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ZX-calculus Hopf algebras and Entanglement Picturing Quantum Entanglement ...in MBQC Aleks Kissinger March 7, 2015 Q UANTUM G ROUP ZX-calculus Hopf algebras and Entanglement ZX-calculus The ZX-calculus is a formalism that studies

SLIDE 1

ZX-calculus Hopf algebras and Entanglement

Picturing Quantum Entanglement ...in MBQC

Aleks Kissinger March 7, 2015

QUANTUM GROUP

SLIDE 2

ZX-calculus Hopf algebras and Entanglement

ZX-calculus

The ZX-calculus is a formalism that studies diagrams built from three

kinds of generators:

α

... ... := |0...00...0| + eiα |1...11...1|

α

... ... := |+...++...+| + eiα |−...−−...−| := |+0| + |−1|

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ZX-calculus Hopf algebras and Entanglement

ZX-calculus in QC

Admits an encoding of circuits:

U Z

γ β

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ZX-calculus Hopf algebras and Entanglement

ZX-calculus in QC

Admits an encoding of circuits:

U Z

γ β

...and MBQC:

... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ...

SLIDE 5

ZX-calculus Hopf algebras and Entanglement

ZX-calculus in QC

Admits an encoding of circuits:

U Z

γ β

...and MBQC:

... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ...

...and a means of translating between the two.
⇐

SLIDE 6

ZX-calculus Hopf algebras and Entanglement

Algebraic structure

All of its power comes from its underlying algebraic structures:

frobenius hopf hopf frobenius

...which have been studied extensively in category theory and

representation theory.

SLIDE 7

ZX-calculus Hopf algebras and Entanglement

Algebraic structure

All of its power comes from its underlying algebraic structures:

frobenius hopf hopf frobenius

...which have been studied extensively in category theory and

representation theory.

SLIDE 8

ZX-calculus Hopf algebras and Entanglement

Hopf algebras and Z2-matrices

(Commutative, self-inverse) Hopf algebra expressions are totally

characterised by their Z2-path matrices: ↔  1 0 1

1 0 1

  ↔

SLIDE 9

ZX-calculus Hopf algebras and Entanglement

Hopf algebras and Z2-matrices

(Commutative, self-inverse) Hopf algebra expressions are totally

characterised by their Z2-path matrices: ↔  1 0 1

1 0 1

  ↔

SLIDE 10

ZX-calculus Hopf algebras and Entanglement

Hopf algebras and Z2-matrices

(Commutative, self-inverse) Hopf algebra expressions are totally

characterised by their Z2-path matrices: ↔  1 0 1

1 0 1

  ↔

SLIDE 11

ZX-calculus Hopf algebras and Entanglement

Hopf algebras and Z2-matrices

(Commutative, self-inverse) Hopf algebra expressions are totally

characterised by their Z2-path matrices: ↔  1 0 1

1 0 1

  ↔

SLIDE 12

ZX-calculus Hopf algebras and Entanglement

Hopf algebras and Z2-matrices

(Commutative, self-inverse) Hopf algebra expressions are totally

characterised by their Z2-path matrices: ↔  1 0 1

1 0 1

  ↔

In category-theoretic terms, this means Mat(Z2) is a PROP for

commutative, self-inverse Hopf algebras.

SLIDE 13

ZX-calculus Hopf algebras and Entanglement

Measuring Entanglement

Proposition: The amount of entanglement across any bipartition of a

graph state is equal to its cut-rank (i.e. the rank of the associated adjacency matrix over Z2)1, e.g.

rank

    1 1 1 1 1     = 2 ebits

1Hein, Eisart, Briegel. arXiv:quant-ph/0602096, Prop. 11

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ZX-calculus Hopf algebras and Entanglement

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ZX-calculus Hopf algebras and Entanglement

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ZX-calculus Hopf algebras and Entanglement

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ZX-calculus Hopf algebras and Entanglement

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ZX-calculus Hopf algebras and Entanglement

unitary unitary

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ZX-calculus Hopf algebras and Entanglement

unitary unitary

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ZX-calculus Hopf algebras and Entanglement

unitary unitary

    1 1 1 1 1    

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ZX-calculus Hopf algebras and Entanglement

unitary unitary

    1 1 1 1 1     =     1 1 1 1     1 1 1

SLIDE 22

ZX-calculus Hopf algebras and Entanglement

Measuring Entanglement

When the cut-rank is k, this always yields a factorisation by

isometries through k wires

SLIDE 23

ZX-calculus Hopf algebras and Entanglement

Measuring Entanglement

When the cut-rank is k, this always yields a factorisation by

isometries through k wires

Writing as a bipartite state:

...

k wires

...

∼ =

k Bell pairs

... ... ...

SLIDE 24

ZX-calculus Hopf algebras and Entanglement

Measuring Entanglement

When the cut-rank is k, this always yields a factorisation by

isometries through k wires

Writing as a bipartite state:

...

k wires

...

∼ =

k Bell pairs

... ... ...

V
Computing the entropy of entanglement:

S      

... ... ...

     

SLIDE 25

ZX-calculus Hopf algebras and Entanglement

Measuring Entanglement

When the cut-rank is k, this always yields a factorisation by

isometries through k wires

Writing as a bipartite state:

...

k wires

...

∼ =

k Bell pairs

... ... ...

V
Computing the entropy of entanglement:

S      

... ...      

SLIDE 26

ZX-calculus Hopf algebras and Entanglement

Measuring Entanglement

When the cut-rank is k, this always yields a factorisation by

isometries through k wires

Writing as a bipartite state:

...

k wires

...

∼ =

k Bell pairs

... ... ...

V
Computing the entropy of entanglement:

S      

... ...      

SLIDE 27

ZX-calculus Hopf algebras and Entanglement

Measuring Entanglement

When the cut-rank is k, this always yields a factorisation by

isometries through k wires

Writing as a bipartite state:

...

k wires

...

∼ =

k Bell pairs

... ... ...

V
Computing the entropy of entanglement:

S       ...      