Gaussian optimizers and the additivity problem in quantum information theory

0.4 MB·English

by  A. S. Holevo

Gaussian optimizers and the additivity
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problem in quantum information theory
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r
a
A. S. Holevo
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Steklov Mathematical Institute, Moscow

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h Abstract
p
- We give a survey of the two remarkable analytical problems of
h
t quantuminformationtheory. Themainpartisadetailed reportof the
a
m recent (partial) solution of the quantum Gaussian optimizers problem
which establishes an optimal property of Glauber’s coherent states –
[
a particular instance of pure quantum Gaussian states. We elaborate
3
on the notion of quantum Gaussian channel as a noncommutative
v
2 generalization of Gaussian kernel to show that the coherent states,
5 and under certain conditions only they, minimize a broad class of
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the concave functionals of the output of a Gaussian channel. Thus,
0
0 the output states corresponding to the Gaussian input are “the least
.
1 chaotic”, majorizing all the other outputs. The solution, however, is
0 essentially restricted tothegauge-invariant casewhereadistinguished
5
complex structure plays a special role.
1
: We also comment on the related famous additivity conjecture,
v
i which was solved in principle in the negative some five years ago.
X
Thisrefers to theadditivity or multiplicativity (with respectto tensor
r
a productsof channels) of information quantities related to the classical
capacity of quantum channel, such as (1 p)-norms or the minimal
→
von Neumann or R´enyi output entropies. A remarkable corollary of
the present solution of the quantum Gaussian optimizers problem is
that these additivity properties, while not valid in general, do hold in
the important and interesting class of the gauge-covariant Gaussian
channels.
1
Contents
1 Introduction 2
2 The additivity problem for quantum channels 5
2.1 Definition of channel . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Stinespring-type representation . . . . . . . . . . . . . . . . . 6
2.3 Entropic quantities and additivity . . . . . . . . . . . . . . . . 9
2.4 The channel capacity . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Main conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Majorization for quantum states . . . . . . . . . . . . . . . . . 14
3 Quantum Gaussian systems 16
3.1 Gaussian states and channels . . . . . . . . . . . . . . . . . . 16
3.2 Complex structures and gauge symmetry . . . . . . . . . . . . 19
3.3 Attenuators and amplifiers . . . . . . . . . . . . . . . . . . . . 23
3.4 Gaussian optimizers . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Explicit formulas and additivity . . . . . . . . . . . . . . . . . 33
3.6 The case of quantum-classical Gaussian channel . . . . . . . . 38
4 Appendix 40
5 Acknowledgments 42
1 Introduction
The quantum Gaussian optimizers problem is an analytical problem that
arose in quantum information theory at the end of past century, and which
has an independent mathematical interest. Only recently a solution was
found [23], [53] in a considerably common situation, while in full generality
the problem still remains open. To explain the nature and the difficulty of
theproblemwe startfromtherelatedclassical problem ofGaussianmaximiz-
ers which has been studied rather exhaustively, see Lieb [50] and references
therein. Consider an integral operator G from L (Rs) to L (Rr) given by a
p q
Gaussian kernels (i.e. exponential of a quadratic form) with the (q p)
→ −
norm
G = sup Gf / f = sup Gf . (1)
k kq p k kp k kq k kp
→ f=0 f 1
6 k kq≤
2
Under certain broad enough assumptions concerning the quadratic form
defining the kernel, and also p and q, this operator is correctly defined, and
the supremum in (1) is attained on Gaussian f. Moreover, under some addi-
tional restrictions any maximizer is Gaussian. As it is put in the title of the
paper [50]: “ Gaussian kernels have only Gaussian maximizers”.
Knowledge that the maximizer is Gaussian can be used to compute exact
value of the norm (1); in fact a starting point of the classical Gaussian max-
imizers works were the result of K.I. Babenko [5] and a subsequent paper
of Beckner [6] which established the best constant in the Hausdorff-Young
inequality concerning the (p p′) norm, (p−1 +(p′)−1 = 1, 1 < p 2), of
→ − ≤
the Fourier transform (which is apparently given by a degenerate imaginary
Gaussian kernel).
