Picturing Quantum Entanglement ...in MBQC Aleks Kissinger March 7, - PowerPoint PPT Presentation

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 diagrams built from three kinds of generators: ... : = | 0...0 �� 0...0 | + e i α | 1...1 �� 1...1 | α ... ... : = | + ... + �� + ... + | + e i α |− ... −��− ... −| α ... : = | + �� 0 | + |−�� 1 |

ZX-calculus Hopf algebras and Entanglement ZX-calculus in QC • Admits an encoding of circuits: U α β γ � Z

ZX-calculus Hopf algebras and Entanglement ZX-calculus in QC • Admits an encoding of circuits: U α β γ � Z • ...and MBQC: ... ... ... ... � ... ... ... ... ... ... ... ... ... ... ... ... ... ...

ZX-calculus Hopf algebras and Entanglement ZX-calculus in QC • Admits an encoding of circuits: U α β γ � Z • ...and MBQC: ... ... ... ... � ... ... ... ... ... ... ... ... ... ... ... ... ... ... � � • ...and a means of translating between the two. ⇐

ZX-calculus Hopf algebras and Entanglement Algebraic structure • All of its power comes from its underlying algebraic structures : hopf hopf frobenius frobenius • ...which have been studied extensively in category theory and representation theory.

ZX-calculus Hopf algebras and Entanglement Algebraic structure • All of its power comes from its underlying algebraic structures : hopf hopf frobenius frobenius • ...which have been studied extensively in category theory and representation theory.

ZX-calculus Hopf algebras and Entanglement Hopf algebras and Z 2 -matrices • (Commutative, self-inverse) Hopf algebra expressions are totally characterised by their Z 2 -path matrices:    1 0 1  ↔ ↔ 1 0 1

ZX-calculus Hopf algebras and Entanglement Hopf algebras and Z 2 -matrices • (Commutative, self-inverse) Hopf algebra expressions are totally characterised by their Z 2 -path matrices:    1 0 1  ↔ ↔ 1 0 1

ZX-calculus Hopf algebras and Entanglement Hopf algebras and Z 2 -matrices • (Commutative, self-inverse) Hopf algebra expressions are totally characterised by their Z 2 -path matrices:    1 0 1  ↔ ↔ 1 0 1

ZX-calculus Hopf algebras and Entanglement Hopf algebras and Z 2 -matrices • (Commutative, self-inverse) Hopf algebra expressions are totally characterised by their Z 2 -path matrices:    1 0 1  ↔ ↔ 1 0 1

ZX-calculus Hopf algebras and Entanglement Hopf algebras and Z 2 -matrices • (Commutative, self-inverse) Hopf algebra expressions are totally characterised by their Z 2 -path matrices:    1 0 1  ↔ ↔ 1 0 1 • In category-theoretic terms, this means Mat ( Z 2 ) is a PROP for commutative, self-inverse Hopf algebras.

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 Z 2 ) 1 , e.g.   1 1 0   0 0 1    = 2 ebits rank �  0 0 1 0 0 1 1 Hein, Eisart, Briegel. arXiv:quant-ph/0602096, Prop. 11

ZX-calculus Hopf algebras and Entanglement

ZX-calculus Hopf algebras and Entanglement

ZX-calculus Hopf algebras and Entanglement

ZX-calculus Hopf algebras and Entanglement

ZX-calculus Hopf algebras and Entanglement unitary unitary

ZX-calculus Hopf algebras and Entanglement unitary unitary

ZX-calculus Hopf algebras and Entanglement unitary unitary   1 1 0   0 0 1     0 0 1 0 0 1

ZX-calculus Hopf algebras and Entanglement unitary unitary     1 1 0 1 0 � 1 �     0 0 1 0 1 1 0      =    0 0 1 0 1 0 0 1 0 0 1 0 1

ZX-calculus Hopf algebras and Entanglement Measuring Entanglement • When the cut-rank is k , this always yields a factorisation by isometries through k wires

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: ... ... ... � � � U V U ∼ = ... k wires k Bell pairs ... � V ...

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: ... ... ... � � � U V U ∼ = ... k wires k Bell pairs ... � V ... • Computing the entropy of entanglement:   ... ...     � � V U   S     ...

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: ... ... ... � � � U V U ∼ = ... k wires k Bell pairs ... � V ... • Computing the entropy of entanglement:   ...     � U   S     ...

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: ... ... ... � � � U V U ∼ = ... k wires k Bell pairs ... � V ... • Computing the entropy of entanglement:   ...     � U   S   ...  

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: ... ... ... � � � U V U ∼ = ... k wires k Bell pairs ... � V ... • Computing the entropy of entanglement:     ...     S    

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: ... ... ... � � � U V U ∼ = ... k wires k Bell pairs ... � V ... • Computing the entropy of entanglement: � � k · S = k