on the minimal sufficiency of povm and...

On the Minimal Sufficiency of POVM and Channel

Yui Kuramochi

Kyoto University

[email protected]

Oct. 27, 2016

YK, J. Math. Phys. 56, 102205 (2015)

Yui Kuramochi (Kyoto Univ.) Minimal sufficient POVM and channel Oct. 27, 2016 1 / 33

A problem: a freedom of adding irrelevant informationPOVM is the general description of the outcome statistics of generalquantum measurement. But...

Example. (Ω,Σ,M) : POVM,(Ω1,Σ1, µ) : probability space.Define a POVM (Ω× Ω1,Σ⊗ Σ1,N) by

N(E × E1) := µ(E1)M(E), (E ∈ Σ, E1 ∈ Σ1).

N is realized as follows:1 Perform M (measurement outcome ω ∈ Ω).2 Generate classical random variable ω1 ∈ Ω1 according to the probability

measure µ.3 Then the pair (ω, ω1) ∈ Ω× Ω1 is the measurement outcome for N.

N gives the same information about the system as M.Since the class of probability measures is not a set, neither is the classof POVMs that gives the same information about the system as A.(cf. “The set of all the sets” is ill-defined.)


Motivation

Questions.

Can we reduce such redundancies of an arbitrary POVM?

If possible, then how far such reductions proceed?

Answers: YES, and such reductions proceed in an almost unique way.

The maximally reduced POVM is called a minimal sufficient POVM.


Contents

1 Sufficiency and minimal sufficiency in classical statistics

2 Minimal sufficient POVM


Contents




Probability measure

(Ω,Σ) is a σ-algebra

:def.⇔ Ω is a set and Σ is a set of subsets of Ω s.t.

1 Ω ∈ Σ.2 E ∈ Σ =⇒ Ω \ E ∈ Σ.3 For countable En ⊆ Σ,

⋃n En ∈ Σ.

µ is a measure on a measurable space (Ω,Σ)

:def.⇔ µ : Σ → [0,∞] s.t. µ(∅) = 0 and for any disjoint sequenceEn ⊆ Σ, µ (

⋃nEn) =

∑n µ (En) .

The triple (Ω,Σ, µ) is called a measure space.

A measure P on a measurable space (Ω,Σ) is called a probabilitymeasure iff P (Ω) = 1. Then the measure space (Ω,Σ, P ) is called aprobability space.

An event A(ω) holds µ-almost everywhere (µ-a.e.) iff

µ(N) = 0, N := ω ∈ Ω | A(ω) does not hold ∈ Σ.


IntegrationLet (Ω,Σ, µ) a measure space. We can define the integration formeasurable functions as follows.

Let (Ω1,Σ1) and (Ω2,Σ2) be measurable spaces. A mappingf : Ω1 → Ω2 is called Σ1/Σ2-measurable iff f−1(E) ∈ Σ1 (∀E ∈ Σ2).

f : Ω → R or C is called measurable

:def.⇔ f is Σ/B(R) or B(C)-measurable, where B(R) (resp. B(C)) isthe σ-algebra generated by the family of open sets on R (resp. C).If f : Ω → C is a simple function s.t. f =

∑nk=1 akχEk

(Ek ∈ Σ),then

∫Ω fdµ :=

∑nk=1 akµ(Ek).

For measurable f : Ω → [0,∞], there exists a monotonicallyincreasing sequence of positive simple functions fn convergingpointwise to f. Then

∫Ω fdµ := limn

∫Ω fndµ, which is independent

of the choice of fn.For general measurable f : Ω → C, the integration is defined as thelinear extension of the previous definition if

∫Ω |f |dµ < ∞.

(Such f is called µ-integrable).


Radon-Nikodym theorem

A measure µ on (Ω,Σ,) is called σ-finite iff there is a countablesequence En ⊆ Σ s.t.

⋃nEn = Ω and µ(En) < ∞.

For measures µ and ν on (Ω,Σ), µ is absolutely continuous, writtenµ ν, iff ν(N) = 0 implies µ(N) = 0 for every N ∈ Σ.

(Radon-Nikodym theorem). Let µ and ν be σ-finite measures on(Ω,Σ) satisfying µ ν. Then there exists ν-integrable function f s.t.

µ(E) =

∫Efdν, (E ∈ Σ).