A difficulty in the optimization problem (1) is that it requires maximiza-
tion of a convex function, so the general theory of convex optimization is not
of great use here (it only implies that a maximizer of Gf belongs to a face
k kp
of the convex set f 1). Instead, the solution is based on substantial
k kq ≤
use of the classical Minkovski’s inequality and the related multipicativity of
the classical (q p) norms with respect to tensor products of the integral
→ −
operators.
A notable application of these classical results to a problem in quantum
mathematical physics was Lieb’s solution [51] of Wehrl’s conjecture [63]. Let
ρ be a density operator in a separable Hilbert space representing state of
H
a quantum system; the “classical entropy” of the state ρ is defined as 1
d2z
H (ρ) = p (z)logp (z) ,
cl ρ ρ
− π
C
Z
where p (z) = z ρ z is the diagonal value of the kernel of ρ in the system
ρ
h | | i
of Glauber’s coherent vectors2 z ;z C [44], [33]. The conjecture was
{| i ∈ }
that H (ρ) has the minimal value if ρ is itself a coherent state i.e. projec-
cl
tor onto one of the coherent vectors. Lieb [51] used exact constants in the
Hausdorff-Young inequality for L -norms of Fourier transform [5], [6] and the
p
Young inequality for convolution [6] to prove similar maximizer conjecture
for f(x) = xp and considered the limit lim (1 p) 1(1 xp) = xlogx.
p 1 −
↓ − − −
1 Throughoutthe paper the base of logarithmis a fixed number a>1. In information
theorythe naturalchoiceis a=2, thenall the entropic quantities aremeasuredin“bits”.
2 In analysis, they correspond to complex-parametrized Gaussian wavelets. Notice
that this is the only place in the presentarticle where we formally usedDirac’s notations,
uncommon among mathematicians.
3
Recently, Lieb and Solovej [52], by using a completely different approach
based on study of the spin coherent states, strengthened the result of [51]
by showing that the coherent states minimize any functional of the form
f(p (z))d2z, where f(x),x [0,1] is a nonnegative concave function with
C ρ π ∈
f(0) = 0.
R
In the language of quantum information theory, the affine map G : ρ
→
p (z), taking density operators ρ (quantum states) into probability densities
ρ
p (z) (classical states), is a “ quantum-classical channel” [39]. Moreover, it
ρ
transforms Gaussian density operators ρ (in the sense defined below in Sec.
3.1 ) into Gaussian probability densities, and in this sense it is a “ Gaussian
channel” . From this point of view, Wehrl entropy H (ρ) is the output
cl
entropy of the channel, and Lieb’s result says that it is minimized by pure
Gaussian states ρ. Moreover, the corresponding result for f(x) = xp can be
interpretedas“Gaussianmaximizer”statementforthenorm G .Notice
k k1 p
that the case q = 1, which is excluded in the classical problem f→or obvious
reasons, appears and is the most relevant in the quantum (noncommutative)
case.
ThequantumGaussianoptimizersproblemdescribedinthepresentpaper
refers to Bosonic Gaussian channels – a noncommutative analog of Gaussian
Markov kernels and, similarly, requires maximization of convex functions (or
minimization of concave functions, such as entropy) of the output state of
the channel, while the argument is the input state. A general conjecture
is that the optimizers belongs to the class of pure Gaussian states. The
conjecture, first formulated in [42] in the context of quantum information
theory, however natural it looks, resisted numerous attacks for several years.
Among others, notable achievements were the exact solution for the classical
capacity of pure loss channel [21] and a proof of additivity of the R´enyi en-
tropies of integer orders p [24] for special channels models. Even restricted
to the class of Gaussian input states, the optimization problem turns out to
be nontrivial [56], [31]. There was some hope that in solving the problem,
similarly to Wehrl’s conjecture, one could also use the classical “ Gaussian
maximizers” results. However the solution found recently by Giovannetti,
Holevo, Garcia-Patron [23], and Mari, Giovannetti, Holevo [53] uses com-
pletely different ideas based on a thorough study of structural properties of
quantum Gaussian channels. As it was mentioned, a solution of the classi-
cal problem uses the Minkowski inequality and the implied multiplicativity
of (q p)-norms. However, the noncommutative analog of the Minkowski’s
→
inequality [12] is not powerful enough to guarantee the multipicativity of
4
norms (or additivity of the corresponding entropic quantities). Moreover,
the related long-standing additivity problem in quantum information theory
[34] was recently shown to have negative solution in general [26]. We show
that, remarkably, a solution of the quantum Gaussian optimizers problem
given in [23] implies also a proof of the multipicativity/additivity property
in the restricted class of gauge-covariant or contravariant quantum Gaussian
channels.
