Quantum Information, Entanglement and Entropy

QUANTUM INFORMATION,
ENTANGLEMENT AND ENTROPY

Siddharth Sharma

Abstract

In this review I had given an introduction to axiomatic Quantum Mechanics, Quantum Information and its measure entropy. Also, I had given an introduction to Entanglement and its measure as the minimum of relative quantum entropy of a density matrix of a Hilbert space with respect to all separable density matrices of the same Hilbert space. I also discussed geodesic in the space of density matrices, their orthogonality and Pythagorean theorem of density matrix.

Postulates of Quantum Mechanics

The basic postulate of quantum mechanics is about the Hilbert space formalism:
• Postulates 0: To each quantum mechanical system, has a complex Hilbert space ℍ associated to it:

ℍ
⁄
⟦ ⟧ ⟦ ⟧

{

}

Set of all pure state,

≔ { 푓 | 푓 ≔ ℎ: ℎ ∼ 푓 } and ℎ ∼ 푓 ⇔ ∃ 푧 ∈ ℂ such that
∼

ℎ = 푧 푓 where 푧 is called phase
• Postulates 1: The physical states of a quantum mechanical system are described by statistical operators acting on the Hilbert space.

A density matrix or statistical operator, 휌 is a positive operator of trace 1 on the

〈

〉

〈

〉

Hilbert space. Where 휌 is called positive if 푥, 휌푥 for all 푥 ∈ ℍ and ⋇,⋇ .
• Postulates 2: The observables of a quantum mechanical system are described by selfadjoint operators acting on the Hilbert space.

A self-adjoint operator A on a Hilbert space ℍ is a linear operator 퐴: ℍ → ℍ which satisfies

〈

〉

〈

〉

퐴푥, 푦 = 푥, 퐴푦 for 푥, 푦 ∈ ℍ.

Lemma: The density matrices acting on a Hilbert space form a convex set whose extreme points are the pure states. Proof. Denote by Σ the set of density matrices. It is obvious that a convex combination of density matrices is positive and of trace one. Therefore, Σ is a convex set.

Lemma: Let

|

⟩⟨ |

|

⟩⟨

휌 = ∑ 푥_푒푥_푒= ∑ 푦_ℎ푦_ℎ

푒∈픹

ℎ∈픹

be decompositions of a density matrix and where 픹 is the basis of Hilbert space, ℍ.

(

)

푒ℎ∈픹

Then there exists a unitary matrix 푈_푒ℎ

such that

|

⟩

|

⟩

∑ 푈_푒ℎ푥_푒= 푦_ℎ

푒∈픹

Where,

⟨

|

⟩

푥_푒푦_ℎ

푈_푒ℎ

≔

⟨

|

⟩

√ 푥_푒푥_ℎ

• Postulates 3: Let 풳 be a finite set and for 푥 ∈ 풳 an operator 푂_ꢀ∈ ℬ(ℍ) be given such that

∑ 푂_ꢀ^⋇푂_ꢀ= 푖푑_ℍ

ꢀ∈풳

where 푂_ꢀ^⋇is conjugate of 푂_ꢀ. Such an indexed family of operators is a model of a measurement with values in 풳. If the measurement is performed in a state 휌, then the

⋇

(

)

outcome 푥 ∈ 풳 appears with probability tr 푂_ꢀ휌푂_ꢀand after the measurement the state of the system is

푂_ꢀ^⋇휌푂_ꢀ

⋇

(

)

푡푟 푂_ꢀ휌푂_ꢀ

( )

(

)

If 휑 ∶ ℬ(ℍ) → ℂ is a linear functional such that 휑 퐴 ≥ 0 if 퐴 is positive and 휑 푖푑 = 1, Then there exists a density matrix 휌_ꢁsuch that,

( )
휑 퐴 = tr(휌_ꢁ퐴)

The functional 휑 associates the expectation value to the observables 퐴. The density matrices 휌 and 휌′ are called orthogonal if any eigenvector of 휌 is orthogonal to any eigenvector of 휌′.

