Validity and failure of some entropy inequalities for CAR systems

0.15 MB·English

by  Hajime Moriya

Validity and failure of some entropy inequalities
for CAR systems
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Hajime Moriya
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v Abstract
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BasicpropertiesofvonNeumannentropysuchasthetriangleinequal-
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ityandwhatwecallMONO-SSAarestudiedforCARsystems. Weshow
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that both inequalities hold for every even state by using symmetric pu-
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rificationwhichisapplicabletosuchastate. Weconstructacertainclass
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4 ofnonevenstatesgivingexamplesofthenonvalidityofthoseinequalities.
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1 Introduction
Let H be a Hilbert space and D be a density matrix on H, i.e. a positive trace
class operator on H whose trace is unity. The von Neumann entropy is given
by
−Tr DlogD , (1)
(cid:0) (cid:1)
whereTrdenotesthe tracewhichtakesthevalue1oneachminimalprojection.
Let ̺ be a normal state of B(H), the set of all bounded linear operators on H.
Then ̺ has its density matrix D , and its von Neumann entropy S(̺) is given
̺
by (1) with D =D .
̺
It has been known that von Neumann entropy is useful for description and
characterization of state correlation for composite systems. Among others, the
following inequality called strong subadditivity (SSA) is remarkable:
S(ϕI∪J)−S(ϕI)−S(ϕJ)+S(ϕI∩J)≤0, (2)
where I, J, I∩J and I∪J denote the indexes of subsystems and ϕ denotes the
I
restrictionof a state ϕ to the subsystem indexed by I, and so on. Such entropy
inequalities have been studied for quantum systems, see e.g. Refs. 4, 5, 9, 11,
13, 15, and also their references. However, the composite systems considered
there were mostly tensor product of matrix algebras to which we refer as the
tensor product systems.
We investigate some well known entropy inequalities, the triangle inequal-
ity and MONO-SSA (which will be specified soon), for CAR systems. This
study is relevant to our previous works on state correlations such as quantum-
entanglement7 and separability8 for CAR systems. In a certain sense, the con-
ditions of validity and failure of such entropy inequalities which we are going
to establish will explain the similarities and differences in the possible forms of
state correlations between CAR and tensor product systems.
LetLbeanarbitrarydiscreteset. Thecanonicalanticommutationrelations
(CAR) are
{a∗,a } = δ 1,
i j i,j
∗ ∗
{a ,a } = {a ,a }=0,
i j i j
where i,j ∈ L and {A,B} = AB +BA (anticommutator). For each subset I
of L, A(I) denotes the subsystem on I given as the C∗-algebra generated by
all a∗ and a with i ∈ I. For I ⊂ J, A(I) is naturally imbedded in A(J) as its
i i
subalgebra.
We have already shown that SSA (2) holds for the CAR systems. For the
convenience, we sketch its proof given in Ref. 6 and Theorem 10.1 of Ref. 2.
First we check the commuting square property for a CAR system as follows:
A(I∪J) −−−E−I→ A(I)
EJ EI∩J
 
y y
A(J) −−−−→ A(I∩J),
EI∩J
2
where E denotes the conditional expectation with respect to the tracial state
onto the subsystem with a specified index. From this property SSA follows
for every state (without any assumption on the state, like its evenness) by a
well-known proof method using the monotonicity of relative entropy under the
action of conditional expectations.
We move to entropy inequalities for which CAR makes difference. The fol-
lowing is usually referred to as the triangle inequality:
(cid:12)S(ϕI)−S(ϕJ)(cid:12)≤S(ϕI∪J), (3)
(cid:12) (cid:12)
(cid:12) (cid:12)
whereIandJaredisjoint. Whilethisissatisfiedforthetensorproductsystems,1
it is not valid in general for the CAR systems; there is a counter example.7
We next introduce our main target,
S(ϕI)+S(ϕJ)≤S(ϕK∪I)+S(ϕK∪J), (4)
where I, J, and K are disjoint. We may call (4) “MONO-SSA”, because it is
equivalent to SSA (2) for the tensor product systems at least, and it obviously
implies the monotonicity of the following function
K7→S(ϕK∪I)+S(ϕK∪J),
with respect to the inclusion of the index K. Our question is whether MONO-
SSA holds for the CAR systems, if not, under what condition it is satisfied.