Such f is unique up to ν-a.e. equality and called theRadon-Nikodym derivative, written dµ

dν .


Lp space

Let (Ω,Σ, µ) be a measure space.

For measurable functions f and g, f(ω) = g(ω), µ(ω)-a.e. defines anequivalence relation, and the equivalence class to which f belongs isdenoted by [f ]µ.

For each p ∈ [1,∞), Lp(µ) := [f ]µ |∫Ω |f |pdµ < ∞ ,

||f ||p :=(∫

Ω |f |pdµ)1/p

.

||f ||∞ := infa ∈ [0,∞]||f | ≤ a, µ-a.e..L∞(µ) := [f ]µ | ||f ||∞ < ∞ .(Riesz-Fischer). (Lp(µ), || · ||p) (p ∈ [1,∞]) is a Banach space, i.e.complete normed linear space.


Conditional expectation and conditional probability

Let (Ω,Σ, P ) be a probability space and let Σ1 ⊆ Σ be a sub σ-algebra.

For each [f ]P ∈ L1(P ), the complex measure

µf : Σ1 3 E1 7−→∫E1

fdP

is absolutely continuous w.r.t. P |Σ1 , and the Radon-Nikodym

derivativedµf

dP |Σ1(ω) is called the conditional expectation of f w.r.t.

Σ1, denoted by E(f |Σ1, ω) or E(f |Σ1).E(f |Σ1) is a Σ1-measurable function s.t.∫

E1

fdP =

∫E1

E(f |Σ1)dP, (∀E1 ∈ Σ1).

P (E|Σ1) := E(χE |Σ1) (E ∈ Σ) is called the conditional probabilityw.r.t. Σ1.


Conditional expectation and conditional probability

If (ΩT ,ΣT ) is a measurable space and T : Ω → ΩT isΣ/ΣT -measurable, then T is often called a statistic on Ω.

P T (F ) := P (T−1(F )) (F ∈ ΣT ).

For [f ]P ∈ L1(P ) and E ∈ Σ, the conditional expectation ET (f |t)and the conditional probability P T (E|t) are ΣT -measurable functionss.t. ∫

T−1(F )f(ω)dP (ω) =

∫FET (f |t)dP T (t), (F ∈ ΣT ),

P T (E|t) := E(f |t).


Statistical model

A 4-tuple (Ω,Σ,Θ, (Pθ)θ∈Θ) called a (classical) statisticalexperiment, or statistical model. if (Ω,Σ) is a measurable space,Θ 6= ∅ is a set (called parameter set), (Pθ)θ∈Θ is a family ofprobability measures on (Ω,Σ) indexed by Θ.

Example: N -times coin tossing.Ω = 0, 1N = (x1, · · · , xN ) | xi = 0 or 1 , Θ = [0, 1],P = pθ0≤θ≤1,

pθ((x1, · · · , xN )) = θ∑N

i=1 xi(1− θ)N−∑N

i=1 xi .


Sufficient subalgebra and statistic

Let (Ω,Σ,Θ, (Pθ)θ∈Θ) be a statistical experiment, Σ1 ⊂ Σ be a subσ-subalgebra, and let T : Ω → ΩT be a Σ/ΣT -measurable statistic.

Σ1 is sufficient w.r.t. (Pθ)θ∈Θ iff the conditional probability Pθ(·|Σ1)can be taken independent of θ ∈ Θ.

T is sufficient w.r.t. (Pθ)θ∈Θ iff the conditional probability Pθ(E|t)can be taken independent of θ ∈ Θ.

Intuitively, by taking a sufficient statistic T , we do not lose theinformation about the parameter θ.


Sufficient statistic: coin tossing example

pθ(x1, · · · , xN ) = θ∑N

i=1 xi(1− θ)N−∑N

i=1 xi .

T (x1, · · · , xN ) :=

N∑i=1

xi (total number of heads).

pTθ (t) =∑

x1,··· ,xN

δt,T (x1,··· ,xN )pθ(x1, · · · , xN ) =

(N

t

)θt(1− θ)N−t,

pθ(x1, · · · , xN |t) = δt,T (x1,··· ,xN )pθ(x1, · · · , xN )/pTθ (t)

= δt,T (x1,··· ,xN )/

(N

t

).

Therefore T is sufficient.


