Mixed quantum states in higher categories Linde Wester Department - - PowerPoint PPT Presentation

Mixed quantum states in higher categories

Linde Wester Department of Computer Science, University of Oxford (with Chris Heunen and Jamie Vicary) June 6, 2014

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Table of contents

Existing models for classical and quantum data Special dagger Frobenius algebras 2-categorical quantum mechanics The construction 2(−) The theory of bimodules The 2(−) construction 2(CP∗(−)) Applications A unified description of teleportation and classical encryption A unified security proof

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CP∗(−)

• 1. Special dagger Frobenius algebras in a monoidal category C:

= = = = = =

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CP∗(−)

• 1. Special dagger Frobenius algebras in a monoidal category C:

= = = = = =

• 2. Completely positive maps between Frobenius algebras:

morphisms f in C, for which ∃g such that f = g† g

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2-categories and their graphical language

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2-categories and their graphical language

0-cells Regions Classical information A

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2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems A B

S 4 / 15

2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems A B

S 4 / 15

2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems 2-cells Vertices Quantum dynamics A B

S S′ α 4 / 15

2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems 2-cells Vertices Quantum dynamics A B

S S′ α 4 / 15

2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems 2-cells Vertices Quantum dynamics Horizontal composition A B C

S S′ T α 4 / 15

2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems 2-cells Vertices Quantum dynamics Horizontal composition A B C

S S′ T α 4 / 15

2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems 2-cells Vertices Quantum dynamics Horizontal composition Vertical composition A B C

S S′ T S′′ α γ 4 / 15

2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems 2-cells Vertices Quantum dynamics Horizontal composition Vertical composition A B C

S S′ S′′ T α γ 4 / 15

2-categories and their graphical language

0-cells Regions Classical information 1-cells Lines Quantum systems 2-cells Vertices Quantum dynamics Horizontal composition Vertical composition C2 C C

H1 H2

• S′

S′′

• H3

H4

• H1 ⊗ H3 ⊕ H2 ⊗ H4
• α

γ

The standard example is 2Hilb:

◮ 0-cells given by natural numbers ◮ 1-cells given by matrices of finite-dimensional Hilbert spaces ◮ 2-cells given by matrices of linear maps

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Quantum systems interacting with their environment

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Quantum systems interacting with their environment

Let (A, , ) and (B, , ) be classical structures in C. A dagger C-D-bimodule is a morphism M satisfying:

M

= M

M A

M

B B

M

B M A A M 5 / 15

Quantum systems interacting with their environment

Let (A, , ) and (B, , ) be classical structures in C. A dagger C-D-bimodule is a morphism M satisfying:

M

= M

M A

M

B B

M

B M A A M M M

M =

M 5 / 15

Quantum systems interacting with their environment

Let (A, , ) and (B, , ) be classical structures in C. A dagger C-D-bimodule is a morphism M satisfying:

M

= M

M A

M

B B

M

B M A A M M M

M =

M M A M A M B B

M† M =

M 5 / 15

Quantum systems interacting with their environment

Let (A, , ) and (B, , ) be classical structures in C. A dagger C-D-bimodule is a morphism M satisfying:

M

= M

M A

M

B B

M

B M A A M M M

M =

M M A M A M B B

M† M =

M

A bimodule homomorphism is a morphism f ∈ C, such that

M′ M

M = f M′

M′ M

f

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Quantum systems interacting with their environment

Let (A, , ) and (B, , ) be classical structures in C. A dagger C-D-bimodule is a morphism M satisfying:

M

M =

M

M M

M M A A B B M M

M =

M M M

M =

M

M†

M A A B B

A bimodule homomorphism is a morphism f ∈ C, such that

M′ M

M = f M′

M′ M

f

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

Some properties of 2(−) are:

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

Some properties of 2(−) are:

◮ 2(C) is a 2-category.

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

Some properties of 2(−) are:

◮ 2(C) is a 2-category. ◮ 2(−) preserves the dagger.

