Quantum structures and their transformations
Lajos Molnár
University of Debrecen, Hungary
Operator Theory and The Principles of Quantum Mechanics
CIMPA-MOROCCO School
Faculté des Sciences, Meknès, Morocco, September 8-17, 2014
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Plan
Notation, basic concepts;
Operator structures in the foundations of QM, quantum structures;
Automorphisms, isomorphism of structures of self-adjoint operatotrs (observables)
and Hilbert space effects;
Automorphisms of structures of rank-one projections (pure states) and density
operators (mixed states).
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Notation, basic concepts
H: complex Hilbert space;
h., .i: inner product, conjugate linear in the 2nd variable;
k.k: norm;
B(H): algebra of all bounded linear operators on H;
0, I: zero, identity;
x ⊗ y : x, y ∈ H, (x ⊗ y )(z) = hz, y ix, z ∈ H;
A∗ : adjoint of A ∈ B(H), hA∗ x, y i = hx, Ay i, x, y ∈ H;
self-adjoint operator: A∗ = A;
Bs (H): set of all self-adjoint operators on H;
projection: self-adjoint idempotent; P ∗ = P, P 2 = P;
P(H): set of all projections on H;
order on Bs (H): A ≤ B iff hAx, xi ≤ hBx, xi, x, y ∈ H;
positive operator: 0 ≤ A;
Hilbert space effect: 0 ≤ A ≤ I;
E(H): set of all effects on H, Hilbert space effect algebra;
Tr: for any
P 0 ≤ A, A ∈ Bs (H) and orthonormal basis (ei ) in H let
Tr A = i hAei , ei i; well-defined;
Density operator: 0 ≤ A, A ∈ Bs (H) with Tr A = 1;
D(H): set of all density operators on H;
P1 (H): set of all rank-one projections on H; P1 (H) ⊂ D(H);
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Related QM concepts
self-adjoint operator ←→ (bounded) observable
Hilbert space effect ←→ two-valued quantum measurement that can be unsharp
density operator ←→ (mixed) state
rank-one projection ←→ pure state
Other representations of pure states: unit rays (original approach), one-dimensional
subspaces
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Quantum structures
Bs (H):
Jordan algebra with the Jordan product (AB + BA)/2. Meaning that it is a (real)
linear space with a compatible multiplication • that satisfies A • B = B • A and
(A • B) • (A • A) = A • (B • (A • A));
Partially ordered set with the usual order ≤.
Another important relation: commutativity (compatibility)
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Quantum structures
E(H):
(E1) Partial addition ⊕ on E(H): if A, B ∈ E(H) are such that A + B (the usual sum of
operators) belongs to E(H), then A ⊕ B := A + B.
(E2) The usual order ≤ gives rise to a partial order on E(H). The map
⊥: A 7→ A⊥ = I − A defines a kind of orthocomplementation on E(H).
(E3) E(H) is a convex subset of the linear space Bs (H). A convex combination
λA + (1 − λ)B of effects A, B is called a mixture.
(E4) Sequential product: if A, B ∈ E(H), then their sequential product is
A ◦ B := A1/2 BA1/2 .
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Quantum structures
P1 (H):
transition probability: Tr PQ.
D(H):
operation: convex combination;
various numerical quantities of two variables: relative entropies, divergences, etc.
Details will be given later.
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Corresponding automorphisms
Bs (H):
Jordan automorphism: φ : Bs (H) → Bs (H) a bijective (real) linear transformation
which satisfies φ(AB + BA) = φ(A)φ(B) + φ(B)φ(A), A, B ∈ Bs (H). Also called
Segal automorphism.
Order automorphism: φ : Bs (H) → Bs (H) a bijective map (not necessarily linear)
such that for any A, B ∈ Bs (H) we have
A ≤ B ⇐⇒ φ(A) ≤ φ(B).
Commutativity (compatibility) preservers: φ : Bs (H) → Bs (H) a bijective map (not
necessarily linear) such that for any A, B ∈ Bs (H) we have
AB = BA ⇐⇒ φ(A)φ(B) = φ(B)φ(A).
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Corresponding automorphisms
Four different kinds of automorphisms of E(H):
(EA1) The automorphisms of E(H) which correspond to the partial addition ⊕ are called
E-automorphisms (or effect automorphisms). A bijective map φ : E(H) → E(H) is
an E-automorphism if for any A, B ∈ E(H) it satisfies
A + B ∈ E(H) ⇐⇒ φ(A) + φ(B) ∈ E(H)
and, in this case, we have
φ(A + B) = φ(A) + φ(B).