Neyman factorization theorem (Halmos and Savage (1949), Bahadur (1954))

Let (Ω,Σ,Θ, (Pθ)θ∈Θ) be a statistical experiment, let Σ1 be a subσ-algebra, and let T : Ω → ΩT be a Σ/ΣT -measurable statistic.

(Pθ)θ∈Θ is dominated iff there exists a σ-finite measure ν s.t. Pθ νfor all θ ∈ Θ.

If (Pθ)θ∈Θ is dominated, then there exists a countable subsetθii≥1 ⊂ Θ s.t. P∗ =

∑i≥1 ciPθi (ci > 0,

∑i≥1 ci = 1) dominates

(Pθ)θ∈Θ.Let (Pθ)θ∈Θ be dominated by P∗ =

∑i≥1 ciPθi .

1 Σ1 is sufficient w.r.t. (Pθ)θ∈Θ iff there exist Σ1-measurable functionsgθ(·) such that

dPθ

dP∗(ω) = gθ(ω), P∗(ω)-a.e.

for all θ ∈ Θ.2 T is sufficient w.r.t. (Pθ)θ∈Θ iff

dPθ

dP∗(ω) =

dPTθ

dPT∗(T (ω)), P∗(ω)-a.e.

for all θ ∈ Θ.


Minimal sufficient subalgebra

Let (Pθ)θ∈Θ be dominated by P∗ =∑

i≥1 ciPθi .

For sub σ-algebras Σ1 and Σ2,

Σ1 ⊆P∗ Σ2 :def.⇔ ∀E1 ∈ Σ1, ∃E2 ∈ Σ2 s.t. P∗(E14E2) = 0

(or equivalently Pθ(E14E2) = 0 (∀θ ∈ Θ));

Σ1 ≡P∗ Σ2 :def.⇔ Σ1 ⊆P∗ Σ2 and Σ2 ⊆P∗ Σ1.

A sub σ-algebra Σ0 is minimal sufficient iff Σ0 is sufficient andΣ0 ⊆P∗ Σ1 holds for all sufficient sub σ-algebra Σ1.

The σ-algebra generated by dPθdP∗

θ∈Θ is a minimal sufficient sub

σ-algebra.


Minimal sufficient statistic

A statistic T is called minimal sufficient iff T is sufficient and forany sufficient statistic S there exists a measurable f s.t.

T (ω) = f(S(ω)), P∗(ω)-a.e.

If (Pθi)i≥1 is dense in (Pθ)θ∈Θ w.r.t. the total variation norm, thenΣ/B(R∞)-measurable mapping

T : Ω 3 ω 7−→(dPθi

dP∗

)i≥1

∈ R∞

is a minimal sufficient statistic. (Bahadur 1954).


Contents




Hilbert space

A Hilbert space H is an inner product space complete w.r.t. the norm.

A topological space S is separable :def.⇔ S has a countable dense

subset.

A Hilbert space H is separable ⇐⇒ H has a countable orthonormalbasis.

L(H) denotes the set of bounded linear operators on H.

T ∈ L(H) is trace class :def.⇔

∑i 〈xi,

√T ∗Txi〉 < ∞ for a

orthonormal basis xi.T (H) denotes the set of trace class operators on H.The trace of T ∈ T (H) is defined by tr[T ] :=

∑i 〈xi, Txi〉 , which

converges and is independent from the choice of the orthonormalbasis xi.

ρ ∈ T (H) is a density operator or state :def.⇔ ρ ≥ 0 and tr[ρ] = 1.

S(H) denotes the set of density operators on H.


POVM: mathematical definition

ρ M// outcome ω//

Let H be a Hilbert space.

(Ω,Σ,M) is a POVM on H iff M is a mapping M : Σ → L(H) s.t.1 M(E) ≥ O (E ∈ Σ);2 M(Ω) = 1H (identity operator);3 for each disjoint and countable Ei ⊂ Σ, M(∪iEi) =

∑i M(Ei) (in

the weak operator topology).

For each state ρ ∈ S(H) the outcome probability measure PMρ on

(Ω,Σ) is defined by PMρ (E) := tr[ρM(E)] (E ∈ Σ).

If a mapping Φ maps each ρ ∈ S(H) to a probability measure on(Ω,Σ) is affine on S(H), then there exists a unique POVM (Ω,Σ,M)s.t. Φ(ρ) = PM

ρ .


