kG
E such that Φn (ρ) =
∗
TrE Vn ρVn for all n > 0 and
q
G
G
(34)
kVn − V0 kE ≤ 2βE (Φn , Φ0 ) + ε ≤ 2 kΦn − Φ0 kG
cb,E + ε.
16
Factor ”2” in (34) is a cost of the possibility to take the sequence {Vn } of Stinespring
operators representing the sequence {Φn } for given HE and V0 : HA → HB ⊗ HE .
If the operator G is discrete and unbounded (Def.1) then the norm k · kG
cb,E
generates the strong convergence topology on bounded subsets of the set F(A, B) of all
CP linear maps from T(HA ) to T(HB ) (by Proposition 3 in [17]), while the norm k·kG
E
generates the strong operator topology on subsets of linear maps from HA to HB ⊗ HE
bounded by the operator norm (by Proposition 2B). Thus, in this case Corollary 1 gives
representation of bounded strongly converging sequences of CP linear maps via strongly
converging sequence of Stinespring operators. For sequences of quantum channels such
representation is obtained in [16].
5
Operator E -norms for unbounded operators
In this section we will extend the operator E -norms to unbounded operators. We
will assume that G is a positive semidefinite unbounded 13 operator on H with dense
domain satisfying condition (2). The case of discrete type operator G will be considered
separately after formulations of general results.
Speaking about extension of the operator E -norms to unbounded operators we √
may
restrict attention to linear operators √
on H relatively bounded w.r.t. the operator G,
i.e. linear operators A defined on D( G) such that
√
√
kAϕk2 ≤ a2 kϕk2 + b2 k Gϕk2 , ∀ϕ ∈ D( G),
(35)
for some nonnegative numbers a √
and b (depending on A but not depending on ϕ)[7].
Such operators are briefly
called
G-bounded. Indeed, it is easy to see that the r.h.s.
√
of (3) is finite for any G-bounded operator A and all E > 0. The following lemma
contains the converse statement (in strengthened form).
√
Lemma 3. Let A be a linear operator on H such that D(A) ⊇ D( G). If the
quantity kAkG
E defined in (3) is finite for some E > 0 then
2
• the function E 7→ kAkG
is finite and concave on R+ ;
E
√
• the operator A is G-bounded.
.
Proof. Consider the set SGf = {ρ ∈ S(H) | TrGρ < +∞, rankρ < +∞}. A state ρ
belongs to this set if and only if it has a finite decomposition
X
ρ=
|ϕi ihϕi |,
(36)
i
√
where {ϕi } is a set of vectors
in D( G). Moreover, any such decomposition of ρ
√
consists of vectors in D( G).
13
2A.
If G is a bounded operator then the norm k·kG
E is equivalent to the operator norm by Proposition
17
For any state ρ in SGf with representation (36) define the operator AρA∗ as follows
. X
AρA∗ =
|αi ihαi |, where |αi i = A|ϕi i.
(37)
i
By using Schrodinger’s mixture theorem (see [3, Ch.8]) it is easy to show that the r.h.s.
of (37) does not depend on representation (36). This implies that
. X
ρ 7→ TrAρA∗ =
kAϕi k2
(38)
i
is an affine function on SGf .
Thus, to prove the first assertion of the lemma it suffices to show that
p
TrAρA∗
kAkG
=
sup
E
(39)
f :TrGρ≤E
ρ∈SG
for any E > 0. This equality means that the supremum in the r.h.s. of (39) can be
taken only over pure states ρ in SGf such that TrGρ ≤ E. Since the function (38) is
affine, this can be shown easily by using the fact that all the extreme points of the
convex set of states ρ such that TrGρ ≤ E are pure states [26].
The second assertion
ofGthe
2 lemma is derived from the first one, since the concavity
of the function E 7→ kAkE implies existence of numbers a and b such that
2
2
2
[kAkG
E] ≤ a + b E
∀E > 0. G
Thus, in what follows
√ we will consider the operator E-norm k·kE defined by formula
(3) on the set of all G-bounded operators. Below we will show that the quantity
∗
TrAρA
can be defined correctly (without using the notion of adjoint operator) for any
√
G-bounded operator A and any state ρ with finite TrGρ (not only for a finite rank
state as in the√proof of Lemma 3). This will allows to show that the operator E-norm
G-bounded operator A can be also defined by formula (4).
kAkG
E of any
Denote by Π√G (A) the set of all pairs (a, b) for which (35) holds. It is easy to see
√
that Π√G (A) is a closed subset of R2+ . The G-bound of A (denoted by b√G (A) in
what follows) is defined as
b√G (A) = inf b | (a, b) ∈ Π√G (A) .