(EA2) The automorphisms of E(H) with respect to the partial order ≤ and the
orthocomplementation ⊥ are called ortho-order automorphisms. A bijective map
φ : E(H) → E(H) is an ortho-order automorphism if for any A, B ∈ E(H) we
A ≤ B ⇐⇒ φ(A) ≤ φ(B)
and
φ(A⊥ ) = φ(I − A) = I − φ(A) = φ(A)⊥ .
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Corresponding automorphisms
(EA3) The automorphisms of E(H) corresponding to the operation of mixtures are called
mixture automorphisms. A bijective map φ : E(H) → E(H) is a mixture
automorphism if it is affine, i.e., satisfies
φ(λA + (1 − λ)B) = λφ(A) + (1 − λ)φ(B)
for all A, B ∈ E(H) and λ ∈ [0, 1].
(EA4) The automorphisms of E(H) with respect to the sequential product ◦ are called
sequential automorphisms. A bijective map φ : E(H) → E(H) is a sequential
automorphism if
φ(A ◦ B) = φ(A) ◦ φ(B)
holds for all A, B ∈ E(H).
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Corresponding automorphisms
P1 (H):
(Traditional) quantum mechanical symmetry transformation: φ : P1 (H) → P1 (H) a
bijective map such that for any P, Q ∈ P1 (H) we have Tr φ(P)φ(Q) = Tr PQ.
D(H):
Mixture automorphism, or S-automorphism, or Kadison automorphism: affine
bijection of D(H).
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Structure of those automorphisms
Unitary operator on H: surjective linear isometry.
Conjugate linear operator on H: additive transformation which satisfies A(λx) = λAx
for all x ∈ H and λ ∈ C.
Antiunitary operator on H: surjective conjugate-linear isometry.
"Almost all" of the mentioned automorphisms are of the form
A 7−→ UAU ∗
with some unitary or antiunitary operator U on H.
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Structure of those automorphisms
Exceptions:
Order automorphisms of Bs (H): In case dim H ≥ 2 they are exactly the maps of the
form
A 7−→ TAT ∗ + X ,
where T is a bounded invertible linear or conjugate-linear map and X ∈ Bs (H).
Commutativity preservers of Bs (H): In the case where H is separable and dim H ≥ 3
any such map is of the form
A 7−→ UfA (A)U ∗
with real valued bounded Borel functions fA on the spectrum of A and a unitary or
antiunitary operator U on H.
Mixture automorphisms of E(H): They are the maps of the forms
A 7−→ UAU ∗ ,
A 7−→ U(I − A)U ∗ .
Remarks on the ortho-order automorphisms of E(H).
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Jordan automorphisms of Bs (H)
Jordan homomorphisms, algebra homomorphisms, antihomomorphisms of algebras.
Prime algebra: aAb = {0} implies a = 0 or b = 0.
Herstein’s theorem: A Jordan homomorphism from an algebra onto a prime algebra is
necessarily a homomorphism or an antihomomorphism.
Theorem 1
Every Jordan *-isomorphism of B(H) is either of the form A 7→ UAU ∗ with a unitary
U : H → H or of the form A 7→ UA∗ U ∗ with an antiunitary U : H → H
Corollary: Jordan (Segal) automorphisms of Bs (H) are all of the form A 7→ UAU ∗ with
a unitary or antiunitary operator U : H → H.
Much more on different sorts of automorphisms of Bs (H) and E(H) will be given in the
lectures by P. Šemrl.
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Isomorphisms between observables and effects in von Neumann algebras
In what follows let A, B be von Neumann algebras.
As : space of all self-adjoint elements of A.
Positive elements in A, order on As .
E(A): effects, the set of all elements A in As such that 0 ≤ A ≤ I.
Operations, relations and the corresponding concepts of automorphisms,
isomorphisms previously formulated for Bs (H), E(H) are extended for As , E(A)
trivially.
Easy observation: Jordan (Segal) isomorphisms between As and Bs are bijective
linear transformations which extend to Jordan *-isomorphisms from A onto B.
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Isomorphisms between observables and effects in von Neumann algebras
Jordan triple maps: maps which are homomorphisms under the triple product ABA.
Here no linearity is assumed! Example: Jordan homomorphisms are Jordan triple
maps.
Theorem 2
Let A, B be von Neumann algebras. Suppose that A does not have a commutative
direct summand. Let φ : As → Bs be a Jordan triple isomorphism, i.e., bijective map
which satisfies
φ(ABA) = φ(A)φ(B)φ(A),
A, B ∈ As .