The MONO-SSA for the tensor product systems is shown by what is called
purificationimplyingtheequivalenceofMONO-SSAandSSAforthosesystems
(see 3.3 of Ref. 11). We note that the purification is a sort of state extension,
and is not automatic for the CAR systems. We shall review the basic concept
of state extension.
In the descriptionofa quantumcomposite system, the totalsystemis given
by a C∗-algebra A, and its subsystems are described by C∗-subalgebras A of
i
A indexed by i=1,2,···. Let ϕ be a state of A. We denote its restrictions to
A by ϕ . Surely ϕ is a state of A . Conversely, suppose that a set of states
i i i i
ϕ of A , i = 1,2,··· , are given. Then a state ϕ of A is called an extension of
i i
{ϕ } if its restriction to each A coincides with ϕ .
i i i
For tensor product systems, there always exists a state extension for any
given prepared states {ϕ } on disjoint regions, at least their product state ex-
i
tension ϕ = ϕ ⊗ ··· ⊗ ϕ ⊗ ···, and generically other extensions. On the
1 i
contrary, it is not always the case for CAR systems. When two (or more than
two) prepared states on disjoint regions are not even, there may be no state
extension. We have shown that if all of them are noneven pure states, then
there exists no state extension7.3
Weexplaintheabove-mentionedpurificationintermsofstateextension. We
are givena state ̺ of A(I). We then preparesome state ̺ on some A(J) with
1 2
J∩I=∅ such that it has the same nonzero eigenvalues and their multiplicities
as̺ fortheirdensitymatrices. Wewanttoconstructtheirpurestateextension
1
toA(I∪J). Weusetheterm“symmetricpurification”torefertothisprocedure
3
where the “symmetric” may indicate the above specified property of ̺ . For
2
the tensor product systems, symmetric purification exists for every ̺ . On the
1
contraryfortheCARsystems,thoughwecaneasilymakeapurestateextension
of ̺ , its pair ̺ cannot be always chosen among those states which have the
1 2
same nonzero eigenvalues and their multiplicities as ̺ . In the above and what
1
follows,weshallidentifystateswiththeirdensitymatriceswhenthereisnofear
of confusion.
We will show that MONO-SSA is not satisfied in generalin § 3. However it
is shown to hold for every even state in § 2. TABLE 1 shows the truth ((cid:13))
and the falsity (×) of the entropy inequalities.
TABLE 1. Truth and the falsity of von Neumann entropy inequalities
Property Tensor-product systems CAR systems
SSA (cid:13) (cid:13)
Triangle (cid:13) × in general, but (cid:13) for every even state
MONO-SSA (cid:13) × in general, but (cid:13) for every even state
We fix our notation. The even-odd grading Θ is determined by
∗ ∗
Θ(a )=−a , Θ(a )=−a . (5)
i i i i
The even and odd parts of A(I) are given by
A(I)± ≡ nA∈A(I)(cid:12)Θ(A)=±Ao.
(cid:12)
For an element A∈A(I) we have the decomposition
1
A=A++A−, A± ≡ A±Θ(A) ∈A(I)±.
2(cid:0) (cid:1)
For a finite subset I, define
∗ ∗
v ≡ v , v ≡a a −a a . (6)
I Y i i i i i i
i∈I
By a simple computation, v is a self-adjoint unitary operator in A(I) imple-
I +
menting Θ, namely
Ad(v )(A)=Θ(A), A∈A(I). (7)
I
For a finite subset I, every even pure state of A(I) is given by an eigenvectorof
v as its vector state.
I
The following is a simple consequence of the CAR givene.g. in § 4.5 of Ref.
2.
Lemma 1. Let I be a finite subset and J be a (finite or infinite) subset disjoint
with J. Let A(I′|I∪J) ≡ A(I)′ ∩A(I∪J) and A(I′|J) ≡ A(I)′ ∩A(J), the
commutant of A(I) in A(I∪J) and that in A(J), respectively. Then
′
A(I |I∪J) = A(J)++vIA(J)−, (8)
′
A(I |J) = A(J) . (9)
+
4
In this note we restrict our discussion to finite-dimensional systems so as
to exclude from the outset the cases where our statements themselves on von
Neumannentropydo notmakesense;forinfinite-dimensionalsystemsadensity
matrix does not exist in general for a given state. (However in the proof of
Proposition8 we shall mention possible infinite-dimensionalextensions of some
results.)