It would then be interesting to investigate a possible development of
such an approach to obtain noncommutative generalizations of the classi-
cal “ Gaussian maximizers” results for (q p) norms. Such generalization
→ −
couldshedanewlighttothehypercontractivityproblemforquantumdynam-
ical semigroups and related noncommutative analogs of logarithmic Sobolev
inequalities, see e.g. [62].
2 The additivity problem for quantum chan-
nels
2.1 Definition of channel
Let be a separable complex Hilbert space, L( ) the algebra of all bounded
H H
operators in and T( ) the ideal of trace-class operators. The space T( )
H H H
equippedwiththetracenorm isBanachspace, whichisusefultoconsider
k·k1
as a noncommutative analog of the space L . The convex subset of T( )
1
H
S( ) = ρ : ρ = ρ 0,Trρ = 1 ,
∗
H { ≥ }
is a base of the positive cone in T( ). Operators ρ from S( ) are called
H H
density operators or quantum states. The state space is a convex set with
the extreme boundary
P( ) = ρ : ρ 0,Trρ = 1,ρ2 = ρ .
H ≥
ThusextremepointsofS( )(cid:8),whicharecalled pure sta(cid:9)tes,areone-dimensional
H
projectors, ρ = P for a vector ψ with unit norm, see, e.g. [55].
ψ
∈ H
The class of maps we will be interested is a noncommutative analog of
Markov maps (linear, positive, normalized maps) in classical analysis and
probability. Let , be the two Hilbert spaces, which will be called input
A B
H H
and output space, correspondingly. A map Φ : T( ) T( ) is positive
A B
H → H
5
if X 0 implies Φ[X] 0, and it is completely positive [61], [54] if the maps
≥ ≥
Φ Id are positive for all d = 1,2,..., where Id is the identity map of
(d) (d)
⊗
the algebra L = L(Cd) of complex d d matrices. Equivalently, for every
d
× −
nonnegative definite block matrix [X ] the matrix [Φ[X ]] is
jk j,k=1,...,d jk j,k=1,...,d
nonnegative definite.
A linear map Φ is trace-preserving if TrΦ[X] = TrX for all X T( ).
A
∈ H
DefinitionQuantumchannel isalinearcompletelypositivetrace-preserving
map Φ : T( ) T( ). Letter A will be always associated with the input
A B
H → H
of the channel, while B with the output. Sometimes, to abbreviate notations,
we will write simply Φ : A B. (cid:4)
→
Apparently, every channel is a positive map taking states into states:
Φ[S( )] S( ). Since T( ) is a base-normed space, this implies [17]
A B
H ⊆ H H
that Φ is a bounded map from the Banach space T( ) to T( ). The dual
A B
H H
Φ of the map Φ is uniquely defined by the relation
∗
TrΦ[X]Y = TrXΦ [Y]; X T( ), Y L( ), (2)
∗ A B
∈ H ∈ H
and it is called dual channel. The dual channel is linear completely positive
weakly continuous map from L( ) to L( ), which is unital: Φ[I ] =
∗− HB HA HB
I .Here andinwhat follows I withpossible index denotes theunit operator
A
H
in the corresponding Hilbert space.
There are positive maps that are not completely positive, a basic example
provided by matrix transposition X X in a fixed basis.
⊤
→
From the definition of complete positivity one easily derives [39] that
composition of channels Φ Φ defined as
2 1
◦
Φ Φ [X] = Φ [Φ [X]],
2 1 2 1
◦
and naturally defined tensor product of channels
Φ Φ = (Φ Id ) (Id Φ )
1 2 1 2 1 2
⊗ ⊗ ◦ ⊗
are again channels.
2.2 Stinespring-type representation
The notion of completely positive map was introduced by Stinespring [61] in
a much wider context of C*-algebras. This allows also to cover the notion of
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hybrid channel where the input is quantum while the output is classical or
vice versa. An example of such channel was mentioned in Sec. 1. We will
not pursue this topic further here, see [39], but only mention that complete
positivity reduces to positivity in such cases.