Then we have direct decompositions
A = A1 ⊕ A2 ⊕ A3 ⊕ A4
and B = B1 ⊕ B2 ⊕ B3 ⊕ B4
and bijective maps
Φ1 : A1 → B1 ,
Φ2 : A2 → B2 ,
Φ3 : A3 → B3 ,
Φ4 : A4 → B4
such that Φ1 , Φ2 are linear *-isomorphisms, Φ3 , Φ4 are linear *-antiisomorphisms and
φ = Φ1 ⊕ (−Φ2 ) ⊕ Φ3 ⊕ (−Φ4 )
holds on As .
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Isomorphisms between observables and effects in von Neumann algebras
Commutative case: easy counterexample.
Pick a discontinuous bijective additive function a : R → R (consider R as a linear space
over the rationals) and set m(x) = exp(a(ln(x))), x > 0. Define φ : R → R by
sgn(x)m(|x|), if x 6= 0;
φ(x) =
0,
if x = 0.
φ is a wide multiplicative bijection of R.
Order automorphisms, commutativity preservers: no result in the setting of von
Neumann algebras.
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Isomorphisms between observables and effects in von Neumann algebras
State: positive linear functional ω : A → C normalized by ω(I) = 1.
S(A): set of all states, convex set.
Pure state: extreme point of S(A).
Important numerical quantities corresponding to classical random variables: mean
value, moments, variance, etc. Similar quantities for observables:
Mean value: The mean value of A ∈ As in the state ω ∈ S(A) is m(A, ω) = ω(A).
Observe kAk = sup{|m(A, ω)| : ω ∈ S(A)} for any A ∈ As .
Variance: The variance of A in the state ω is
var(A, ω) = m((A − m(A, ω)I)2 , ω) = ω((A − ω(A)I)2 ) = ω(A2 ) − ω(A)2 .
Maximal deviation:
kAkv = sup{(var(A, ω))1/2 : ω ∈ S(A)}.
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Isomorphisms between observables and effects in von Neumann algebras
What are the linear transformations on As preserving mean values, or more generally,
the norm?
Theorem 3
Let A, B be unital C ∗ -algebras. Any surjective real-linear isometry φ : As → Bs can be
written as
φ(A) = UJ(A)
(A ∈ As ),
where J : A → B is a Jordan *-isomorphism and U is a central symmetry (self-adjoint
unitary) in B.
What are the linear transformations that preserve variances, or more generally, the
maximal deviation?
Meaning of maximal deviation:
Proposition 4
For any A ∈ As we have
kAkv = (diam σ(A))/2 = inf kA + λIk.
λ∈R
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Isomorphisms between observables and effects in von Neumann algebras
Theorem 5
Let φ : As → Bs be a bijective linear transformation which preserves the maximal
deviation, i.e., which has the property that
kφ(A)kv = kAkv
(A ∈ As ).
If A is a factor, then there is an algebra *-isomorphism or *-antiisomorphism θ : A → B
and a real-linear functional f : As → R such that φ is either of the form
φ(A) = θ(A) + f (A)I
(A ∈ As )
or of the form
φ(A) = −θ(A) + f (A)I
(A ∈ As ).
If A is not a factor and it does not have a type I2 direct summand, then a similar
representation holds for φ as above with the only difference that then θ : A → B is a
Jordan *-isomorphism.
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Isomorphisms between observables and effects in von Neumann algebras
Isomorphisms of effects in von Neumann algebras.
E-isomorphisms: can be reduced to the case of mixture isomorphisms.
Ortho-order isomorphisms: no result.
Mixture isomorphisms: we introduce the more general concept of maps preserving
mixtures in both directions.
The bijective map φ : E(A) → E(B) is said to preserve mixtures in both directions if it
has the property that for any A, B, C ∈ E(A) we have that
A is a mixture (i.e. a convex combination) of B and C
if and only if
φ(A) is a mixture of φ(B) and φ(C).
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Isomorphisms between observables and effects in von Neumann algebras
Theorem 6
Let A, B be von Neumann algebras. Suppose that one of them has dimension at least
2. Let φ : E(A) → E(B) be a bijective map which preserves mixtures in both
directions. Then there are direct sum decompositions
A = A1 ⊕ A2 ⊕ A3 ⊕ A4 ,
B = B1 ⊕ B2 ⊕ B3 ⊕ B4
and bijective linear maps φk : Ak → Bk (k = 1, . . . , 4) such that φ1 , φ2 are algebra
*-isomorphisms, φ3 , φ4 are algebra *-antiisomorphisms and for any A ∈ E(A) with
decomposition A = A1 + A2 + A3 + A4 (Ak ∈ E(Ak ), k = 1, . . . , 4) we have
φ(A) = φ1 (A1 ) + φ2 (I − A2 ) + φ3 (A3 ) + φ4 (I − A4 ).
In particular, φ is a mixture isomorphism.
Remark: one dimensional case, E-automorphisms are mixture-automorphisms.
