The Shannon-McMillan theorem (AEP) for quantum sources and related - PowerPoint PPT Presentation

The Shannon-McMillan theorem (AEP) for quantum sources and related topics I.Bjelakovic, T.K., A. Szkola, R.Siegmund-Schultze

Motivation • Transfer of fundamental theorems of classical information theory to quantum information theory • In a wider context: how a quantum ergodic theory and quantum dynamical system theory looks like

The classical Shannon-McMillan-(Breiman) theorem • Given ( Σ , µ , σ ), Σ sequence space over finite alphabet, µ ergodic measure, σ shift-transformation, Σ � x, x(n) = (x 1 , , x 2 , x 3 ,…, x n ) • a.s. for ergodic µ : the individual information rate equals the average information rate ( ) ( ) − µ   − log x n 1 ∑ ( ) ( ) = = µ µ   lim h lim w log w µ →∞ →∞ n n   n n ( ) n ∈Σ w • This is a law of large numbers under very mild assumptions

Typical subspaces and data compression Reformulation in terms of typical subspaces: { } ( ) ⊂ Σ n there is a family of typical sets T s.t. n ( ) ⊃ T n T (filtration property) and n+1 n ( ) µ → T 1 and n 1 log #T → h and µ n n ( ) ( ) ( ) − −ε ∀ε > µ ∈ ≤ n h > ε µ 0 one ha s: w T e for n n n 0

{ } furthermore for any family B s.t. n 1 ( ) < limsup log #B h it follows that µ n n ( ) µ → B 0 ( strong converse ) n in other w or ds : to cover a positive fraction of the whole space one needs asymptot i ca lly nh µ e cylinder -sets of lengt n

( ) ( ) + Σ n 1 Σ n + − − T typical subspace T typical subspace n 1 n ( ) µ µ > − ε for : T 1 n n ( ) ( ) + + Σ Σ n 2 n 3 + − + − T typical subspace T typical subspace n 2 n 3

Ap plication to da ta compression: ( ) { } n ∈ given a typical long symbol sequence x ,x ,.......,x 0,1 1 2 n 1 < Codeboo k : typical words of length , k k lo g n 2 h µ kh ⇒ µ � there are abou t 2 typical words Codebooksize n and kh bits needed to specify a word from the codebook µ Spl i tting : x ,x ,...,x x ,....,x ..............x ,... .,x ( ) + − + �� �� � �� �� � 1 2 k k 1 2k n k n 1 1 �� � ��� � code with kh bits code µ ��������� � ���������� � code n codewords k ( ) (only o n fr act io n of blocks does not belong to the codebook) n kh [ ] ⇒ ⋅ = ∈ nh bits needed to code the whole sequence ( h 0,1 ) µ µ µ k

The quantum setting κ � A : matrix-a lgebra over Hilbert space H= C* ( -algebra) A : copy of A at site x x { } ∞ = = ⊗ � n A norm-closure of A : A x { } ∈ x 1,2,...,n n σ : shif t transformat ion ∞ ϕ : positive, normed, linear functional on A (me asure) ϕ ϕ = ϕ σ � nvari i an t : ϕ ϕ erg odi c : is extremal among the invar ia nt f unctiona ls

ϕ = ϕ for : there is a density matrix D n n n A ( ) ( ) ϕ = = s.t. a tr D a and D tr D (consistency) + + n n n 1 n 1 ( t r partial trace with respect to site n+1 ) n+1 ( ) ( ) ϕ = − entro py : S tr D lo g D ( von Neuma nn ) n n n 1 ( ) ( ) ϕ = ϕ entropy rate : s lim S n →∞ n n ( ) β ε co v ering expo nent : : { } 1 ( ) ϕ > − ε n lim min log t rP : P project or fr om A s.t . P 1 →∞ n n

The quantum Shannon-McMillan theorem (Ref.: Inventiones Mathematica, 2003) ϕ ∞ ⇒ Let be an ergodic state on A { } ∃ ∈ n family of orthogonal projectors Q A s.t.: n 1 ( ) ( ) ( ) ϕ → = ϕ i) Q 1 and lim log tr Q s n n →∞ n n { } < ⇒ ii ) for any sequence of minimal projectors p Q n n − 1log ( ) ( ) ϕ → ϕ p s n n { } ′ ∈ n ii i) for any seque n e c o f pr ojectors Q A s.t. n 1 ( ) ( ) ( ) ′ ′ < ϕ ⇒ ϕ → li m lo g tr Q s Q 0 n n →∞ n n

Comments ν � • The theorem holds for - lattices as well • Covering exponent is for all ε > 0: β ( ε ) = s( φ ) • The typical projectors (subspaces) can be explicitly constructed from the eigenspaces of D n ( ) − ϕ ns corresponding to eigenvalues of order e • The relation between the typical subspaces for different n is still unclear • Extensions to other group actions are possible • The typical subspaces can be chosen to be universal (not depending on φ but only on s( φ )) due to a result by Kalchenkov