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

Some properties of 2(−) are:

◮ 2(C) is a 2-category. ◮ 2(−) preserves the dagger. ◮ If C is compact, so is 2(C): 1-cells have ambidextrous duals.

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

Some properties of 2(−) are:

◮ 2(C) is a 2-category. ◮ 2(−) preserves the dagger. ◮ If C is compact, so is 2(C): 1-cells have ambidextrous duals. ◮ If C has dagger biproducts, so do all hom-categories of 2(C).

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

Some properties of 2(−) are:

◮ 2(C) is a 2-category. ◮ 2(−) preserves the dagger. ◮ If C is compact, so is 2(C): 1-cells have ambidextrous duals. ◮ If C has dagger biproducts, so do all hom-categories of 2(C). ◮ The subcategory of scalars of 2(C) corresponds to C.

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

Some properties of 2(−) are:

◮ 2(C) is a 2-category. ◮ 2(−) preserves the dagger. ◮ If C is compact, so is 2(C): 1-cells have ambidextrous duals. ◮ If C has dagger biproducts, so do all hom-categories of 2(C). ◮ The subcategory of scalars of 2(C) corresponds to C. ◮ 2(FHilb) is isomorphic to the category 2Hilb.

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The 2(−) construction

How can we construct the 2-category 2(C) from C?

◮ 0-cells: classical structures in C ◮ 1-cells: bimodules of classical structures in C ◮ 2-cells: module homomorphisms in C

In representation theory: The orbifold completion of a monoidal category

Some properties of 2(−) are:

◮ 2(C) is a 2-category. ◮ 2(−) preserves the dagger. ◮ If C is compact, so is 2(C): 1-cells have ambidextrous duals. ◮ If C has dagger biproducts, so do all hom-categories of 2(C). ◮ The subcategory of scalars of 2(C) corresponds to C. ◮ 2(FHilb) is isomorphic to the category 2Hilb.

For proofs see LW (2013), Masters’s thesis, ’Categorical Models for Quantum Computing’.

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Horizontal composition in 2(−)

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Horizontal composition in 2(−)

Horizontal composition is defined by the following coequaliser in C: M ⊗ N M ⊗B N K π f ˜ f M ⊗ B ⊗ N

MB ⊗ idN idM⊗ BN

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Horizontal composition in 2(−)

Horizontal composition is defined by the following coequaliser in C: M ⊗ N M ⊗B N π M ⊗ B ⊗ N

MB ⊗ idN idM⊗ BN

π

M N

π

=

K M N M N A B M N B C K 8 / 15

Horizontal composition in 2(−)

Horizontal composition is defined by the following coequaliser in C: M ⊗ N M ⊗B N K π f M ⊗ B ⊗ N

MB ⊗ idN idM⊗ BN

f

M N

f

=

K M N M N A B M N B C K 8 / 15

Horizontal composition in 2(−)

Horizontal composition is defined by the following coequaliser in C: M ⊗ N M ⊗B N K π f ˜ f M ⊗ B ⊗ N

MB ⊗ idN idM⊗ BN

f

M N

f

=

K M N M N A B M N B C K

Can we find this module explicitly?

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Horizontal composition in 2(−)

Horizontal composition is defined by the following coequaliser in C: M ⊗ N M ⊗B N K π f ˜ f M ⊗ B ⊗ N

MB ⊗ idN idM⊗ BN

f

M N

f

=

K M N M N A B M N B C K

Can we find this module explicitly? Yes!

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Horizontal composition in terms of dagger splittings

Any such f factorizes through M N:

f

M N

f

=

K M N M N A B M N C K M K

f

A

=

N M

M

B A M B A M

M

N K

f

M

=

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Horizontal composition in terms of dagger splittings

Any such f factorizes through M N:

f

M N

f

=

K M N M N A B M N C K M K

f

A

=

N M

M

B A M B A M

M

N K

f

M

=

Theorem

Finding the dagger coequaliser is equivalent to finding a dagger splitting of the following morphism:

N

M

M B C A M N

N

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2(CP∗(−))

We would like to understand the 2-category ’?’ FHilb CP∗(FHilb) 2(FHilb) ? CP∗(−) 2(−) CP∗(−) 2(−)

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2(CP∗(−))

We would like to understand the 2-category ’?’ FHilb CP∗(FHilb) 2(FHilb) ? CP∗(−) 2(−) CP∗(−) 2(−) This is not obvious!