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Isomorphisms between observables and effects in von Neumann algebras
For curiosity: The proof of the theorem relies essentially on the following key
observation. Denote by P(A) the set of all projections in A.
Lemma 7
A projection in P(A) is central (i.e., commutes with every element of A) if and only if it
can not be connected with a different projection in P(A) by a continuous path in P(A).
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Isomorphisms between observables and effects in von Neumann algebras
Sequential isomorphisms:
Theorem 8
Let A, B be von Neumann algebras and let φ : E(A) → E(B) be a sequential
isomorphism. Then there are direct decompositions
A = A1 ⊕ A2 ⊕ A3
and B = B1 ⊕ B2 ⊕ B3
and bijective maps
φ1 : E(A1 ) → E(B1 ),
Φ2 : A2 → B2 ,
Φ3 : A3 → B3
such that
(i) A1 , B1 are commutative von Neumann algebras and the algebras A2 ⊕ A3 ,
B2 ⊕ B3 have no commutative direct summands;
(ii) φ1 is a multiplicative bijection, Φ2 is a *-isomorphism, Φ3 is a *-antiisomorphism
and φ = φ1 ⊕ Φ2 ⊕ Φ3 holds on E(A).
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Isomorphisms between observables and effects in von Neumann algebras
For factor von Neumann algebras we have the following immediate corollary.
Corollary 9
Let A, B 6= CI be factors and let φ : E(A) → E(B) be a sequential isomorphism. Then
φ extends either to a *-isomorphism or to a *-antiisomorphism between the algebras A
and B.
1-dim case: not true.
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Isomorphisms between observables and effects in von Neumann algebras
Short topological detour:
Between the ’commutative parts’ of the underlying algebras the behavior of φ can be
quite ’irregular’.
Consider arbitrary compact Hausdorff spaces X , Y . Let C(X ) denote the algebra of all
continuous complex valued functions on X . Assume ϕ : Y → X is a homeomorphism
and take any strictly positive continuous function p : Y →]0, ∞[ and define
φ : E(C(X )) → E(C(Y )) by
φ(f )(y ) = f (ϕ(y ))p(y )
(y ∈ Y , f ∈ E(C(X ))).
Then φ is a sequential isomorphism between E(C(X )) and E(C(Y )) which, in general,
does not extend to an algebra isomorphism between C(X ) and C(Y ). (If A, B are
commutative, then the sequential product on E(A), E(B) coincides with the ordinary
product and hence the sequential isomorphisms between E(A) and E(B) are just the
multiplicative bijections or, in other words, semigroup isomorphisms.)
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Isomorphisms between observables and effects in von Neumann algebras
J. Marovt: If X , Y are first countable compact Hausdorff spaces, then every sequential
isomorphism between E(C(X )) and E(C(Y )) are of the standard form that appears
above.
Marovt’s conjecture: First countability can in fact be dropped.
Z. Ercan and S. Önal: Not true, counterexample (X = Y = β]0, 1[)
J. Araujo: Characterization of spaces X for which there exists a sequential
automorphism of E(C(X )) which is not standard. Roughly speaking, he proved that
this is the case if and only if X is the Stone-Čech compactification of a proper subset of
itself.
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Isomorphisms between observables and effects in von Neumann algebras
For curiosity: sequential endomorphisms of finite dim Hilbert space effect algebras.
Above we have described sequential isomorphisms between von Neumann algebra
effects.
Problem: what about sequential endomorphisms of E(H)?
In what follows, H is a finite dimensional complex Hilbert space with dim H ≥ 3 and we
denote n = dim H.
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Isomorphisms between observables and effects in von Neumann algebras
Theorem 10
Assume φ : E(H) → E(H) is a continuous sequential endomorphism. Then we have
the following four possibilities:
(i) there exists a unitary or an antiunitary operator U on H and a non-negative real
number c such that
φ(A) = (det A)c UAU ∗ ,
A ∈ E(H);
(ii) there exists a unitary or an antiunitary operator V on H such that
φ(A) = V (adj A)V ∗ ,
A ∈ E(H);
(iii) there exists a unitary or an antiunitary operator W on H and a real number d > 1
such that
(det A)d WA−1 W ∗ , if A ∈ E(H) is invertible;
φ(A) =
0,
otherwise;
(iv) there are pairwise orthogonal rank-one projections P1 , . . . , Pm on H
(m ≤ n = dim H) and non-negative real numbers c1 , . . . , cm such that
φ(A) =
m
X
k =1
(det A)ck Pk ,
A ∈ E(H).
Problem: What if dim H = 2?
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Wigner’s theorem
Original formulation, different interpretations. Proof.