√
If b√G (A) = 0 then A is called G-infinitesimal operator (infinitesimally bounded
√
w.r.t. G). These notions are widely used in the modern operator theory, in particular,
in analysis of perturbations of unbounded operators on a Hilbert space [7, 14, 21].
We will use the following simple lemmas.14
14
I would be grateful for direct references to these results.
18
√
Lemma 4. If A is a G-bounded operator on H then
√ for any separable Hilbert
space K the operator A⊗IK naturally defined
√ on the set D( G)⊗K has a unique linear
√
G ⊗ IK -bounded extension to the set D( G ⊗ IK ).15 This extension (also denoted by
A ⊗ IK ) has the following property
!
X
X
A ⊗ IK
|ϕi i ⊗ |ψi i =
A|ϕi i ⊗ |ψi i
(40)
i
i
√
P √
2
for
any
countable
sets
{ϕ
}
⊂
D(
G)
and
{ψ
}
⊂
K
such
that
i
i
i k Gϕi k < +∞,
P
2
√
√
i kϕi k < +∞ and hψi |ψj i = δij , which implies that Π G⊗IK (A ⊗ IK ) = Π G (A).
√
√
Proof. For any E > 0 the linear spaces D( G) and D( G ⊗ IK ) equipped, respectively, with the inner products
hϕ|ψiG
E = hϕ|ψi + hϕ|G|ψi/E
K
and hη|θiG⊗I
= hη|θi + hη|G ⊗ IK |θi/E
E
G
are Hilbert spaces [13]. Denote the first space by HE
. Then it is easy to see that
G
the second space coincides with the Hilbert space HE ⊗ K. Since the operator
√ A is
G
bounded as an operator from HE into H the operator A ⊗ IK defined on D( G) ⊗ K
G
is uniquely extended to√a bounded operator from HE
⊗ K into√H ⊗ K. Since the linear
G
spaces HE
⊗ K and D( G ⊗ IK ) coincide, this extension is a G ⊗ IK -bounded linear
operator on H ⊗ K.
G
Property (40) follows from continuity of the operator A ⊗ IK : HE
⊗ K → H ⊗ K.
√
P
Any vector
i ⊗ |ψi i, where
√ η in D( G ⊗ IK ) can be represented as |ηi = Pi |ϕi√
G) and {ψi } ⊂ K are collections of vectors such that i k Gϕi k2 < +∞,
{ϕ
P i } ⊂ D(
2
i kϕi k < +∞ and hψi |ψj i = δij . By using property (40) we obtain
X
X
X √
√
kA ⊗ IK ηk2 =
kAϕi k2 ≤ a2
kϕi k2 + b2
k Gϕi k2 = a2 kηk2 + b2 k G ⊗ IK ηk2
i
i
i
for any (a, b) ∈ Π√G (A). This implies Π√G (A) ⊆ Π√G⊗IK (A ⊗ IK ), and hence
Π√G (A) = Π√G⊗IK (A ⊗ IK ), since the converse inclusion is obvious. Remark 5. Property (40) implies that
(A ⊗ IK )(IH ⊗ W )|ϕi = (IH ⊗ W )(A ⊗ IK )|ϕi
√
for any ϕ ∈ D( G ⊗ IK ) and a partial isometry W ∈ B(K) s.t. IH ⊗ W ∗ W |ϕi = |ϕi.
√
Lemma 5. For any G-bounded operators A and B on H the affine function
.
ρ 7→ AρB ∗ ∈ T(H) is well defined on the set T+
G = {ρ ∈ T+ (H) | TrGρ < +∞} by the
formula 16
. X
AρB ∗ =
|αi ihβi |,
|αi i = A|ϕi i, |βi i = B|ϕi i,
(41)
i
√
√
15
D( G) ⊗ K is the linear span of all the vectors ϕ ⊗ ψ, where ϕ ∈ D( G) and ψ ∈ K.
16
We define the operator AρB ∗ in such a way to avoid the notion of adjoint operator, since we make
no assumptions about closability of the operators A and B.
19
where ρ =
+
i |ϕi ihϕi | is any decomposition of ρ ∈ TG into 1-rank positive operators.