2 Symmetric Purification for Even States
Symmetric purification is a useful mathematical technique having a lot of ap-
plications. For example, we can derive MONO-SSA from SSA for the tensor
product systems by using it.
We nowdiscusssymmetricpurificationforthe CARsystems. Weshallshow
its existence for even states.
Lemma2. LetIandJbemutuallydisjoint finitesubsets. Let̺beaneven pure
stateofA(I∪J), andlet̺ and̺ beitsrestrictionstoA(I)andA(J). Then the
1 2
density matrix of ̺ has the same nonzero eigenvalues and their multiplicities
1
as those of ̺ . In particular, S(̺ )=S(̺ ).
2 1 2
Proof. WehaveA(I∪J)=A(I)⊗A(I′|I∪J)by(8). Bysomefinite-dimensional
Hilbert spaces H , H and H ≡ H ⊗H , we can write A(I∪J) = B(H ),
1 2 1,2 1 2 1,2
A(I)=B(H ), and A(I′|I∪J)=B(H ).
1 2
Since ̺ is a pure state of A(I∪J), its density matrix (with respect to the
non-normalizedtraceTrofB(H ))isaone-dimensionalprojectionoperatorof
1,2
B(H ), and hence there exists a unit vector ξ ∈H such that D η =(ξ,η)ξ
1,2 1,2 ̺
for anyη ∈H . By using the Schmidt decomposition,12 we havethe following
1,2
decomposed form:
ξ = λ ξ ⊗ξ , λ >0, (10)
X i 1i 2i i
i
where {ξ } and {ξ } are some orthonormal sets of vectors of H and H .
1i 2i 1 2
For ν = 1, 2, let P(ξ ) denote the projection operator on the one-dimensional
νi
subspace of H containing ξ . We denote the restrictedstates of ̺ onto B(H )
ν νi 2
by ̺ . By (10), the density matrices of ̺ and ̺ have the following symmetric
2 1 2
forms:
e e
D = λ2P(ξ ), D = λ2P(ξ ). (11)
̺1 X i 1i ̺2 X i 2i
i f i
Since̺isanevenstate,itsrestriction̺ isevenandhenceitsdensitymatrix
2
D belongs to A(J) .
̺2 +
Ontheotherhand,theevenstate̺isinvariantundertheactionofAd(vI∪J)=
Ad(v v ):
I J
vI∪JD̺vI∪J =vIvJD̺vIvJ =D̺.
5
Acting the conditionalexpectationonto B(H ) with respect to the tracialstate
2
of B(H ) on the above equality, we obtain
1,2
v D v =D
J ̺2 J ̺2
f f
noting that v belongs to B(H )=B(H )′∩B(H ).
I 1 2 1,2
We denote B(H ) ≡ B(H )∩A(I∪J) , the set of all invariant elements
2 + 2 +
under Ad(v v ) in B(H ). By (8), B(H ) is equal to A(J) , and also to the
I J 2 2 + +
set of all invariant elements under Ad(v ) in B(H ). Therefore both D and
J 2 ̺2
D̺2 belong to B(H2)+. Accordingly, D̺2 is equal to D̺2 as the densityfof the
state ̺ restricted to B(H2)+, and hencef
D =D = λ2P(ξ ). (12)
̺2 ̺2 X i 2i
f i
From (11) and (12), it follows that ̺ and ̺ have the same nonzero eigen-
1 2
values and their multiplicities equal to {λ2}. Thus
i
S(̺ )=S(̺ ).
1 2
For a subset I of L, |I| denotes the number of sites in I.
Lemma 3. Let I be a finite subset and ̺ be a state of A(I). Let J be a finite
1
subset such that J∩I = ∅ and |J| ≥ |I|. Then there exists a pure state ̺ on
A(I∪J) satisfying
̺|A(I) =̺1. (13)
Moreover, if ̺ is even, then the above ̺ can be taken to be even.