To be presented on board.
Remarks:
non-surjective version;
Uhlhorn’s version on bijective orthogonality preserving maps on P1 (H)
(dim H ≥ 3).
Generalizations:
Wigner’s theorem in Hilbert modules;
preserving principal angles on Grassmannians. Problem: structure of (not
necessarily surjective) maps on Pn (H) which preserve the trace of product;
Győry-Šemrl’s generalization of Uhlhorn’s version on orthogonality preservers on
Grassmannians;
Li-Plevnik-Šemrl’s generalization: bijective maps on P1 (H), H is finite
dimensional, preserving a fixed transition probability;
Wigner’s theorem in indefinite inner product spaces;
Šemrl’s version of Wigner’s theorem in quaternionic indefinite inner product
spaces;
Uhlhorn’s version in indefinite inner product spaces;
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Wigner’s theorem
Generalization in another direction:
√
Observe that kP − Qk = 1 − Tr PQ, P, Q ∈ P1 (H). Therefore, any map
φ : P1 (H) → P1 (H) preserves the transition probability iff φ is an isometry.
Let n be a fixed positive integer and denote by Pn (H) the set of all rank-n projections
on H.
Theorem 11
Let dim H ≥ 4n. If the surjective map φ : Pn (H) → Pn (H) satisfies
kφ(P) − φ(Q)k = kP − Qk,
P, Q ∈ Pn (H)
then there exists either a unitary or an antiunitary operator U on H such that
φ(P) = UPU ∗ ,
P ∈ Pn (H).
For curiosity: in the proof we need a characterization of orthogonality in Pn (H).
Lemma 12
For any two commuting projections P, Q in Pn (H) we have that P and Q are
orthogonal if and only if the the commutant {P, Q}c as a subspace of the metric space
Pn (H) has a pathwise connected component K such that the maximal number of
pairwise commuting projections of rank n in K c is exactly 2n
.
n
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Mixture automorphisms of D(H)
Mixture (Kadison) automorphisms:
Theorem 13
Every bijective affine map φ of D(H) is of the form
φ(A) = UAU ∗ ,
A ∈ D(H)
where U : H → H is a unitary or antiunitary operator.
Proof to be presented on board.
He-Hou-Li have recently showed that assuming dim H ≥ 2, a bijective map
φ : D(H) → D(H) preserves mixture (in one or in both directions) iff it is of the form
φ(A) =
TAT ∗
,
Tr TAT ∗
A ∈ D(H),
where T : H → H is an invertible bounded linear or conjugate-linear operator.
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Transformations on density ops preserving q rel entropy or related quantities
In what follows we assume dim H < ∞.
von Neumann entropy:
S(A) = − Tr A log A,
A ∈ D(H).
Relative entropies: measures of distinguishability between states
Umegaki relative entropy: For any pair A, B ∈ D(H), the Umegaki relative entropy
S(A||B) is defined by
Tr[A(log A − log B)], if supp A ⊂ supp B;
S(A||B) =
+∞,
otherwise.
S(A||B) is always nonnegative and equals zero if and only if A = B.
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Transformations on density ops preserving q rel entropy or related quantities
Theorem 14
Let φ : D(H) → D(H) be a bijective map which satisfies
S(φ(A)kφ(B)) = S(AkB),
A, B ∈ D(H)
Then there exists either a unitary or an antiunitary operator U on H such that φ is of the
form
φ(A) = UAU ∗ ,
A ∈ D(H).
Proof to be presented.
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3
operator U
on H is a symmetry transformation.
that the converse statement is also true:
Wigner’s theorem says
every symmetry transformation
can be obtained in that way. Hence, our present result can be considered as
a Wigner-type result concerning state transformations leaving the relative
entropy invariant. Similar results of the same spirit on state transformations
preserving Uhlmann’s fidelity, trace distance, Bures distance, or a measure
of compatibility can be found in the papers [2], [4], [5] (also see Sections 2.3
and 2.4 in our recent book [3]).
In what follows we formulate and prove the already announced result.
Theorem. Let φ : S (H ) → S (H ) be a bijective map which preserves the
relative entropy, i.e., which satisfies
S (φ(ρ)||φ(σ)) = S (ρ||σ)
for every ρ, σ ∈ S (H ). Then there is an either unitary or antiunitary operator U on H such that φ is of the form
φ(ρ) = U ρU ∗
(ρ ∈ S (H )).
Proof. As the relative entropy S (ρ||σ) is finite if and only if supp ρ ⊂ supp σ,
we deduce that φ has the property
(2)
supp ρ ⊂ supp σ ⇐⇒ supp φ(ρ) ⊂ supp φ(σ)
for any ρ, σ ∈ S (H ).