P
P
Proof. If ρ =
i } is any set of orthogonal unit
i |ϕi ihϕi | and {ψ
√ vectors in a
P
separable Hilbert space K then |ηi = i |ϕi i ⊗ |ψi i is a vector in D( G ⊗ IK ) such
that ρ = TrK |ηihη|. By Lemma 4 the operators
√ A ⊗ IK and B ⊗ IK have unique linear
√
G ⊗ IK -bounded extensions to the set D( G ⊗ IK ) satisfying (40). Hence
X
|Aϕi ihBϕi | = TrK |A ⊗ IK ηihB ⊗ IK η|.
(42)
i
So, by using the well known relation between different purifications of a given state
[6, 27] and Remark P
5, it is easy to show that the r.h.s. of (42) does not depend on the
representation ρ = i |ϕi ihϕi |. It follows that the r.h.s. of (42) correctly defines an
affine function ρ 7→ AρB ∗ on the set T+
G. +
Lemma 5 implies, in particular, that ρ 7→ AρA∗ is an affine
√ function from TG into
T+ (H) (well defined by formula (41) with B = A) for any G-bounded operator A.
Hence the r.h.s. of (4) is well defined for any such operator.
The√following proposition shows that we may also define the operator E -norm kAkG
E
of any G-bounded operator A by formula (4).
√
Proposition 6. Let A be an arbitrary G-bounded operator and E > 0.
A) The right hand sides of (3) and (4) coincide (provided that AρA∗ is defined by
formula (41) with B = A).
B) The suprema in definitions (3) and (4) can be taken, respectively, over all vectors in
H≤1 satisfying the condition hϕ|G|ϕi ≤ E and over all operators in T+,1(H) satisfying
the condition TrGρ ≤ E.
2
Proof. A) The concavity of the function E 7→ kAkG
(Lemma 3A) and the proof
E
of Proposition 3A show that the supremum in the r.h.s. of (39) can be taken over all
finite rank positive
P+∞ operators in T1,+ such that TrGρ ≤ E.
Let
Pρ = i=1 |ϕi ihϕi | be an arbitrary state in S(H) such that TrGρ ≤ E. Then
ρn = ni=1 |ϕi ihϕi | is a finite rank positive operator in T1,+ such that TrGρn ≤ E for
each n, and
lim TrAρn A∗ = TrAρA∗ ≤ +∞.
n→+∞
Thus, assertion A follows from equality (39) and the remark at the begin of the proof.
2
B) This assertion is proved by using concavity of the function E 7→ kAkG
(Lemma
E
3A) and the proof of Proposition 3A. √
By Proposition 6B for any vector ϕ in D( G) such that kϕk ≤ 1 we have
G
(43)
kAϕk ≤ kAkG
Eϕ ≤ Kϕ kAkE ,
√
p
where Eϕ = k Gϕk2 and Kϕ = max{1, Eϕ /E} . This implies the following
Lemma
√ 6. Let PE be the spectral projector of G corresponding to the interval [0,G E].
For any G-bounded operator A the operator APE is bounded and kAPE k ≤ kAkE .
20
G
√ Proof. 2It follows from (43) that kAPE ϕk ≤ kAkE for any unit vector ϕ in H, since
k GPE ϕk ≤ E and kPE ϕk ≤ 1. The following lemma shows that the set Π√G (A) is completely determined by the
2
function E 7→ kAkG
and vice versa.
E
√
a2 + b2 E
Lemma 7. A pair (a, b) belongs to the set Π√G (A) if and only if kAkG
E ≤
for all E > 0.
√
a2 + b2 E then it follows from (43) that
Proof. If kAkG
≤
E
q
√
G√
kAϕk ≤ kAkk Gϕk2 ≤ a2 + b2 k Gϕk2
√
for any unit vector ϕ in D( G). Hence (a, b) ∈ Π√G (A). If (a, b) ∈ Π√G (A) then
o √
n
√
√
sup kAϕk ϕ ∈ D( G), kϕk ≤ 1, k Gϕk2 ≤ E ≤ a2 + b2 E
√
for any E > 0. So, definition (3) implies that kAkG
a2 + b2 E. E ≤
√
Denote by BG (H) the linear space of all G-bounded operators equipped with the
norm k · kG
(we identify operators
E defined
√ by the equivalent expressions (3) and (4)
G
coinciding on D( G)). We will √
also consider the norm k|·k|E defined in (10), which is
commonly used on the space of G-bounded operators.
Theorem 3. Let G be a positive semidefinite unbounded densely defined operator
on H satisfying condition (2).
G
A) BG (H) is a nonseparable Banach space. The norms k · kG
E and k| · k|E satisfy the
equivalence relations (5), (11) and (12) on BG (H) . For any A ∈ BG (H) and E > 0
the following expressions hold
p
√
G
G
k|Ak|G
kAkG
1 + 1/t.