Let 픹 be an orthonormal basis in a Hilbert space ℍ . The unit vector 휉 ∈ ℍ is

complementary to the given basis if

1

|〈 〉|

ꢂ, 휉

=

dim ℍ

√

For all ꢂ ∈ 픹, where dim ℍ is dimension of ℍ. Two orthonormal bases are called complementary if all vectors in the first basis are complementary to the other basis.

• Postulates 4: The composite system is described by the tensor product Hilbert space

⨂ ℍ_휗

휗

Given a density matrix 휌 on ⨂_휗ℍ_휗there are density matrices 휌_휗∈ ℬ(ℍ_휗) such that

( )

(

)

tr (휌 ⨂ 풜_훾ꢃ ) = tr 휌_휗퐴_휗

훾

Where,

퐴_휗, ꢄ = ꢃ

( )

풜_훾ꢃ ≔ {

푖푑_ℍ_ꢅ, ꢄ ≠ ꢃ

and 퐴_휗∈ ℬ(ℍ_휗), here 휌_휗is called reduced density matrices.

Let,

휌 = ⨂ 휌_휗

휗

( )

Then we define 퐭퐫_휸흆 as follows:

( )

tr_훾휌 ≔ tr(휌_훾) ⨂ 휌_휗

휗≠훾

Information and its Measures

• Shannon entropy: In his revolutionary paper Shannon proposed a statistical approach and he posed the problem in the following way: “Suppose we have a set of possible events whose probabilities of occurrence are 푝₁, 푝₂, . . . , 푝_푛. These probabilities are known but that is all we know concerning which event will occur. Can we find a measure of how much “choice” is involved in the selection of the event or how uncertain we are of the outcome?” Denoting such a measure by 퐻(푝₁, 푝₂, . . . , 푝_푛), he listed three very reasonable requirements which should be satisfied, those three postulates are as follows:

a. Continuity: 퐻(푝, 1 − 푝) is a continuous function of 푝. b. Symmetry: 퐻(푝₁, 푝₂, . . . , 푝_푛), is a symmetric function of its variables.

(

)

c. Recursion: For every 0 ≤ 휆 < 1 the recursion 퐻 푝₁, . , 휆푝_ꢆ, . , 1 − 휆 푝_ꢆ. , 푝_푛
=

(

)

퐻 푝₁, 푝₂, . , 푝_ꢆ. . , 푝_푛+ 푝_ꢆ퐻(휆, 1 − 휆), holds

According to him there is only one function satisfying all this postulate is:

푛

(

)

( )

퐻 푝₁, 푝₂, . , 푝_ꢆ. . , 푝_푛= −휅 ∑ 푝_ꢇ푙ꢈ 푝_ꢇ

ꢇ=1

푛

Let (푋_ꢇ)^푛_ꢇ=1be random variables with values in the set 풳 ≔ _ꢇ=1풳_ꢇ. The following

∏

notation will be used. a. 푝(푥_ꢇ)^푛_ꢇ=1is probability of (푋_ꢇ)^푛_ꢇ=1at (푥_ꢇ)^푛_ꢇ=1b. 푝(푥_ꢇ|푥 ) is probability of 푋_ꢇat 푥_ꢇknowing the probability of 푋 . at 푥

푗

Then,

퐻(푋_ꢇ)^푛_ꢇ=1≔ −

∑

푝(푥_ꢇ)^푛_ꢇ=1ln 푝(푥_ꢇ)_ꢇ^푛₌₁

ꢊ

(ꢀ ) ∈풳

ꢉ

ꢉ=1

Where if ꢈ = 2

퐻(푋_ꢇ, 푋 ) ≔ − ∑ ∑ 푝(푥_ꢇ, 푥 ) ln 푝(푥_ꢇ, 푥 )

푗

ꢀ ∈풳 ꢀ ∈풳

ꢉ

ꢋ

and

푝(푥_ꢇ)^푛_ꢇ=1= 푝(푥₂)푝(푥₁|푥₂, 푥₃, … . 푥_푛)