As equality of sets can be expressed by two-sided
inclusion, we infer that
supp ρ = supp σ ⇐⇒ supp φ(ρ) = supp φ(σ)
and next that
(3)
supp ρ ( supp σ ⇐⇒ supp φ(ρ) ( supp φ(σ)
4
LAJOS MOLNÁR
for any ρ, σ ∈ S (H ).
It is easy to see that the rank of an operator ρ ∈ S (H ) is n if and only if
there is a chain
supp ρ1 ( supp ρ2 ( . . . ( supp ρn ⊂ supp ρ
of supports of length n and there does not exist another such chain of length
strictly larger than n. By the properties (2), (3) of φ it now follows that φ
preserves the rank of the elements of S (H ).
In particular, ρ ∈ S (H ) is a
rank-one pro jection if and only if so is φ(ρ).
Let σ ∈ S (H ) be a rank-2 operator of the form σ = λp + µq, where p, q ∈
S (H ) are mutually orthogonal rank-one projections and 0 < λ < µ < 1 with
λ + µ = 1. For any rank-one projection r on H we easily have


 −[log λ · tr rp + log µ · tr rq],
if supp r ⊂ supp σ;
S (r||σ) =

 ∞,
otherwise.
(4)
As r varies, the quantities tr rp, tr rq run through the set of all pairs of
nonnegative real numbers with sum 1.
Consequently, the relative entropy
S (r||σ) runs through the closed interval [− log µ, − log λ] plus the possible
value ∞.
This shows that from that set of entropies we can recover the
eigenvalues of the operator σ.
Using this observation and the facts that φ
preserves the relative entropy and φ(r) runs through the whole set of rankone pro jections, we deduce that the rank-2 operator φ(σ) is of the form
φ(σ) = λp0 + µq 0
with some mutually orthogonal rank-one pro jections p0 , q 0 on H . Clearly, any
element of a nontrivial compact real interval is a unique convex combination
of the endpoints. Therefore, we have S (r||σ) = − log λ if and only if tr rp =
5
1.
It follows, for example, from the criterion of equality in the Cauchy-
Schwarz inequality that tr rp = 1 holds if and only if r = p. We now have
the following chain of equivalences
r = p ⇐⇒ S (r||σ) = − log λ ⇐⇒
S (φ(r)||φ(σ)) = − log λ ⇐⇒ S (φ(r)||λp0 + µq 0 ) = − log λ ⇐⇒ φ(r) = p0
yielding φ(p) = p0 . We similarly have φ(q) = q 0 . Therefore, we obtain
(5)
φ(λp + µq) = λφ(p) + µφ(q).
In particular, it follows that φ preserves the mutual orthogonality between
rank-one pro jections.
Let now p, r be different rank-one pro jections which are not orthogonal
to each other.
Pick a rank-one pro jection q which is orthogonal to p and
has the property that supp r ⊂ supp p + supp q. Choose 0 < λ < µ < 1 with
λ + µ = 1 as above. By (4) and (5) we obtain that
(6)
−[log λ · tr rp + log µ · tr rq] = S (r||λp + µq) =
S (φ(r)||λφ(p) + µφ(q)) = −[log λ · φ(r)φ(p) + log µ · tr φ(r)φ(q)].
We know that the rank-one projections φ(p), φ(q) are mutually orthogonal
and
supp φ(r) ⊂ supp φ(λp + µq) = supp(λφ(p) + µφ(q)) = supp φ(p) + supp φ(q).
It follows that tr φ(r)φ(p), tr φ(r)φ(q) are nonnegative real numbers with
sum 1.
Referring again to the easy fact that any element of a nontrivial
compact real interval is a unique convex combination of the endpoints, we
deduce from (6) that
tr φ(r)φ(p) = tr rp.
6
LAJOS MOLNÁR
This shows that φ when restricted to the set of all rank-one pro jections on H
gives rise to a bijective transformation preserving the transition probability,
i.e., a symmetry transformation.
By Wigner’s fundamental theorem we
infer that there is a unitary or antiunitary operator U
φ(p) = U pU ∗ for every rank-one projection p on H .
transformation ψ : ρ 7→
U ∗ φ(ρ)U .
on H
such that
Now, consider the
Clearly, this is a bijective map on S (H )
which preserves the relative entropy and has the additional property that
ψ(p) = p holds for every rank-one projection p.
In what follows we prove
that ψ(ρ) = ρ holds for every ρ ∈ S (H ).