E = sup kAktE / 1 + t,
E = inf k|Ak|tE
t>0
t>0
B) If A ∈ BG (H) and E > 0 then
n√
o
2 + b2 E (a, b) ∈ Π√ (A)
a
kAkG
=
inf
E
G
√
E.
and b√G (A) = lim kAkG
/
E
E→+∞
The limit in the last formula can be replaced by the infimum over all E > 0.
21
C) The completion of B(H) w.r.t. any of the norms√k · kG
E , E > 0, coincides with
0
the closed subspace√BG (H) of BG (H) consisting of all G-infinitesimal operators, i.e.
operators with the G-bound equal to 0. An operator A belongs to BG0 (H) if and only
if
√
kAkG
as E → +∞.
(44)
E = o( E)
If G is a discrete operator (Def.1) then the Banach space BG0 (H) is separable.
D) Any ball in B(H) is complete with respect to any of the norms k·kG
E , E > 0. An
G
operator A belongs to B(H) if and only if the function E 7→ kAkE is bounded. In this
G
case kAk = supE>0 kAkG
E = limE→+∞ kAkE .
√
E) The G-bound is a continuous seminorm on BG (H). Quantitatively,
√
(45)
b√G (A) − b√G (B) ≤ b√G (A − B) ≤ kA − BkG
E/ E
for arbitrary A, B in BG (H) and any E > 0.
17
K
F) If K is a separable Hilbert space then kA ⊗ IK kG⊗I
= kAkG
E for any A ∈ BG (H)
E
√
G) For arbitrary G-bounded operators A and B and any operator ρ in T+ (H) such
.
that Trρ ≤ 1 and Eρ = TrGρ < +∞ the following inequalities hold
G
|TrAρB ∗ | ≤ kAρB ∗ k1 ≤ kAkG
Eρ kBkEρ ,
where AρB ∗ is the trace class operator defined in (41).
H) For any A in BG (H) and E > 0 the suprema in definitions (3) and (4) can be taken,
respectively, over all unit vectors in H satisfying the condition hϕ|G|ϕi = E and over
all states in S(H) satisfying the condition TrGρ = E.
I) If the operator G is discrete (Def.1) then for any A ∈ B0G (H) and E > 0 the suprema
in (3) and (4) are attainable. Moreover, if A is a unbounded operator in B0G (H) or
kAkG
E < kAk < +∞ then the suprema in (3) and (4) are attained, respectively, at
unit vector ϕ0 in H such that hϕ0 |G|ϕ0 i = E and at a state ρ0 in S(H) such that
TrGρ0 = E.
Remark 6. The expressions in Theorem 3A show that the functions√E 7→ kAkG
E
G-bounded
and E 7→ k|Ak|G
for
arbitrary
E are completely determined by each other
√
G
operator A. So, if kAkG
G-bounded operators A and
E = kBkE for all E > 0 for some
G
G
18
B then k|Ak|E = k|Bk|E for all E > 0 and vise versa. These expressions mean that
2
G 2
the concave function E 7→ [kAkG
E ] and the function E 7→ [k|Ak|E ] are related by the
transformations
f (xt)
t>0 t + 1
F [f ](x) = sup
and G[f ](x) = inf f (xt) (1 + 1/t),
t>0
defined on the set of nonnegative functions on (0, +∞). In [12] it is shown that G ◦ F
maps any nonnegative function f on (0, +∞) into its concave hull and hence G[F [f ]] =
17
18
A ⊗ IK denotes the operator mentioned
√ in Lemma 4.
This holds for the operators a and N in the below example.
22
f for any concave nonnegative function f . This shows that the second expression in
2
Theorem 3A follows from the first one and the concavity of the function E 7→ [kAkG
E] .
0
√ Remark 7. The below proof of the density of B(H) in BG (H) shows that the
G-infinitesimality criterion (44) is equivalent to the following one
lim kAP̄n kG
E = 0,
n→+∞
(46)
where P̄n is the spectral projector of G corresponding to the interval (n, +∞).
Proof of Theorem 3. A) By Lemma 3A inequalities (5), (11) and (12) for any A
in BG (H) are proved by the same arguments as for a bounded operator A. By using
definitions (3) and (10) it is easy to show that
√
k|Ak|G
rkAkG1−r E , A ∈ BG (H).
E = sup
r
r∈(0,1)
The first expression in A follows from this one by the change of variables t = (1 − r)/r.