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2(CP∗(−))

We would like to understand the 2-category ’?’ FHilb CP∗(FHilb) 2(FHilb) ? CP∗(−) 2(−) CP∗(−) 2(−) This is not obvious!

◮ This required a classification of classical structures in

CP∗(FHilb).

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2(CP∗(−))

We would like to understand the 2-category ’?’ FHilb CP∗(FHilb) 2(FHilb) ? CP∗(−) 2(−) CP∗(−) 2(−) This is not obvious!

◮ This required a classification of classical structures in

CP∗(FHilb).

◮ There is a correspondence between special dagger Frobenius

algebras on classical structures in FHilb and finite groupoids.

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2(CP∗(−))

We would like to understand the 2-category ’?’ FHilb CP∗(FHilb) 2(FHilb) ? CP∗(−) 2(−) CP∗(−) 2(−) This is not obvious!

◮ This required a classification of classical structures in

CP∗(FHilb).

◮ There is a correspondence between special dagger Frobenius

algebras on classical structures in FHilb and finite groupoids.

◮ CP∗(FHilb) does not have all coequalisers.

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Modelling POVM’s

The following subcategory of 2(CP∗(FHilb)) is a sufficient model for modelling communication protocols:

◮ 0-cells: natural numbers ◮ 1-cells: matrices of dagger Frobenius algebras ◮ 2-cells: matrices of completely positive maps

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Modelling POVM’s

The following subcategory of 2(CP∗(FHilb)) is a sufficient model for modelling communication protocols:

◮ 0-cells: natural numbers ◮ 1-cells: matrices of dagger Frobenius algebras ◮ 2-cells: matrices of completely positive maps

Measurements are defined as counit-preserving 2-cells of type: µ

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Modelling POVM’s

The following subcategory of 2(CP∗(FHilb)) is a sufficient model for modelling communication protocols:

◮ 0-cells: natural numbers ◮ 1-cells: matrices of dagger Frobenius algebras ◮ 2-cells: matrices of completely positive maps

Measurements are defined as counit-preserving 2-cells of type: µ

Theorem

Measurements on algebras Cn are exactly stochastic maps. Measurements on algebras B(H) are exactly POVMs.

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Modelling POVM’s

Proof.

The counit preserving condition gives us    µ Cn =   

†

⇔    µ† Cn =   

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Modelling POVM’s

Proof.

The counit preserving condition gives us    µ Cn =   

†

⇔    µ† Cn =    So we have the following equalities of positive elements:

n

• i=1

µ†

i

= µ† Cn =

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Modelling POVM’s

Proof.

The counit preserving condition gives us    µ Cn =   

†

⇔    µ† Cn =    So we have the following equalities of positive elements:

n

• i=1

µ†

i

= µ† Cn =

◮ On Cn this corresponds to a stochastic map

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Modelling POVM’s

Proof.

The counit preserving condition gives us    µ Cn =   

†

⇔    µ† Cn =    So we have the following equalities of positive elements:

n

• i=1

µ†

i

= µ† Cn =

◮ On Cn this corresponds to a stochastic map ◮ On B(Cn) this corresponds to a POVM

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Classical encryption and quantum teleportation

Quantum teleportation and classical encryption are solutions to the following equation with µ a measurement and ν unitary 2-cell: µ ν =

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Classical encryption and quantum teleportation

Quantum teleportation and classical encryption are solutions to the following equation with µ a measurement and ν unitary 2-cell: µ ν

    C C C C     B(C2) B(C2) B(C2)

• C

C C C

• u1

... u4

• C4

=

B(C2)

C4

    C C C C    

• C

C C C

• This equation corresponds to:

◮ quantum teleportation, if the input is a matrix algebra

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Classical encryption and quantum teleportation

Quantum teleportation and classical encryption are solutions to the following equation with µ a measurement and ν unitary 2-cell: µ ν

    C C C C     C2 C2 C2

• C

C C C

• u1

... u4

• C4

=

C2

C4

    C C C C    

• C

C C C

• This equation corresponds to:

◮ quantum teleportation, if the input is a matrix algebra ◮ classical encryption, if the input is a classical structure

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A unified security proof

When the output is destroyed, all information is lost: µ ν = ⇒ µ =

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A unified security proof

When the output is destroyed, all information is lost: µ ν = ⇒ µ =

◮ We apply the trace map on both sides of the equation

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A unified security proof

When the output is destroyed, all information is lost: µ ν = ⇒ µ =

◮ We apply the trace map on both sides of the equation ◮ On the left-hand-side: ν is a family invertible completely

positive maps, which are trace preserving. So this give a unified security proof

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Overview

The results:

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Overview

The results:

◮ A categorical generalisation of 2Hilb, based on modules:

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Overview

The results:

◮ A categorical generalisation of 2Hilb, based on modules:

The construction 2(−), which preserves daggers, compactness, biproducts, such that the scalars of 2(C) correspond to C.

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Overview

The results:

◮ A categorical generalisation of 2Hilb, based on modules:

The construction 2(−), which preserves daggers, compactness, biproducts, such that the scalars of 2(C) correspond to C.

◮ Horizontal composition in 2(C) is given by dagger splittings.

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Overview

The results:

◮ A categorical generalisation of 2Hilb, based on modules:

The construction 2(−), which preserves daggers, compactness, biproducts, such that the scalars of 2(C) correspond to C.

◮ Horizontal composition in 2(C) is given by dagger splittings. ◮ First steps in understanding 2(CP∗(FHilb)).

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Overview

The results:

◮ A categorical generalisation of 2Hilb, based on modules:

The construction 2(−), which preserves daggers, compactness, biproducts, such that the scalars of 2(C) correspond to C.

◮ Horizontal composition in 2(C) is given by dagger splittings. ◮ First steps in understanding 2(CP∗(FHilb)). ◮ 2(FHilb) contains a subcategory of classical structures,

matrices of special dagger Frobenius algebras, and matrices of completely positive morphisms.

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Overview

The results:

◮ A categorical generalisation of 2Hilb, based on modules:

The construction 2(−), which preserves daggers, compactness, biproducts, such that the scalars of 2(C) correspond to C.

◮ Horizontal composition in 2(C) is given by dagger splittings. ◮ First steps in understanding 2(CP∗(FHilb)). ◮ 2(FHilb) contains a subcategory of classical structures,

matrices of special dagger Frobenius algebras, and matrices of completely positive morphisms.

◮ Unified description of teleportation and classical encryption.

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Overview

The results:

◮ A categorical generalisation of 2Hilb, based on modules:

The construction 2(−), which preserves daggers, compactness, biproducts, such that the scalars of 2(C) correspond to C.

◮ Horizontal composition in 2(C) is given by dagger splittings. ◮ First steps in understanding 2(CP∗(FHilb)). ◮ 2(FHilb) contains a subcategory of classical structures,

matrices of special dagger Frobenius algebras, and matrices of completely positive morphisms.

◮ Unified description of teleportation and classical encryption. ◮ Security proof of teleportation and classical encryption.

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Overview

The results:

◮ A categorical generalisation of 2Hilb, based on modules:

The construction 2(−), which preserves daggers, compactness, biproducts, such that the scalars of 2(C) correspond to C.

◮ Horizontal composition in 2(C) is given by dagger splittings. ◮ First steps in understanding 2(CP∗(FHilb)). ◮ 2(FHilb) contains a subcategory of classical structures,

matrices of special dagger Frobenius algebras, and matrices of completely positive morphisms.

◮ Unified description of teleportation and classical encryption. ◮ Security proof of teleportation and classical encryption.

Thank you!

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