Let ρ ∈ S (H ) be arbitrary. As ψ preserves the relative entropy, similarly
to (2), for any rank-one projection p on H we have
supp p ⊂ supp ρ ⇐⇒ supp p = supp ψ(p) ⊂ supp ψ(ρ)
which gives us that supp ρ = supp ψ(ρ). Moreover, for every such rank-one
pro jection p on H we have
(7)
− tr p log ρ = S (p||ρ) = S (ψ(p)||ψ(ρ)) = S (p||ψ(ρ)) = − tr p log ψ(ρ).
Let px denote the pro jection onto the one-dimensional subspace of H generated by the unit vector x ∈ H . For any unit vector x ∈ supp ρ = supp ψ(ρ),
inserting px into the equality (7) we see that
hx| log ρ|xi = hx| log ψ(ρ)|xi.
From this we deduce that log ρ
=
log ψ(ρ).
This implies ρ
=
ψ(ρ)
=
U ∗ φ(ρ)U
and hence we have φ(ρ) = U ρU ∗ completing the proof of the
theorem.
Transformations on density ops preserving q rel entropy or related quantities
Jensen-Shannon divergence:
S A 21 (A + B) + S B 12 (A + B)
SJS (AkB) =
,
2
A, B ∈ D(H).
A sort of symmetrization of relative entropy.
p
For curiosity: SJs is conjectured to be a true metric D(H). Proved only if dim H = 2.
Theorem 15
Assume that φ : D(H) → D(H) is a bijection satisfying
SJS (φ(A)kφ(B)) = SJS (AkB),
A, B ∈ D(H).
Then there is a unitary or an antiunitary operator U on H such that
φ(A) = UAU ∗ ,
A ∈ D(H).
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Transformations on density ops preserving q rel entropy or related quantities
Theorem 16
The condition of bijectivity of maps on D(H) preserving relative entropy can be relaxed.
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Transformations on density ops preserving q rel entropy or related quantities
Several concepts of relative entropy: Let A, B ∈ D(H).
(i) Umegaki relative entropy: S(AkB) = Tr(A(log A − log B)) if supp A ⊂ supp B, and
S(AkB) = +∞ otherwise;
(ii) Belavkin-Staszewski
entropy:
√ relative
√
√
SBS (AkB) = Tr( A log AB −1 A) if supp A ⊂ supp B, and SBS (AkB) = +∞
otherwise;
(iii) Tsallis relative entropy: For given 0 < q < 1,
ST (AkB) = (1/(1 − q))(1 − Tr Aq B 1−q );
(iv) Quadratic relative entropy: SQ (AkB) = Tr A−1 (A − B)2 if supp B ⊂ supp A, and
SQ (AkB) = +∞ otherwise;
(v) Jensen-Shannon
divergence:
SJS (AkB) = 21 S(A 21 (A + B) ) + S(B 21 (A + B)) .
By the −1-th power of an element of D(H) we mean the inverse of its restriction onto
its support.
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Transformations on density ops preserving q rel entropy or related quantities
Theorem 17
Let X (.k.) denote any of the relative entropies (i)–(v). Suppose that φ : D(H) → D(H)
is a transformation such that
X (φ(A)kφ(B)) = X (AkB)
holds for all A, B ∈ D(H). Then we have either a unitary or an antiunitary operator U
on H such that φ is of the form
φ(A) = UAU ∗ ,
A ∈ D(H).
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Transformations on density ops preserving q rel entropy or related quantities
Csiszár’s f -divergence in classical information theory:
Let f : [0, ∞[→ R be a convex function and let P = (p1 , · · · , pn ), Q = (q1 , · · · , qn ) be
probabtility distributions. Then the f -divergence of P and Q is
X
X
pi
Df (PkQ) =
qi f
+α
pi ,
qi
{i:qi 6=0}
where α = limx→∞
{i:qi =0}
f (x)
.
x
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Transformations on density ops preserving q rel entropy or related quantities
Quantum f -divergences: quantum analogue of Csiszár’s f -divergence.
Denote by Pd(H) the set of all invertible positive operators on H.
Let A, B ∈ Pd(H). Define the linear transformations LA , RB : B(H) → B(H) by
LA X = AX ,
RB X = XB,
X ∈ B(H).
They are commuting positive operators on B(H) equipped with the standard
(Hilbert-Schmidt) inner product hX , Y i = Tr XY ∗ .
Let f : [0, ∞[→ R be a function and K ∈ B(H) an operator. For any positive operator A
and B ∈ Pd(H), the quantity
SfK (A||B) = Tr K ∗ f (LA RB −1 )RB K
is called quasi-entropy that was introduced by D. Petz.
Important properties of quasi-entropy :
monotonicity (assuming f is operator monotone and f (0) ≥ 0);
joint convexity (assuming f is operator convex).