Which leads to:

푛

(

)

푝 ((( 푥₁|푥₂|푥₃)| … . ) |푥_푛)

푝(푥_ꢇ)^푛_ꢇ=1

=

∏ 푝(푥₂)

(

)

푝 푥₁

ꢇ=1

Theorem: If (푋_ꢇ)^푛_ꢇ=1are random variables of finite range, then

푛

퐻(푋_ꢇ)^푛_ꢇ=1≤ ∑ 퐻(푋_ꢇ)

ꢇ=1

Proof:

퐻(푋_ꢇ)^푛_ꢇ=1≔ −

∑

푝(푥_ꢇ)^푛_ꢇ=1ln 푝(푥_ꢇ)_ꢇ^푛₌₁

ꢊ

(ꢀ ) ∈풳

ꢉ

ꢉ=1

And

푛

∑ 퐻(푋_ꢇ) ≔ −

∑

∏ 푝(푥₂) ln ∏ 푝(푥₂)

ꢊ

ꢇ=1

(ꢀ ) ∈풳

ꢉ

ꢉ=1

And

푛

∏ 푝(푥₂) ≤ 푝(푥_ꢇ)^푛_ꢇ=1

ꢇ=1

As

(

)

(

)

푝 푥₁≤ 푝 ((( 푥₁|푥₂|푥₃)| … . ) |푥_푛)

Hence,

푛

퐻(푋_ꢇ)^푛_ꢇ=1≤ ∑ 퐻(푋_ꢇ)

ꢇ=1

Fano’s inequality: Let 푋 and 푌 be random variables such that their range is in a set of cardinality d and let 푝 ≔ 푃푟표푏(푋 ≠ 푌) .Then

퐻(푋|푌) ≤ 푝 푙표푔(푑 − 1) + 퐻(푝, 1 − 푝)

• von Neumann Entropy: Let 휌 be density matrix of a quantum system then von
Neumann Entropy of that quantum system:

( ) )

(

푆 휌 ≔ −휅 tr 휌 ln 휌

Theorem: Let 휌 and 휎 be densities on a 푑 −dimensional Hilbert space and let

‖

1

휌 − 휎
푝 ≔

2

where Then,
₁: ℬ(ℍ) → ℝ⁺₀, is the norm on operator space ℬ(ℍ) of Hilbert space ℍ.

‖ ‖
⋇
|푆(휌) − 푆(휎)| ≤ 푝 푙표푔(푑 − 1) + 퐻(푝, 1 − 푝)

Holds.

• Quantum Relative Entropy:

The relative entropy, or I-divergence of the probability distributions 푝(푥) and 푞(푥), is defined as

∞

( )
푝 푥

(

)

( )

퐷 푝|| 푞 ≔ ∫ 푝 푥 ln

푑푥

( )

푞 푥

−∞

Assume that 휌 and 휎 are density matrices on a Hilbert space ℍ , then

(

)

( )

tr 휌 ln 휌 − ln 휎 if supp ρ ≤ supp 휎

(

)

푆 휌|| 휎 ≔ {
+∞ 표푡ℎꢂ푟푤푖푠ꢂ

Hilbert–Schmidt inner product

훥푎 = 휌푎휎⁻¹
( )
For all 푎 ∈ ℬ ℍ

12
12

(

)

〈

(

)

〉

tr 휌 ln 휌 − ln 휎 = − 휌 , ln 훥 휌

⋇

〈

〉

(

)

where, 퐴, 퐵 ≔ tr 퐴 퐵 and 퐴 is conjugate of 퐴

⋇

12
12

(

)

(

)

tr 휌 ln 휌 − ln 휎 = −tr ((휌 ) (휌 ) ln 훥 )

Let 휌 ≡ ꢂ^ꢌ, 휔 and 휎 be three invertible densities. The ꢂ − 푔ꢂ표푑ꢂ푠푖푐 connecting 휌 and 휎 is the curve

ꢂ^ꢌ+ꢍꢎ

( )

ꢄ_푒푡 ≔

ꢌ+ꢍꢎ

(

)

tr ꢂ

( )

for all 푡 ∈ [0,1], where 퐴 ≔ ln 휎 − ln 휌. Then, ꢄ_푒0 = 휌 and ꢄ_푒1 = 휎. The 푚 −

푔ꢂ표푑ꢂ푠푖푐 connecting 휌 and 휔 is the curve.