Weak Markov kernel

Let (Ω1,Σ1,M) and (Ω2,Σ2,N) be POVMs. A mapping

κ(·|·) : Σ1 × Ω2 → [0, 1]

is an M-N weak Markov kernel iff

1 κ(E|·) is Σ2-measurable for each E ∈ Σ1.

2 κ(Ω1|·) = 1 N-a.e.

3 If M(N) = 0 then κ(N |·) = 0 N-a.e. (N ∈ Σ1).

4 For each disjoint and countable Ej ⊂ Σ1, κ(∪jEj |·) =∑

j κ(Ej |·)N-a.e.

κ(·|ω2) is a conditional probability.


Equivalence relation by the classical postprocessing

Definition (H. Martens et al. (1990), S. Dorofeev et al. (1997), T. Heinonen (2005), A. Jencova et al. (2008) )

Let (Ω1,Σ1,M) and (Ω2,Σ2,N) be POVMs.

1 M N (M is a postprocessing of N)

:def.⇔ ∃κ(·|·) : M-N weak Markov kernel s.t.

M(E) =

∫Ω2

κ(E|ω2)dN(ω2) (E ∈ Σ2).

2 M ' N (M is postprocessing equivalent to N)

:def.⇔ M N and N M.

M N indicates that M is realized by a classical postprocessing of N.


Equivalence relation by the classical postprocessing

is a preorder relation, i.e.1 M M for any POVM M.2 M1 M2 and M2 M3 imply M1 M2.

' is an equivalence relation, i.e.1 M ' M for any POVM M;2 M ' N implies N ' M.3 M1 ' M2 and M2 ' M3 imply M1 ' M3.

' is defined for arbitrary POVMs on H and not a set theoreticalequivalence relation.


Sufficient statistic for a POVM

Let H be a separable Hilbert space.

T (H) and S(H) are separable w.r.t. the trace norm||T ||1 := tr[

√T ∗T ].

There exists a faithful state ρ∗ ∈ S(H), i.e. for any A ≥ 0,tr[ρ∗A] = 0 implies A = 0.

Let (Ω,Σ,M) be a POVM on H and let T : Ω → ΩT be aΣ/ΣT -measurable statistic.Define a POVM (ΩT ,ΣT ,MT ) by MT (F ) := M(T−1(F )) (F ∈ ΣT ).Then the following conditions are equivalent.

1 M ' MT .2 T is sufficient w.r.t. (PM

ρ )ρ∈S(H).

3dPM

ρ

dPMρ∗

(ω) =dPMT

ρ

dPMTρ∗

(T (ω)), M(ω)-a.e.


Observation

The first example:(Ω,Σ,M): POVM, (Ω1,Σ1, µ): probability space.(Ω× Ω1,Σ⊗ Σ1,N): POVM s.t.

N(E × F ) = µ(F )M(E).

N can be “reduced” to M by the following map

f : Ω× Ω1 3 (ω, ω1) 7→ ω ∈ Ω.

In fact, Nf := N(f−1(·)) = M.

Bearing this in mind, we define the minimal sufficient POVM.


Relabeling minimal sufficient POVM

Definition

A POVM (Ω1,Σ1,M) is relabeling minimal sufficient ifffor any POVM (Ω2,Σ2,N) s.t. M ' N,there exists measurable f : Ω2 → Ω1 s.t. Nf := N(f−1(·)) = M.

The mapping f in the above definition is a sufficient statistic for N sinceNf = M ' N.


Main result (1)

Theorem

Let (Ω,Σ,M) be an arbitrary POVM on a separable Hilbert space H. Thenthere exists a relabeling minimal sufficient POVM M with a standard Boreloutcome space s.t. M ' M.Furthermore such M is unique up to almost isomorphism (next slide).

(Ω,Σ) is standard Borel

:def.⇔ (Ω,Σ) is Borel isomorphic (isomorphic as a measurable space) toa complete separable metric space equipped with the σ-algebragenerated by the family of open sets.Any countable direct product of standard Borel spaces is againstandard Borel.Any countable discrete space and the real line (R,B(R)) are standarBorel.Conversely, any standard Borel space is Borel isomorphic to either acountable discrete space or (R,B(R)).(Kuratowski’s Borel isomorphism theorem).