If K = I, f (t) = t log t, t > 0, we have
SfI (A||B) = Sf (A||B) = Tr[A(log A − log B)],
A, B ∈ Pd(H),
i.e., the Umegaki relative entropy of A, B.
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Transformations on density ops preserving q rel entropy or related quantities
Important particular case of quasi-entropy: f -divergence of density operators.
Assume f is continuous on the open interval ]0, ∞[ and the limit
α := lim
x→∞
f (x)
x
exists in [−∞, ∞]. Then the limit
lim Sf (A||B + I)
&0
exists for all positive A, B and is called the f -divergence of A and B. The following
formula can be derived.
For any λ ∈ R denote by Pλ , respectively by Qλ the projection on H projecting onto the
kernel of A − λI, respectively onto the kernel of B − λI. The f -divergence of A and B is


X
X
a

Sf (A||B) =
bf
Tr Pa Qb + αa Tr Pa Q0  .
b
a∈σ(A)
b∈σ(B)\{0}
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Transformations on density ops preserving q rel entropy or related quantities
1
If f (x) = x log x, x > 0 and f (0) = 0, then
Tr A(log A − log B),
Sf (AkB) = S(AkB) =
∞,
supp A ⊂ supp B
otherwise
which is the standard Umegaki relative entropy of A, B.
2
3
Let q ∈]0, 1[ and define the functions fq : [0, ∞[→ R by fq (x) = (1 − x q )/(1 − q),
x ≥ 0. Then
1 − Tr Aq B 1−q
Sfq (AkB) =
1−q
is the quantum Tsallis relative entropy of A, B.
√ 2
√
√
If f (x) = ( x − 1)2 , x ≥ 0, then Sf (AkB) = A − B HS , where k.kHS stands
for the Hilbert-Schmidt norm (Frobenius norm of matrices).
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Transformations on density ops preserving q rel entropy or related quantities
Theorem 18
Assume that f : [0, ∞[→ R is a strictly convex function and φ : D(H) → D(H) is a
transformation satisfying
Sf (φ(A)||φ(B)) = Sf (A||B),
A, B ∈ D(H).
Then there is either a unitary or an antiunitary operator U on H such that φ is of the
form
φ(A) = UAU ∗ ,
A ∈ D(H).
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Transformations on density ops preserving q rel entropy or related quantities
A fresh result:
Theorem 19
Let K , L ∈ B(H) be invertible, f (t) = t log t, t > 0. Assume φ : Pd(H) → Pd(H) is a
bijective map satisfying
SfL (φ(A)||φ(B)) = SfK (A, B),
A, B ∈ Pd(H).
Then we have a unitary or antiunitary operator U on H and a positive scalar λ such that
φ(A) = λUAU ∗ ,
√
and L = (1/ λ)UKU ∗ .
A ∈ Pd(H)
Problem: What about general strictly convex f ?
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Transformations on density ops preserving q rel entropy or related quantities
Holevo bound:
Let (λ1 , . . . , λm ) be a probability distribution. The Holevo bound (or information) of a
collection (or ensemble) (A1 , . . . , Am ) of density operators is
!
m
m
X
X
χ(A1 , . . . , Am ) = S
λk Ak −
λk S(Ak ).
k =1
k =1
This quantity plays an important role in quantum communication. According to a
fundamental result of Holevo, this provides an upper bound for the information that can
be sent over a quantum channel.
The Holevo bound is nonnegative and equals 0 iff A1 = . . . = Am . Moreover
m
!
m
X
X
χ(A1 , . . . , Am ) =
λk S Ak λl Al .
k =1
Observe that if λ1 = λ2 =
SJS (AkB) =
1
,
2
l=1
we get the Jensen-Shannon divergence:
S A 21 (A + B) + S B 12 (A + B)
2,
A, B ∈ D(H).
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Transformations on density ops preserving q rel entropy or related quantities
Theorem 20
Let (λ1 , . . . , λm ) be a fixed probability distribution of positive numbers. Assume that
φ : D(H) → D(H) is a map satisfying
χ(φ(A1 ), . . . , φ(Am )) = χ(A1 , . . . , Am ),
A1 , . . . , Am ∈ D(H).
(1)
Then there is a unitary or antiunitary operator U on H such that
φ(A) = UAU ∗ ,
A ∈ D(H).
Equation (1) means that φ preserves the Holevo bound only for a fixed probability
distribution not for all.
Consequence: structure of transformations on density operators which preserve the
von Neumann entropy of a fixed convex combination of n-tuples of density operators.
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Transformations on density ops preserving q rel entropy or related quantities
Several other results are known concerning e.g., isometries (trace-norm, Bures metric)
or maps preserving other numerical quantities (Uhlmann’s fidelity and related
quantities) on density operators.
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