Dagger Compact Categories

A dagger compact category is a compositional setting with adjoints and compatible dual objects. It supports a precise version of bending an input wire into an output wire using an entangled state. In finite-dimensional quantum theory, the resulting identities explain algebraic features of teleportation, while normalization determines their operational probabilities.

Daggers and duals

A dagger is identity on objects, reverses the direction of morphisms, and obeys

$$(g\circ f)^\dagger=f^\dagger\circ g^\dagger, \qquad (f^\dagger)^\dagger=f.$$

It respects tensor products. In Hilbert spaces the dagger is the Hermitian adjoint, not generally a physical time-reversal operation.

Compact structure supplies a dual object $A^*$ and cup and cap maps. With one consistent ordering convention,

$$\eta:I\to A^*\otimes A,\qquad \epsilon:A\otimes A^*\to I.$$

They satisfy the snake identity

$$(\epsilon\otimes 1_A)\circ (1_A\otimes\eta)=1_A,$$

with associativity and unit identifications understood. There is a corresponding identity on $A^*$. Dagger compatibility relates cap and cup through the adjoint and symmetry.

A basis calculation

For a $d$-dimensional Hilbert space, use its conjugate space as the dual and set

$$\eta(1)=\sum_{i=1}^d\lvert\bar i\rangle\otimes\lvert i\rangle, \qquad \epsilon(\lvert j\rangle\otimes\lvert\bar i\rangle)=\delta_{ij}.$$

Applied to $\lvert j\rangle$, the snake composition retains the one term $i=j$, giving $\lvert j\rangle$. This direct calculation shows what the compact identity asserts.

The cup has squared norm $d$. It is deliberately unnormalized. Replacing both cup and cap by normalized maximally entangled vectors multiplies the resulting map by $1/d$. For a normalized input, the selected Bell projection then has probability $1/d^2$.

Deterministic teleportation includes all measurement outcomes, classical communication, and conditional corrections. A single cup-cap identity does not itself implement the entire protocol with unit probability.

Scope and limitations

Finite-dimensionality matters: the sum defining a maximally entangled cup is not a normalizable vector in an infinite-dimensional Hilbert space. One cannot import compact closure unchanged into all quantum systems.

Dagger compact axioms also do not uniquely select complex Hilbert spaces. They characterize a common compositional structure shared by several models. Reconstruction programs must add the ingredients that determine a physical probability theory.

Exercises

If a closed cup-cap loop uses the unnormalized basis sum above, what scalar does it produce?

Solution

It gives $\sum_i1=d$. With a normalized Bell state and its adjoint it instead gives one. Tracking this factor is essential when comparing a graphical equality of linear maps with the success probability of a physical measurement branch.

References

• Abramsky, Samson, and Bob Coecke, “A categorical semantics of quantum protocols,” Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 415–425 (2004). Paper.
• Selinger, Peter, “Finite dimensional Hilbert spaces are complete for dagger compact closed categories,” Logical Methods in Computer Science 8(3:6), 1–12 (2012), doi:10.2168/LMCS-8(3:6)2012. Author manuscript.