Categories and Quantum Informatics Week 4: Dual objects Chris Heunen 1 / 37
Dual objects Idea: ◮ Quantum mechanically: maximally entangled states ◮ Graphically: bending wires 2 / 37
Dual objects Idea: ◮ Quantum mechanically: maximally entangled states ◮ Graphically: bending wires An object L is left-dual to an object R , and R is right-dual to L , written L ⊣ R , when there is a unit morphism I η R ⊗ L and a counit morphism L ⊗ R ε I such that: ρ − 1 id L ⊗ η L L ⊗ ( R ⊗ L ) L L ⊗ I α − 1 id L L , R , L I ⊗ L ( L ⊗ R ) ⊗ L L λ L ε ⊗ id L λ − 1 η ⊗ id R R ( R ⊗ L ) ⊗ R R I ⊗ R α R , L , R id R R ⊗ I R ⊗ ( L ⊗ R ) R ρ R id R ⊗ ε 2 / 37
Snake equations Draw an object L as a wire with an upward-pointing arrow, and a right dual R as a wire with a downward-pointing arrow. L R 3 / 37
Snake equations Draw an object L as a wire with an upward-pointing arrow, and a right dual R as a wire with a downward-pointing arrow. L R The unit I η R ⊗ L and counit L ⊗ R ε I are drawn as bent wires: R L L R Duality equations become: = = Also called the snake equations. 3 / 37
Dual Hilbert spaces FHilb has all duals: any finite-dimensional Hilbert space H is both right and left dual to its dual Hilbert space H ∗ , in a canonical way. 4 / 37
Dual Hilbert spaces FHilb has all duals: any finite-dimensional Hilbert space H is both right and left dual to its dual Hilbert space H ∗ , in a canonical way. The counit H ⊗ H ∗ ε C is: | φ � ⊗ � ψ | �→ � ψ | φ � The unit C η H ∗ ⊗ H is defined like so, for any orthonormal basis | i � : � 1 �→ � i | ⊗ | i � i 4 / 37
Dual Hilbert spaces FHilb has all duals: any finite-dimensional Hilbert space H is both right and left dual to its dual Hilbert space H ∗ , in a canonical way. The counit H ⊗ H ∗ ε C is: | φ � ⊗ � ψ | �→ � ψ | φ � The unit C η H ∗ ⊗ H is defined like so, for any orthonormal basis | i � : � 1 �→ � i | ⊗ | i � i Is η basis-dependent, but ε not? No. (Will prove shortly.) Infinite-dimensional spaces do not have duals. (Will prove later.) 4 / 37
Dual matrices In Mat C , every object n is its own dual, with a canonical choice of η and ε given as follows: � η : 1 �→ | i � ⊗ | i � ε : | i � ⊗ | j � �→ δ ij 1 i 5 / 37
Dual relations In Rel , every object is its own dual, even infinite sets. Unit 1 η S × S and counit S × S ε 1 are: • ∼ η ( s , s ) for all s ∈ S ( s , s ) ∼ ε • for all s ∈ S 6 / 37
Names and conames Set only has duals for singleton sets. Let A f B be a morphism in a monoidal category with dualities A ⊣ A ∗ and B ⊣ B ∗ . Its name I � f � A ∗ ⊗ B and coname A ⊗ B ∗ � f � I are: A ∗ B f f A B ∗ 7 / 37
Names and conames Set only has duals for singleton sets. Let A f B be a morphism in a monoidal category with dualities A ⊣ A ∗ and B ⊣ B ∗ . Its name I � f � A ∗ ⊗ B and coname A ⊗ B ∗ � f � I are: A ∗ B f f A B ∗ Morphisms can be recovered from their names or conames: B B f f = A A 7 / 37
Names and conames Set only has duals for singleton sets. Let A f B be a morphism in a monoidal category with dualities A ⊣ A ∗ and B ⊣ B ∗ . Its name I � f � A ∗ ⊗ B and coname A ⊗ B ∗ � f � I are: A ∗ B f f A B ∗ Morphisms can be recovered from their names or conames: B B f f = A A In Set I is terminal, and so all conames A ⊗ B ∗ � f � I must be equal. If Set had duals this would imply all functions A B were equal. 7 / 37
Duals are unique up to iso In a monoidal category with L ⊣ R , then L ⊣ R ′ if and only if R ≃ R ′ . Similarly, if L ⊣ R , then L ′ ⊣ R if and only if L ≃ L ′ . 8 / 37
Duals are unique up to iso In a monoidal category with L ⊣ R , then L ⊣ R ′ if and only if R ≃ R ′ . Similarly, if L ⊣ R , then L ′ ⊣ R if and only if L ≃ L ′ . R ′ and R ′ Proof: If L ⊣ R and L ⊣ R ′ , define maps R R by: R ′ R L L R ′ R The snake equations imply that these are inverse. 8 / 37
Duals are unique up to iso In a monoidal category with L ⊣ R , then L ⊣ R ′ if and only if R ≃ R ′ . Similarly, if L ⊣ R , then L ′ ⊣ R if and only if L ≃ L ′ . R ′ and R ′ Proof: If L ⊣ R and L ⊣ R ′ , define maps R R by: R ′ R L L R ′ R The snake equations imply that these are inverse. Conversely, if L ⊣ R and R f R ′ is invertible, we can construct a duality L ⊣ R ′ : R ′ L R f -1 f R R ′ L An iso L ≃ L ′ lets us produce duality L ′ ⊣ R in a similar way. 8 / 37