( )

ꢄ_ꢏ푡 ≔ 휌 + 푡퐵

( )

for all 푡 ∈ [0,1] where 퐵 ≔ 휔 − 휌. Then ꢄ_ꢏ0 = 휌 and ꢄ_ꢏ1 = ω

Assume that the ꢂ − 푔ꢂ표푑ꢂ푠푖푐 connecting 휌 and 휎 is orthogonal to the 푚 − 푔ꢂ표푑ꢂ푠푖푐 connecting 휌 and 휔 with respect to the inner product

∞

−1

⋇

−1

〈

〉

((

)

(

)

퐸, 퐹 ≔ ∫ tr 푠핀 + 휌 퐸 푠핀 + 휌 퐹 푑푠

ꢐ

0

A plain computation yields

(

)

〈

〉

푆(휔||휌) + 푆(휌||휎) − 푆(휔||휎) = tr 퐴퐵 = 퐴, 퐵

ꢌꢑ
∞

−1

푇 ∶ 푋 ↦ ∫ 푠핀 + 휌 푋 푠핀 + 휌 ⁻¹푑푠

(

)

(

)

ꢐ

0

Them According Dénes Petz And,

〈

〉

〈

( ) 〉

= 푇 푋 , 푌

ꢐ

푋, 푌

ꢌꢑ

ꢐ

( )
푇 퐴 = ꢄ_푒
( )

0

̇

ꢐ

It is obvious that: Therefore

( )

0

퐵 = ꢄ_ꢏ

̇

〈

〉

〈

( )

〉

〈 ( ) ( )〉

0 , ꢄ_ꢏ

ꢐ

퐴, 퐵

= 푇 퐴 , 퐵 = ꢄ_푒

̇

0

ꢌꢑ

ꢐ

and we can conclude that if our assumption holds then that implies:

〈 ( ) ( )〉 | )

ꢄ_푒0 , ꢄ_ꢏ= 0 ⇒ 푆(휔||휌) + 푆(휌| 휎 = 푆(휔||휎)

̇

0

ꢐ

This is some time called Pythagorean theorem of density matrix.

• Renyi Entropy, Quantum Renyi Entropy and Quantum Relative Renyi Entropy:

The Renyi entropy of order 훼 ≠ 1 of the probability distribution (푝₁, 푝₂, . . . , 푝_푛) is defined by

푛

1

(

)

퐻_ꢒ푝₁, 푝₂, . . . , 푝_푛

=

ln ∑ 푝_푘^ꢒ

1 − 훼

푘=1

The Quantum Renyi entropy of order 훼 ≠ 1 of the density matrix 휌 is defined by

1

ꢒ

( )

푆_ꢒ휌 =

(

)

ln tr 휌

1 − 훼

The Quantum Renyi entropy of order 훼 ≠ 1 of the density matrix 휌 with respect 휎 to is defined by

1

ꢒ

1−ꢒ

(

)

(

)

푆_ꢒ휌||휎 =

ln tr 휌 휎

훼 − 1

Entanglement

Let ℬ(ℍ_ꢎ) and ℬ(ℍ_ꢓ) be the algebras of bounded operators acting on the Hilbert spaces ℍ_ꢎand ℍ_ꢓ. The Hilbert space of the composite system is ℍ_ꢎꢓ≔ ℍ_ꢎ⊗ ℍ_ꢓ. The

(

)

algebra of the operators acting on ℍ_ꢎꢓis ℬ ℍ_ꢎꢓ≔ ℬ(ℍ_ꢎ) ⊗ ℬ(ℍ_ꢓ).