1
Proof. We use the same notation as in the proof of the preceding lemma and
write A(I ∪ J) = B(H ), A(I) = B(H ), and A(I′|I ∪ J) = B(H ). Let
1,2 1 2
̺ = λ2P(ξ ), where λ >0, {ξ } is an orthonormal set of H , and P(ξ )
1 Pi i 1i i 1i 1 1i
istheprojectionoperatorontheone-dimensionalsubspaceofH containingξ .
1 1i
Since |J|≥|I| and hence dimH ≥dimH , we can take an orthonormal set of
2 1
vectors {ξ } of H having the same cardinality as {ξ }. Define a unit vector
2i 2 1i
ξ ∈ H by the same formula as (10) and let ̺ be its vector state, namely the
1,2
state whose density matrix is the projection operator on the one-dimensional
subspaceof H containing ξ. This ̺ is a pure state extensionof̺ to A(I∪J)
1,2 1
by its definition.
Assume now that ̺ is even, and hence its density matrix is in A(I) . For
1 +
each eigenvalue, the associated spectral projection is also even and commutes
with v , and its range is invariant under v . Therefore we can choose an or-
I I
thonormal basis of the range of the projection which consists of eigenvectors of
v . We take {ξ } to be a set of eigenvectors of v .
I 1i I
Since v belongs to B(H ) (=A(J) ), there exists an orthonormalbasis of
J 2 + +
H consistingofeigenvectorsofv . Duetotheassumption|J|≥|I|,wecantake
2 J
6
a set of different eigenvectors{ξ } of v such that for each i its eigenvalue, +1
2i J
or −1, is equal to that of ξ for v . Define a unit vector ξ by (10) using these
1i I
{ξ1i}and{ξ2i}. Since thisξ isaneigenvectorofvI∪J byitsdefinition,itsvector
state ̺ is even.
Combining the above two lemmas we obtain the following.
Proposition 4. Let I be a finite subset and ̺ be an even state of A(I). Let J
1
beafinitesubsetsuchthatJ∩I=∅and|J|≥|I|. Then thereexistsaneven pure
state̺ on A(I∪J) such that its restriction to A(I) is equal to̺ and the density
1
matrix of its restricted state ̺2 ≡ ̺|A(J) has the same nonzero eigenvalues and
their multiplicities as those of ̺ .
1
We may call the above state extension from ̺ to ̺ the symmetric purifica-
1
tion. Thanks to this, we obtain the following two theorems.
Theorem 5. Let I, J and K be mutually disjoint finite subsets. For every even
state ϕ, MONO-SSA
S(ϕI)+S(ϕJ)≤S(ϕK∪I)+S(ϕK∪J) (14)
is satisfied.
Proof. The equivalence of MONO-SSA and SSA for even states follows from
Proposition4 in the same way as (3) p164 of Ref. 1. Since SSA holds for every
state, MONO-SSA is valid for every even state.
Similarly, by using Proposition4 we immediately obtain the triangle inequality
for even states in much the same way as (3.1) of Ref. 1. We omit its proof.
Theorem 6. Let I and J be mutually disjoint finite subsets. For every even
state ϕ, the triangle inequality
(cid:12)S(ϕI)−S(ϕJ)(cid:12)≤S(ϕI∪J) (15)
(cid:12) (cid:12)
(cid:12) (cid:12)
holds.
3 Violation of MONO-SSA
In this section we give a certain class of noneven states.
pure on I∪K
tracial pure, noneven ◦ even
I K
J
FIG.1. A state not satisfying MONO-SSA.
7
We shallgivea sketchofourmodel indicatedby FIG.1. We cantakea pure
state ̺I∪K on I∪K whose restriction ̺K is a pure state, but ̺I is non-pure,
saythe tracialstate. Such̺I∪K doesnotsatisfythe triangleinequality,because
the entropies on I and on K are different, whereas the entropy on I∪K is zero.
It can be said that the pure state ̺I∪K has the asymmetric restrictions in our
terminology. This asymmetry is due to the large amount of the oddness of ̺ ,
K
whose precise meaning will be given soon. (Note however that for the infinite-
dimensional case, the GNS representations π and π should be unitarily
̺K ̺KΘ
equivalent,seeProposition8(i).) We takeanarbitraryevenstate̺ onJ. The
J
desired state on ̺I∪K∪J on I∪K∪J is given by the product state extension of
̺I∪K and ̺J, which will be denoted by ̺I∪K◦̺J.