Unit determines counit If ( L , R , η, ε ) and ( L , R , η, ε ′ ) both exhibit duality, then ε = ε ′ . If ( L , R , η, ε ) and ( L , R , η ′ , ε ) both exhibit duality, then η = η ′ . 9 / 37
Unit determines counit If ( L , R , η, ε ) and ( L , R , η, ε ′ ) both exhibit duality, then ε = ε ′ . If ( L , R , η, ε ) and ( L , R , η ′ , ε ) both exhibit duality, then η = η ′ . Proof: ε ε ε ′ ε ′ iso = = = ε ε ′ 9 / 37
Duals respect tensors In a monoidal category, I ⊣ I , and L ⊗ L ′ ⊣ R ⊗ R ′ if L ⊣ R and L ′ ⊣ R ′ . 10 / 37
Duals respect tensors In a monoidal category, I ⊣ I , and L ⊗ L ′ ⊣ R ⊗ R ′ if L ⊣ R and L ′ ⊣ R ′ . Proof: Taking η = λ − 1 : I I ⊗ I and ε = λ I : I ⊗ I I shows that I I ⊣ I . Snake equations follow from the coherence theorem. 10 / 37
Duals respect tensors In a monoidal category, I ⊣ I , and L ⊗ L ′ ⊣ R ⊗ R ′ if L ⊣ R and L ′ ⊣ R ′ . Proof: Taking η = λ − 1 : I I ⊗ I and ε = λ I : I ⊗ I I shows that I I ⊣ I . Snake equations follow from the coherence theorem. Now suppose L ⊣ R and L ′ ⊣ R ′ . We make the new unit and counit maps from the old ones, and compute as follows: R ′ R iso = = L ′ L ′ L L ′ L L 10 / 37
Duals respect braiding In a braided monoidal category, L ⊣ R ⇒ R ⊣ L . 11 / 37
Duals respect braiding In a braided monoidal category, L ⊣ R ⇒ R ⊣ L . Construct a new duality as follows: I η ′ L ⊗ R R ⊗ L ε ′ I 11 / 37
Duals respect braiding In a braided monoidal category, L ⊣ R ⇒ R ⊣ L . Construct a new duality as follows: I η ′ L ⊗ R R ⊗ L ε ′ I Test the snake equations: = = 11 / 37
Duals for morphisms For a morphism A f B and chosen dualities A ⊣ A ∗ , B ⊣ B ∗ , the right dual B ∗ f ∗ A ∗ is defined in the following way: A ∗ A ∗ A ∗ f ∗ := f =: f B ∗ B ∗ B ∗ Represent this graphically by rotating the box for f . 12 / 37
Sliding For all morphisms A f B in a monoidal category with chosen duals A ⊣ A ∗ and B ⊣ B ∗ : f f = = f f 13 / 37
Duals are functorial If a monoidal category has chosen right duals, ( − ) ∗ is a functor. Proof: Let A f B and B g C . g f ∗ ( g ◦ f ) ∗ g = = f = g ∗ f Similarly, ( id A ) ∗ = id A ∗ follows from the snake equations. 14 / 37
Examples f W is W ∗ f ∗ ◮ In FVect and FHilb , right dual of V V ∗ , acting as e C is an arbitrary element of W ∗ . f ∗ ( e ) := e ◦ f , where W ◮ In Mat C , the dual of a matrix is its transpose. ◮ In Rel , the dual of a relation is its converse. So the right duals functor and the dagger functor have the same action: R ∗ = R † for all relations R . 15 / 37
Double duals In monoidal category with chosen right duals, A ∗∗ ⊗ B ∗∗ ≃ ( A ⊗ B ) ∗∗ . 16 / 37
Double duals In monoidal category with chosen right duals, A ∗∗ ⊗ B ∗∗ ≃ ( A ⊗ B ) ∗∗ . Proof: ( A ⊗ B ) ∗∗ ε A ⊗ B η ( A ⊗ B ) ∗ A ∗∗ B ∗∗ 16 / 37
Teleportation In a monoidal category with right duals, a teleportation procedure isa finite family of effects e i : A ⊗ A ∗ I and unitaries U i : A A with: A A U i e i = A A 17 / 37
Teleportation In a monoidal category with right duals, a teleportation procedure isa finite family of effects e i : A ⊗ A ∗ I and unitaries U i : A A with: A A U i e i = A A e i This can be solved to give = . U i 17 / 37
Teleportation Simplify the history: L U i U i L 18 / 37
Teleportation Simplify the history: L L U i U i = U i U i L L 18 / 37
Teleportation Simplify the history: L L L U i U i = = U i U i L L L 18 / 37
Teleportation Simplify the history: L L L L U i U i = = = U i U i L L L L So if the original history occurs, the result is for the state of the original system to be transmitted faithfully. 18 / 37
Teleportation Simplify the history: L L L L U i U i = = = U i U i L L L L So if the original history occurs, the result is for the state of the original system to be transmitted faithfully. If { e i } is a complete set of effects, this will always succeed. 18 / 37
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