The second expression in A is derived from the first formula in part B proved
below by noting√that the infimum in that formula can be taken over all the pairs
G
(k|Ak|G
E , k|Ak|E / E), E > 0. This follows from density of the set
n
o
√
G
k|Ak|G
+
x,
k|Ak|
/
E
+
y
E
>
0,
x,
y
≥
0
E
E
√
√
in Π√G (A), which can be proved by noting that k|Ak|G
E = min{a | (a, a/ E) ∈ Π G (A)}.
√
G
Denote by HE
the Hilbert space obtained by equipping the linear space D( G)
with the inner product
hϕ|ψiG
E = hϕ|ψi + hϕ|G|ψi/E.
√
G-bounded operator A is the operator
Since the norm k|Ak|G
E of any
√ norm of A
G
treated as a bounded operator from HE
into H, the linear space of all G-bounded
operators equipped with the norm k| · k|G
E is a nonseparable Banach space [13]. Hence,
the equivalence of the norms k| · k|G
and
k · kG
E
E implies that BG (H) is a nonseparable
Banach space.
2
B) Since E 7→ [kAkG
E ] is a concave nonnegative function on R+ , it coincides with
2
2
2
the infimum of all linear functions E 7→ a2 + b2 E such that [kAkG
E ] ≤ a + b E for
2
all E > 0. The concavity of the function E 7→ [kAkG
E ] implies that the function
G 2
E 7→ [kAkE ] /E is non-increasing. So, both formulae in part B follow from Lemma 7.
C) The continuity and the seminorm properties of the function A 7→ b√G (A) stated
√ (0) is a closed subspace of BG (H). The
in part E proved below show that BG0 (H) = b−1
G
characterizing property (44) follows from the second formula in part B.
To prove density of B(H) in BG0 (H) it suffices, by Lemma 6, to show that for any
A ∈ BG0 (H) the sequence {APn }, where Pn is the spectral projector of G corresponding
to the interval [0, n], converges to A with respect to the norm k·kG
E . For given Pn let ϕ
23
be any unit vector such that hϕ|G|ϕi ≤ E and xn = hϕ|P̄n |ϕi > 0, where P̄n = IH −Pn .
−1/2
Let |ϕn i = xn P̄n |ϕi. We have
2
G 2
≤
(E/n)
kAk
.
kAP̄n ϕk2 = xn kAϕn k2 ≤ xn kAkG
n
E/xn
The first inequality follows from definition (3) of the operator E -norm and the inequality hϕn |G|ϕn i ≤ E/xn , the second one follows from concavity of the function E 7→
2
kAkG
E , Lemma 2 and the inequality xn ≤ E/n (which holds, since hϕ|G|ϕi ≤ E).
The above estimate implies that
p
.
kA − APn kG
sup
kAP̄n ϕk ≤ E/nkAkG
E =
n.
ϕ∈H1 :hϕ|G|ϕi≤E
So, condition (44) guarantees that kA − APn kG
E tends to zero as n → +∞.
The above arguments and Lemma 6 imply that (44) is equivalent to (46).
If G is a discrete operator then the separability of BG0 (H) follows from separability
of B(H) w.r.t. any of the operator E -norms, which can be easily shown by using
Proposition 2B and separability of B(H) w.r.t. the strong operator topology.
D) We begin with the second assertion. The ”only if” part of this assertion and the
G
expression kAk = supE>0 kAkG
E follow from Proposition 1. If kAkE ≤ M < +∞
√ for
all E > 0√then it follows from (43) that kAϕk ≤ M for any unit vector ϕ in D( G).
Since D( G) is dense in H, this implies that A ∈ B(H).
To prove the first assertion assume that {An } is a sequence in B(H) converging to an
operator A0 ∈ BG0 (H) such that kAn k ≤ M < +∞ for all n. Since kAn kG
E ≤ kAn k ≤ M
for all n and E > 0 and the right hand side of the inequality
G
G
kAn kG
E − kA0 kE ≤ kAn − A0 kE
tends to zero as n → +∞ for any E > 0, it is easy to see that kA0 kG
E ≤ M for all E.
Thus, kA0 k ≤ M by the assertion proved before.
E) The seminorm properites of b√G (·) follow from the second formula in part B of
2
the theorem. So, since the function E 7→ [kAkG
E ] /E is non-increasing for any given
A ∈ BG (H), the inequality (45) follows from the triangle inequality for b√G (·).
F) This assertion follows from Lemma 4 and the first formula in part B of the
theorem.