Almost isomorphism (relabeling up to null sets)

Definition

Let (Ω1,Σ1,M) and (Ω2,Σ2,N) (i = 1, 2) be POVMs.

(i) (Ω1,Σ1,M) ≈ (Ω2,Σ2,N) (strictly isomorphic)

:def.⇔ ∃f : Σ1/Σ2-bimeasurable bijection s.t. (M)f = N.

(ii) (Ω1,Σ1,M) ∼ (Ω2,Σ2,N) (almost isomorphic)

:def.⇔ ∃Ωi ∈ Σi (i = 1, 2) s.t.

M(Ω1) = N(Ω2) = 1,

and(Ω1, Ω1 ∩ Σ1,M|Ω1

) ≈ (Ω2, Ω2 ∩ Σ2,N|Ω2).

Here Ωi ∩ Σi :=Ωi ∩ E

∣∣∣ E ∈ Σi

and M|Ω1

(resp. N|Ω2) is the

restriction of M (resp. N) to Ω1 ∩ Σ1 (resp. Ω2 ∩ Σ2).


The separability of H

From the separability of H there exists a countable subset ρjj≥1

s.t. ρjj≥1 is dense in S(H) w.r.t. the trace norm ||ρ||1 := tr[√ρ∗ρ].

Define a density operator ρ∗ by

ρ∗ :=∑j≥1

cjρj ,

where cj > 0 and∑

j≥1 cj = 1. (e.g. cj = 2−j).

ρ∗ is faithful, i.e. for any bounded operator a ≥ 0, tr[ρ∗a] = 0 impliesa = 0.

PAρ PA

ρ∗ for any ρ ∈ S(H), and we can define dPMρ /dPM

ρ∗ .


Construction of M (sketch)

We define T : Ω → R∞ (Lehmann-Scheffe-Bahadur statistic) by

T (x) :=

(dPM

ρj

dPMρ∗

(ω)

)j≥1

,

which is Σ/B(R∞)-measurable. We define M := MT .

From

|PMρ (E)− PM

σ (E)| ≤ ||ρ− σ||1||M(E)|| ≤ ||ρ− σ||1,

PMρj j≥1 is dense w.r.t. the total variance norm. Therefore T is

(minimal) sufficient and A = AT ' A.

Since (R∞,B(R∞)) is a standard Borel, this construction shows thatany POVM on a separable Hilbert space is equivalent to a POVMwith a standard Borel outcome space.


The set of equivalence classes of POVMs

We can define the set of equivalence classes of POVMs on a separableHilbert space as follows.

A POVM (R∞,B(R∞),M) is an LSB POVM

:def.⇔ M is induced by an LSB statistic of a POVM

⇔(

dPMρj

dPMρ∗(t)

)j≥1

= t, M(t)-a.e.

For any LSB POVMs M and N, M ' N ⇒ M = N.

Thus, for any POVM on H, there exists a unique equivalent LSBPOVM.This shows that the set of LSB POVMs can be identified with the setof equivalence classes of POVMs on H.


Minimal sufficiency condition for discrete POVM

For a countable set Ω, (Ω, 2Ω) is called a discrete space.

A POVM M on a discrete space (Ω, 2Ω) is called discrete POVM,which can be identified with a positive-operator valued function

M(ω) := M(ω)

satisfying the completeness condition∑

ω∈ΩM(ω) = 1.

A discrete POVM M : Ω → L(H) satisfying M(ω) 6= 0 (∀ω ∈ Ω) isrelabeling minimal sufficient iff A is pairwise linearly independent, i.e.M(ω),M(ω′) is linearly independent for any pair ω 6= ω′.

For any discrete POVM M : Ω → L(H), a relabeling minimalsufficient POVM eqivalent to M is induced by a mappingS : Ω → Ω/ ∼ defined by

ω ∼ ω′ :def.⇔ ∃c > 0, M(ω) = cM(ω′),

S(ω) := equivalence class to which ω belongs.


Summary

YK, J. Math. Phys. 56, 102205 (2015)

A is minimal sufficient :def.⇔ ∀B ' A ∃f s.t. Bf = A.

A minimal sufficient POVM A is the least redundant POVM amongPOVMs equivalent to A.

Any POVM can be reduced to a minimal sufficient POVM almostuniquely.

The set of equivalence classes of POVMs on a separable Hilbert spaceis well-defined.


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