Motivated by the famous Naimark’s dilation theorem, Stinespring estab-
lished a representation for completely positive maps of C*-algebras which in
the case of quantum channel reduces [39] to
Proposition 1 Let Φ : A B be a channel. There exist a Hilbert space
→
and an isometric operator V : , such that
E A B E
H H → H ⊗H
Φ[ρ] = Tr VρV ; ρ T( ), (3)
E ∗ A
∈ H
where Tr denotes partial trace with respect to . The representation (3)
E E
H
is not unique, however any two representations with V :
1 HA → HB ⊗ HE1
and V : are related via partial isometry W :
2 HA → HB ⊗HE2 HE1 → HE2
such that V = (I W)V and V = (I W )V .
2 B 1 1 B ∗ 2
⊗ ⊗
Consider a representation (3) for the channel Φ; the complementary chan-
nel [37], [48] is then defined by the relation
Φ˜[ρ] = Tr VρV ; ρ T( ). (4)
B ∗ A
∈ H
Fromtherelationbetween thedifferentrepresentations (3), itfollowsthatthe
complementary channel is unique in the following sense: any two channels
Φ˜ ,Φ˜ complementary to Φ are isometrically equivalent in the sense that
1 2
there is a partial isometry W : such that
HE1 → HE2
Φ˜ [ρ] = WΦ˜ [ρ]W , Φ˜ [ρ] = W Φ˜ [ρ]W, (5)
2 1 ∗ 1 ∗ 2
forallρ.ItfollowsthattheinitialprojectorW W satisfiesΦ˜ [ρ] = W WΦ˜ [ρ],
∗ 1 ∗ 1
˜
i.e. its support contains the support of Φ [ρ], while the final projector WW
1 ∗
has similar property with respect to Φ˜ [ρ]. The complementary to comple-
2
mentary can be shown isometrically equivalent to the initial channel, so that
Φ,Φ˜ are called mutually complementary channels.
In general, we will say that two density operatorsρ andσ (possibly acting
in different Hilbert spaces) are isometrically equivalent if there is a partial
isometry W such that ρ = WσW , σ = W ρW. Apparently, this is the case
∗ ∗
if and only if nonzero spectra (counting multiplicity) of the density operators
ρ and σ coincide. We denote this fact with the notation ρ ∼ σ. We have just
shown that Φ˜ [ρ] ∼ Φ˜ [ρ] for arbitrary ρ.
1 2
7
Lemma 2 Let Φ˜ be a complementary channel (4 ), then Φ[P ] ∼ Φ˜[P ] for
ψ ψ
all ψ .
A
∈ H
Proof. Let V : be the isometry from the representations
A B E
H → H ⊗H
(3), (4), then ρ = VP V is a pure state in , and the statement
BE ψ ∗ B E
H ⊗H
follows from a basic result in quantum information theory (“Schmidt decom-
position”): ifρ isapurestatein andρ = Tr ρ , ρ = Tr ρ
BE B E B E BE E B BE
H ⊗H
are its partial states, then ρ ∼ ρ (see e.g. Proposition 3 in [34])
B E
A different name for channel is dynamical map – in nonequilibrium quan-
tum statistical mechanics they arise as irreversible evolutions of an open
quantum system interacting with an environment [39]. Assume that there is
a composite quantum system AD = BE in the Hilbert space
= , (6)
A D B E
H H ⊗H ≃ H ⊗H
which is initially prepared in the state ρ ρ and then evolves according
A D
⊗
to the unitary operator U. Then the output state ρ depending on the input
B
state ρ = ρ is
A
Φ [ρ] = Tr U(ρ ρ )U , (7)
B E D ∗
⊗
while the output state of the “environment” E is the output of the channel
Φ [ρ] = Tr U(ρ ρ )U . (8)
E B D ∗
⊗
If the initial state of D is pure, ρ = P , then by introducing the isometry
D ψD
V : , which acts as
A B D
H → H ⊗H
Vψ = U(ψ ψ ), ψ ,
D A
⊗ ∈ H
we see that the relations (7), (8) convert into (3), (4), andΦ is just the com-
E
plementary of Φ . Notice also that both partial trace and unitary evolution
B
are completely positive operators, hence the maps (7), (8) are completely
positive; vice versa, any quantum channel has a representation of such a
form, see, e.g. [39].