P
G) Let ρ = i |ϕi ihϕi | be a decomposition into 1-rank positive operators
P and {ψi }
a set of orthogonal
√ unit vectors in a separable Hilbert space K then |ηi = i |ϕi i ⊗|ψi i
is a vector in D( G ⊗ IK )√such that ρ = TrK |ηihη|. By Lemma 4 the operators
√ A ⊗ IK
and B ⊗ IK have unique G ⊗ IK -bounded linear extensions to the set D( G ⊗ IK )
satisfying (40). By the monotonicity of the trace norm we have
K
K
.
kB ⊗IK kG⊗I
kAρB ∗ k1 ≤ k|A⊗IK ηihB ⊗IK η|k1 ≤ kA⊗IK ηkkB ⊗IK ηk ≤ kA⊗IK kG⊗I
Eρ
Eρ
G
By part F of the theorem the r.h.s. of this inequality is equal to kAkG
Eρ kBkEρ .
24
H) If A is a bounded operator then the possibility to take the suprema in (3) and
(4) over all unit vectors in H satisfying the condition hϕ|G|ϕi = E and over all states
in S(H) satisfying the condition TrGρ = E correspondingly follows Proposition 3B. If
A is a unbounded operator then this possibility can be easily shown by noting that the
function E 7→ kAkG
E is strictly increasing on R+ (since it is concave on R+ and tends
to +∞ as E → +∞).
I) If A ∈ BG0 (H) then ρ 7→ TrAρA∗ is an affine continuous function on the set
.
CG,E = {ρ ∈ T+ (H) | Trρ ≤ 1, TrGρ ≤ E } for any E > 0 by Corollary 2 below (proved
independently). So, both assertions are proved by repeating the arguments from the
proof of Proposition 3C. Example: the operators associated with the Heisenberg Commutation
Relation
Let H = L2 (R) and S(R) be the set of infinitely differentiable rapidly decreasing
functions with all the derivatives tending to zero quicker than any degree of |x| when
|x| → +∞. Consider the operators q and p defined on the set S(R) by setting
(qϕ)(x) = xϕ(x) and (pϕ)(x) =
1 d
ϕ(x).
i dx
These operators are essentially self-adjoint. They represent (sharp) real observables of
position and momentum of a quantum particle in the system of units where Planck’s
constant ~ equals to 1 [6, Ch.12]. On the domain S(R) these operators satisfy the
Heisenberg commutation relation
[q, p] = iIH .
For given ω > 0 consider the operators
√
√
a = (ωq + ip)/ 2ω and a† = (ωq − ip)/ 2ω
(47)
(48)
defined on S(R). Via these operators the commutation relation (47) can be rewritten
as [a, a† ] = IH . The operator
N = a† a = aa† − IH
(49)
is positive and essentially self-adjoint. It represents (sharp) real observable of the
number of quanta of the harmonic oscillator with the frequency ω. The selfadjoint
extension of N has the form (1) with En = n and the basic {τn } of eigenvectors of N
which can be described as follows
r
ω
1
ωx2
4
, |τn i = √ [a† ]n |τ0 i, n ≥ 1.
exp −
τ0 (x) =
π
2
n!
So, N is a positive unbounded discrete (Def.1) operator satisfying condition (2).
25
The operators a and a† = a∗ are called annihilation and creation operators correspondingly, since
√
√
(50)
a|τ0 i = 0, a|τn i = n|τn−1 i and a† |τn i = n + 1|τn+1 i.
So, the operators a and a† are correctly extended to the set
)
(
∞
X
√
D( N) = ϕ ∈ H
n|hϕ|τn i|2 < +∞
n=0
√
By using relations (48) the operators p and q are also extended to the set D( N ).
We will estimate the operator E -norm of the operators q, p, a and a† induced by
the operator N (which up to the constant summand coincides with the√Hamiltonian
√
N
of a quantum oscillator).
By using (49) it is easy to show that kakN
E
E = k Nk
E =
√
√
† N
†
and ka
p kE = E + 1 for any E > 0. For the operators q = (a + a)/ 2ω and
p = i ω/2(a† − a) one can obtain the following estimates
r
r
p
p
2E + 1/2
2E + 1
N
< kqkN
≤
,
(2E
+
1/2)ω
<
kpk
≤
(2E + 1)ω (51)
E
E
ω
ω
(the E -norms of q and p depend on ω, since the operator N depends on ω). The
right inequalities in (51) directly follow from the triangle inequality and the above
† N
expressions for kakN
E and ka kE . To prove the left inequalities in (51) it suffices to
show that
√
sup
k(a† ± a)ϕk > 4E + 1.
kϕk=1, hϕ|N |ϕi≤E
√
P
± n/2
|τn i, where
This can be easily done by using the unit vectors |ϕ± i = 1 − r +∞
n=0 cn r
−
iπn/2
+
r = E/(E + 1), cn = e
and cn = 1 for all n.