We recallthe definitionofthetransitionprobability.14 Fortwostatesϕand
ψ of A(I) (where |I| is finite or infinite), take any representationπ of A(I) on a
Hilbert space H containing vectors Φ and Ψ such that
ϕ(A)=(Φ, π(A)Φ), ψ(A)=(Ψ, π(A)Ψ), (16)
for all A∈A(I). The transition probability between ϕ and ψ is given by
P(ϕ, ψ)≡sup|(Φ, Ψ)|2, (17)
where the supremum is taken over all H, π, Φ and Ψ as described above. For a
state ϕ of A(I), we define
p (ϕ)≡P(ϕ, ϕΘ)1/2, (18)
Θ
where ϕΘ denotes the state ϕΘ(A) = ϕ(Θ(A)), A ∈ A(I). Intuitively, p (ϕ)
Θ
quantifies the amount of oddness of the state ϕ. If p (ϕ) = 0 or nearby, then
Θ
we may say that the difference between ϕ and ϕΘ is large. If ϕ is even, p (ϕ)
Θ
takes obviously the maximum value 1.
The following is Lemma 3.1 of Ref. 2.
Lemma 7. If ̺ is a pure state of A(K) and π and π are unitarily equiv-
1 ̺1 ̺1Θ
alent, then there exists a self-adjoint unitary u ∈π (A(K) )′′ satisfying
1 ̺1 +
u π (A)u =π (Θ(A)), A∈A(K). (19)
1 ̺1 1 ̺1
The next proposition is a basis of our construction. It is a generalizationof
Ref. 7. The first paragraph is in principle excerpted from Theorem 4 (4) and
(5) of Ref. 2. The second paragraph is necessary for the argument of entropy.
Proposition 8. Let K and I be mutually disjoint subsets. Assume that ̺ is
1
a (noneven) pure state of A(K) satisfying p (̺ ) = 0. Assume that ̺ is an
Θ 1 2
even state of A(I). There exists a joint extension of ̺ and ̺ other than their
1 2
product state extension if and only if ̺ and ̺ satisfy the following pair of
1 2
conditions:
(i) π and π are unitarily equivalent.
̺1 ̺1Θ
(ii) There exists a state ̺ of A(I) such that ̺ 6=̺ Θ and
2 2 2
e 1 e e
̺ = ̺ +̺ Θ . (20)
2 2 2
2(cid:0) (cid:1)
e e
8
For each ̺ above, there exists the joint extension of ̺ and ̺ to A(K ∪I)
2 1 2
denoted by ψ which satisfies
e ̺2
f
ψ̺2(A1A2)=̺1(A1)̺2(A2+)+̺1(π̺1(A1)u1)̺2(A2−), (21)
f e
where ̺ is the GNS-extension of ̺ to π (A(K))′′.
1 1 ̺1
If K and I are finite subsets, then the entropy of ̺ is equal to that of ψ .
2 ̺2
e f
Proof. We shall show only the sufficiency of the pair of conditions (i) and (ii).
For the necessity of (i) see 5.2 in Ref. 2, and for that of (ii) see (d) in the proof
of its Theorem 4(4).
Let (H , π , Ω ) be a GNS triplet for ̺ and (H , π , Ω ) be that for
̺1 ̺1 ̺ 1 ̺2 ̺2 ̺
̺2. Define f f f
e
H ≡ H ⊗H , Ω ≡Ω ⊗Ω , (22)
̺1 ̺2 ̺ ̺
π(A1A2) ≡ π̺1(A1)⊗fπ̺2(A2+)+π̺1(Af1)u1⊗π̺2(A2−), (23)
f f
forA1 ∈A(K), A2 =A2++A2−, A2± ∈A(I)±. Let11 be the identity operator
of H and 1 be that of H .