Vast literature is devoted to study of quantum dynamical semigroups
(noncommutative analog of Markov semigroups) and quantum Markov pro-
cesses. Stinespring-type representation (3) underlies dilations of quantum
dynamical semigroups to the unitary dynamics of open quantum system in-
teracting with an environment [17], [35].
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2.3 Entropic quantities and additivity
Consider the norm of the map Φ defined similarly to (1):
Φ = sup Φ[X] / X = sup Φ[X] , (9)
k k1 p k kp k k1 k kp
→ X=0 X 1
6 k k1≤
where is the Schatten p norm [55]. As shown in [4],
k·kp −
Φ p = sup TrΦ[ρ]p = sup TrΦ[P ]p, (10)
k k1 p ψ
→ ρ∈S(HA) ψ∈HA
where the second equality follows from convexity of the function xp,p > 1..
The quantum R´enyi entropy of order p > 1 of a density operator ρ is
defined as
1 p
R (ρ) = logTrρp = log ρ , (11)
p 1 p 1 p k kp
− −
Define the minimal output R´enyi entropy of the channel Φ
p
Rˇ (Φ) = inf R (Φ[ρ]) = log Φ (12)
p ρ S( ) p 1 p k k1→p
∈ H −
and the minimal output von Neumann entropy
Hˇ(Φ) = inf H(Φ[ρ]). (13)
ρ S( )
∈ H
In the limit p 1 the quantum R´enyi entropies monotonely nondecreasing
→
converge to the von Neumann entropy
limR (ρ) = Trρlogρ = H(ρ).
p
p 1 −
→
In finite dimensions the set of quantum states is compact, hence by Dini’s
Lemma the minimal output R´enyi entropies converge to the minimal output
von Neumann entropy3.
Multiplicativity of the norm (9) for some channels Φ ,Φ ,
1 2
Φ Φ = Φ Φ (14)
k 1 ⊗ 2k1 p k 1k1 p ·k 2k1 p
→ → →
is equivalent to the additivity of the minimal output R´enyi entropies
Rˇ (Φ Φ ) = Rˇ (Φ )+Rˇ (Φ ). (15)
p 1 2 p 1 p 2
⊗
3The corresponding statement is not valid for infinite-dimensional channels (even for
classical channels with countable set of states), M. E. Shirokov,private communication.
9
Closely related is the similar property for the minimal output von Neumann
entropy:
Hˇ(Φ Φ ) = Hˇ(Φ )+Hˇ(Φ ). (16)
1 2 1 2
⊗
In finite dimensions, the validity of (15) for certain channels Φ ,Φ and p
1 2
close to 1 implies (16) for these channels.
In the last two relations the inequality (similarly to the inequality
≤ ≥
in (14)) is obvious because the right-hand side is equal to the infimum over
the subset of product states ρ = ρ ρ . On the other hand, existence of
1 2
⊗
“entangled” pure states which are not reducible to product states, is the
cause for possible violation of the equality for quantum channels.
2.4 The channel capacity
The practical importance of the additivity property (16) is revealed in con-
nection with the notion of the channel capacity. To explain it we assume
that , are finite dimensional for the moment.
A B
H H
For a quantum channel Φ, a noncommutative analog of the Shannon
capacity, which we call χ capacity, is defined by
−
C (Φ) = sup H Φ π ρ π H(Φ[ρ ]) , (17)
χ j j j j
−
{πj,ρj} "Xj #! Xj !
wherethesupremumisoverallquantum ensembles,thatisfinitecollectionsof
states ρ ,...,ρ with corresponding probabilities π ,...,π . The quan-
1 n 1 n
{ } { }
tity (17) is closely related to the capacity C(Φ) of quantum channel Φ for
transmitting classical information [34]. The classical capacity of a quantum
channel is defined as the maximal transmission rate per use of the channel,
with coding and decoding chosen for increasing number n of independent
uses of the channel
Φ n = Φ Φ
⊗
⊗···⊗
n
such that the error probability goes to zero asn (fora precise definition
| {z }
→ ∞
see [39]). A basic result of quantum information theory, HSW Theorem [32],
says that such defined capacity C(Φ) is related to C (Φ) by the formula
χ
C(Φ) = lim(1/n)C (Φ n).
χ ⊗
n
→∞
10