By using the first expression in Theorem 3A and
estimates√of the norms
√ the above
† N
N
N
N
† N
kakN
,
ka
k
,
kpk
and
kqk
we
obtain
k|ak|
=
E,
k|a
k|
=
max{1, E},
E
E
E
E
E
E
r
r
p
p
l(E)
u(E)
N
< k|qk|N
≤
and
l(E)ω
<
k|pk|
≤
u(E)ω,
E
E
ω
ω
where l(E) = max{1/2, 2E} and u(E) = max{1, 2E}.
The second formula in Theorem 3B and the above estimates of the norms kakN
E,
N
N
ka† kN
,
kpk
and
kqk
imply
that
E
E
E
p
√
b√N (q) = 2/ω and b√N (p) = 2ω.
b√N (a) = b√N (a† ) = 1,
So, the operators q, p, a and a† belong to the Banach space BN (H) but not lie in
the completion BN0 (H) of B(H) w.r.t. the norm k·kN
E.
√
†
For any t < 1 let at and at be the operators defined on the set D( N) by settings
at |τ0 i = 0,
at |τn i = nt/2 |τn−1 i and a†t |τn i = (n + 1)t/2 |τn+1 i.
26
(52)
It is easy to show that
lim at |ϕi = a|ϕi and
t→1−0
√
lim a†t |ϕi = a† |ϕi for any ϕ ∈ D( N ).
t→1−0
(53)
Since a†t at = N t and at a†t = (N + IH )t , by using concavity of the function x 7→ xt ,
we obtain
r
r
† N
t = E t/2 ,
sup
[TrNρ]
ka
k
≤
sup [Tr(N + IH )ρ]t = (E + 1)t/2 .
kat kN
≤
t E
E
TrN ρ≤E
TrN ρ≤E
So, the operators at and a†t belong to the space BN0 (H) for all t < 1 (since they
satisfy condition (44)), while the ”limit” operators a and a† lie in BN (H) \ BN0 (H). So,
at and a†t do not tend to a and a† as t → 1 w.r.t. the norm k·kN
E in spite of the strong
operator convergence (53).
Remark 8. It follows from (43) that
√
k·kG
⇒ lim An |ϕi = A0 |ϕi ∀ϕ ∈ D( G)
(54)
E - lim An = A0
n→∞
n→∞
for a sequence {An } ⊂ BG (H). The above example shows that the converse implication
is not valid even in the case of discrete operator G (in this case ” ⇔ ” holds in (54) for
any bounded sequence {An } ⊂ B(H) by Proposition 2B).
In the last part of this section we consider properties of the Banach space BG0 (H).
√
Proposition 7. If A ∈ BG0 (H) then the extension of A ⊗ IK to the set D( G ⊗ IK )
mentioned in Lemma 4 is uniformly continuous on the set
√
√
.
VE = {η ∈ D( G ⊗ IK ) | k G ⊗ IK ηk2 ≤ E }
(55)
for any E > 0. Quantitatively,
+
kA ⊗ IK (η − θ)k ≤ εkAkG
4E/ε2 = o(1) as ε → 0
(56)
for any vectors η and θ in VE such that kη − θk ≤ ε.
If A ∈ BG (H) \ BG0 (H) then the operator A ⊗ IK is not continuous on the set VE
for any E > 0.
√
Proof. By Theorem 3F for any unit vector η in D( G ⊗ IK ) we have
√
G
K
(57)
,
where
E
=
k
=
kAk
kA ⊗ IK ηk ≤ kA ⊗ IK kG⊗I
G ⊗ IK ηk2 .
η
E
Eη
η
√
Assume
that
η
and
θ
are
vectors
in
V
such
that
kη−θk
≤
ε.
Since
k
G⊗IK ηk2 ≤ E
E
√
√
2
2
and k G⊗IK θk ≤ E we have k G⊗IK (η−θ)k ≤ 4E. So, by using (57), the concavity
2
of the function E 7→ kAkG
and Lemma 2 we obtain
E
kA ⊗ IK (η − θ)k = kη − θk A ⊗ IK
η−θ
G
≤ kη − θkkAkG
4E/kη−θk2 ≤ εkAk4E/ε2 .
kη − θk
27
By condition (44) the r.h.s. of this inequality tends to zero as ε → 0+ . Thus, the
function η 7→ A ⊗ IK |ηi is uniformly continuous on VE .