History • Josza&Schumacher: typical subspace theorem for product states (Bernoulli case, 1996) • Petz&Mosonyi: weak version of the Shannon- McMillan under the assumption of complete ergodicity (2001) and strong form for Gibbs states (with Hiai, 1993) • Neshveyev&Størmer: Shannon-McMillan for finitely generated C*-algebras but only tracial states (2002) • Datta&Shuchov: Shannon-McMillan for spin lattices with restrictions on the interaction (2002)

Extensions A pointwise variant (Shan non-McMillan-Brei man ): ϕ ∞ ⇒ Let be an ergodic state on A { } ∀ε > ∃ ∈ > ε n 0 family of orthogonal projectors Q A s.t. for n n( ) : ε n, ( ) ( ) 1 ( ) ϕ > − ε < ϕ + ε i ) Q 1 and li m lo g tr Q s ε ε n, n, →∞ n n { } < ⇒ ii ) for any s eq u ence of minimal projectors p Q ε n n, − 1log ( ) ( ) ϕ < ϕ − ε p s n n = i ii) R[tr (Q )] Q (her e R[.] is the range projector) ε ε n+1 n+1, n,

• The relation between the typical projectors for different ε is unclear • For abelian algebras (classical case) the above theorem is equivalent to the Shannon-McMillan- Breiman theorem

A theorem for the relative entropy (I.Bjelakovich, R.Siegmund-Schultze) ω τ R elative entropy of two states and on finite dimensional algebra: ( ) ( )  − ω ≤ τ  tr D log D log D for supp supp ( ) ω ω τ ω τ =  S , : ∞   otherwise Relative entropy rat e : ∞ ψ ϕ an invariant state and an invariant product stat e o n A 1 ( ) ( ) ψ ϕ = ψ ϕ ϕ = ϕ s , lim S , ( : | ) n n n n A →∞ n n

Rel ative exponent : { } ( ) ( ) ( ) β ψ ϕ = ϕ ∈ ψ > − ε n , : min log Q :Q A , projector s.t. Q 1 ε ,n n n ∞ ψ ϕ ⇒ For an ergodic state and an invariant product state on A 1 ( ) ( ) β ψ ϕ = ψ ϕ ∀ε > l im , s , for 0 ε ,n →∞ n n { } ψ ⇒ equivalently for typical subspace pr oject ors Q of n ( ) ( ) ( ) ( ) ( ) ( ) − ψ ϕ +ε − ψ ϕ −ε ≤ ϕ ≤ > ε n s , n s , e Q e ; n n n

Relative entropy typical and untypical subspaces ψ ϕ From point of view: From point of vi ew : ( ) ( ) ψ > −ε ( ) − ψ ϕ ϕ Q 1 ns , � Q e n n ψ ( ) ψ � Q T n n typical subspace for ψ ( ) ψ � Q T n n

• Complete analogy to the classical case • The proof is similar to the one for the Shannon- McMillan theorem but more technical involved • New simple proof of the monotonicity of the relative entropy can be derived from this result • Starting point for developing a large deviation theory (Sanov’s theorem)

Proof strategy Idea : want to u se abelian approximations to lift the classical results to the quantum case Natural c andida t e : algebra B generated by the eigenspace projectors n ϕ = ϕ of D (density matrix corresp ondi ng to : ) { } n n 1,..,n ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ A A ...... A A A ...... A ..... A A ...... A ������� ������� ������� ⊗ ⊗ ⊗ ⊂ ⊂ ⊂ n n n B A B A B A ������������ � ������������� � n n n ∞ m → B B n n ( ) ( ) ∞ ϕ σ B * B , , is an abelian syste m n σ σ ∞ * n corresponds to on A

( ) ( ) ( ) ∞ ϕ σ Σ µ σ B * B , , is isomorphic to a classical system , , n B n ( ) 1 1 ( ) ( ) ( ) ( ) ϕ ≤ ϕ σ = ≤ ϕ + ε B * s s h s n µ n n ( ) Σ µ σ What can be said about the ergodic properties of , , ? B n µ ϕ is ergodic under the assumption of complete ergodicity o f ( Pet z ) µ In the general case splits into at most k n ergodic co mpon ents . All components are isomorphic under some shift-power and have the same entropy. To prove this one needs an ergodic d ec omposition ( ) ∞ ϕ σ n theo rem fo r A , , :

{ } ( ) ( ) ∞ ϕ σ ≤ ≤ ϕ i n i) A , , splits into 1 k n ergodic components ≤ ≤ 1 i k ( ) ( ) ϕ = ϕ σ i i-1 � ii) k n and ( ) ( ) ( ) ( ) ( ) ϕ σ = ϕ σ = ϕ i j n n iii) s s ns finite size entropy estimation: ( ) 1 ( ) ( ) ( ) ∀η > → ∞ ⇒ ϕ ≤ ϕ ≤ ϕ + η i iv) 0 a nd n s S s n A n for almost every ergodic com ponen t Ne x s t t ep : combining the different levels of approximation