̺1 2 ̺2
We cancheckthatthe opferatorsπ(A(K∪I)) satisfythe CAR byusing(19),
and hence π extends to a representation of A(K∪I). We define the state ψ
̺2
on A(K∪I) as f
ψ (A)≡(Ω, π(A)Ω) (24)
̺2
f
for A ∈ A(K∪I). The von Neumann algebra π(A(K ∪I))′′ is generated by
π̺1(A(K))′′ ⊗12, 11⊗π̺2(A(I)+)′′, and the weak closure of 11⊗π̺2(A(I)−),
where we have noted u1 ∈fπ̺1(A(K)+)′′ =B(H̺1). Therefore f
′′ ′′
π(A(K∪I)) =B(H )⊗π (A(I)) . (25)
̺1 ̺2
f
FromthisitfollowsthatthevectorΩiscyclicfortherepresentationπofA(K∪I)
in H. Hence (H, π, Ω) gives a GNS triplet for the state ψ on A(K∪I).
̺2
We have f
ψ̺2(A1A2)=(Ω, π(AA)Ω)=̺(A)̺(A+)+̺(π̺(A)u)̺(A−), (26)
f e e
for A1 ∈ A(K), A2 = A2+ +A2−, A2± ∈ A(I)±. Taking A2 = 1 in (26), we
obtain
ψ (A )=̺ (A ). (27)
̺2 1 1 1
f
We will then show
ψ (A )=̺ (A ). (28)
̺2 2 2 2
f
Under the condition of Lemma 7 (which is our case), we have
p (̺ )=|̺ (u )|, (29)
Θ 1 1 1
9
because the transition probability between the vector states of the algebra
B(H ) = π (A(K))′′ is equal to the (usual) transition probability of their
̺1 ̺1
vectors and hence p (̺ ) = |(Ω , u Ω )|. By the assumption p (̺ ) = 0,
Θ 1 ̺  ̺ Θ 1
(29) implies
̺ (u )=0. (30)
1 1
Setting A =1 in (26), we obtain
1
ψ̺2(A2) = ̺2(A2+)+̺1(u1)̺2(A2−)
f = ̺e(A ). e (31)
2 2+
e
By (20),
̺ (A )=̺ (A )=̺ (A ). (32)
2 2 2 2+ 2 2+
e
From (31) and (32), (28) follows. We have now shown that ψ is an extension
̺2
of ̺1 and ̺2. f
We will show the second paragraph. By (25) and the commutant theorem,
π(A(K∪I))′ = 1 ⊗π (A(I))′, where the commutant is taken in each GNS
1 ̺2
space. Thus we have thfe following isomorphism:
b7→1 ⊗b, b∈π (A(I))′ (33)
1 ̺2
f
from π (A(I))′ onto π(A(K∪I))′. Furthermore by (22) we obtain
̺2
f
(Ω, (1 ⊗b)Ω)=(Ω ⊗Ω , Ω ⊗bΩ )=(Ω , bΩ ). (34)
 ̺ ̺ ̺ ̺ ̺ ̺
f f f f
FromtheassumptionthatKandIarefinitesubsets(whichwehavenotused
so far), π (A(I))′ and π(A(K∪I))′ are both finite-dimensional type I factors.
̺2
Now wfe note the following basic fact about GNS representations for states
of finite-dimensional type I factors which can be considered as the counterpart
ofLemma2 forthe usualcase,namelyfor apairofisomorphicsystemscoupled
by tensor product. Let ω be a state of a finite-dimensional type I factor A and
(H , π , Ω ) denote a GNS triplet of ω. The GNS vector Ω of ω induces a
ω ω ω ω
state onthe commutantπ (A)′ whose expectationvalue for a∈π (A)′ is given
ω ω
by (Ω , aΩ ). We call this state on π (A)′ “ω on the commutant”. (In our
ω ω ω
terminology, the pure state with respect to Ω on B(H ) = π (A)⊗π (A)′
ω ω ω ω
gives a symmetric purification of ω. Also ω on the commutant is symmetric
to ω.) Then the entropy of ω on A (equivalently that on π (A)) is equal to
ω
the entropy of ω on the commutant by the same reason described in Lemma 2.
We note that this holds for a general C∗-algebra if the GNS representation of
a given state generates a type I von Neumann algebra with a discrete center.
Similarly the extension of Lemma 2 is possible under the above condition on
the state.
From the above fact with (33) and (34), we deduce the equality of the en-
tropies of ̺ and of ψ
2 ̺2
e f
10