The last assertion of the proposition follows from the proof of the last assertion of
Corollary 2 below, √
since TrK |A ⊗ IK ηihA ⊗ IK η| = TrAρη A∗ , where ρη = TrK |ηihη|, for
any vector η in D( G ⊗ IK ). Corollary 2. For any operators A and B in BG0 (H) the function ρ 7→ AρB ∗ from
.
T+
G into T(H) (defined by formula (41)) is uniformly continuous on the set CG,E =
{ρ ∈ T+ (H) | Trρ ≤ 1, TrGρ ≤ E } for any E > 0. Quantitatively,
√
G
G
G
+
kAρB ∗ − AσB ∗ k1 ≤ ε kAkG
(58)
E kBk4E/ε + kBkE kAk4E/ε = o(1) as ε → 0
for any operators ρ and σ in CG,E such that kρ − σk1 ≤ ε.
If A ∈ BG (H) \ BG0 (H) then the function ρ 7→ AρA∗ is not continuous on the set
CG,E for any E > 0.
Remark 9. Corollary 2 shows that the operators A in BG0 (H) are characterized by
continuity of the function ρ 7→ AρA∗ on the set CG,E for any given E > 0.
Proof. Let ρ and σ be operators in CG,E such that kρ − σk1 ≤ ε. If K ∼
= H then
one can find vectors η and θ in
the
set
V
(defined
in
(55))
such
that
ρ
=
TrK |ηihη|,
E
√
σ = TrK |θihθ| and √
kη − θk ≤ ε [6]. By Lemma 4 the operators
√ A ⊗ IK and B ⊗ IK
have unique linear G ⊗ IK -bounded extensions to the set D( G ⊗ IK ) satisfying (40).
By using the monotonicity of the trace norm, the inequality
k|αihβ| − |ϕihψ|k1 ≤ kαkkβ − ψk + kψkkα − ϕk,
where |αi = A ⊗ IK |ηi, |βi = B ⊗ IK |ηi, |ϕi = A ⊗ IK |θi, |ψi = B ⊗ IK |θi, and
continuity bound (56) we obtain
√
√
kAρB ∗ − AσB ∗ k1 ≤ εkBkG
εkAkG
4E/ε kA ⊗ IK ηk +
4E/ε kB ⊗ IK θk.
By inequality (57) this implies (58).
The r.h.s. of (58) tends to zero as ε → 0+ , since A and B satisfy condition (44).
If A ∈ BG (H) \ BG0 (H) then, by Remark 7, the sequence kAP̄n kG
E , where P̄n is the
spectral projector of G corresponding to the interval (n, +∞), does not tend to zero.
Hence there is a sequence {ρn } of states in CG,E such that the sequence {TrAP̄n ρn P̄n A∗ }
does not tend to zero. Since the condition TrGρn ≤ E implies TrP̄n ρn ≤ E/n, the
sequence {P̄n ρn P̄n } ⊂ CG,E tends to zero. This shows discontinuity of the function
ρ 7→ AρA∗ on the set CG,E . 28
Since B(H) is dense in BG0 (H) by Theorem 3C, Proposition 4E implies the following
Proposition 8. Let G be a positive densely defined operator on H satisfying condition (2) and E > 0. Any 2-positive linear map Φ : B(H) → B(H) such that
Φ(IH ) ≤ IH having the predual map Φ∗ : T(H) → T(H) with finite19
.
YΦ (E) = sup {TrGΦ∗ (ρ) | ρ ∈ S(H), TrGρ ≤ E }
(59)
is uniquely extended to the bounded linear operator ΦG : BG0 (H) → BG0 (H) such that
p
p
kΦG (A)kG
≤
kΦ(IH )k kAkG
kΦ(IH )kKΦ kAkG
(60)
E
YΦ (E) ≤
E,
where KΦ = max{1, YΦ (E)/E}.
The assertion of Proposition 8 can be strengthened substantially by assuming complete positivity of Φ. The corresponding result is considered in [20].
Different applications of the operator E -norms are presented in [18, 19, 20]. In [18]
the version of the Kretschmann-Schlingemann-Werner theorem for unbounded completely positive linear maps is obtained by using the results from Section 5.
I am grateful to A.S.Holevo, G.G.Amosov, A.V.Bulinsky and M.M.Wilde for discussion and useful remarks. I am also grateful to V.Zh.Sakbaev for consultation concerning
unbounded operators and to T.V.Shulman for the help and useful discussion.
Special thanks to S.Weis for the help in proving the coincidence of definitions (3)
and (4).
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31