Process Theories

A process theory describes physical operations by how they can be composed. Systems specify input and output types; processes transform inputs into outputs. The language is useful because it exposes which manipulations depend only on connectivity and which require specifically quantum assumptions.

Sequential and parallel composition

Write $f:A\to B$ for a process with input system $A$ and output system $B$. If $g:B\to C$, their sequential composition is $g\circ f:A\to C$. Parallel composition is $f\otimes h:A\otimes D\to B\otimes E$ for $h:D\to E$.

Associativity, identity processes, and the interchange law ensure consistent assembly:

$$(g\otimes k)\circ(f\otimes h) =(g\circ f)\otimes(k\circ h).$$

The unit object $I$ represents absence of a physical wire. A state is a process $I\to A$, an effect a process $A\to I$, and a closed experiment a scalar $I\to I$.

A symmetric monoidal category formalizes this structure when exchanging independent system wires is available. Merely drawing boxes and wires does not establish that a theory is quantum: classical stochastic processes also obey these composition laws.

A classical calculation in the same language

For a classical bit, take a state vector $p=(p_0,p_1)^T$ and a stochastic matrix

$$T=\begin{pmatrix}1-q&q\\q&1-q\end{pmatrix}.$$

The effect $e=(0,1)$ asks whether the final bit is one. The closed circuit evaluates to $eTp=qp_0+(1-q)p_1$. Replacing vectors by density operators, stochastic maps by channels, and effects by POVM elements gives the quantum counterpart $\operatorname{Tr}[E\mathcal E(\rho)]$.

This parallel demonstrates what the compositional notation organizes while leaving each theory’s allowed processes distinct.

Discarding and causality

A discard effect is denoted by $u_A$. For classical normalized distributions it sums probabilities; for quantum states it is the trace. A deterministic process obeys

$$u_B\circ f=u_A.$$

A selected outcome may instead be a subnormalized process. Its scalar value is the probability of that outcome. Postselecting and renormalizing is a nonlinear operation on conditional descriptions; it should not be silently substituted for a deterministic channel.

The adjoint of a channel is unital, but is not generally another trace-preserving channel. Thus “reverse the diagram” requires specifying whether one is changing mathematical representation, taking an adjoint, or implementing a physical reversal.

An adjoint updates the test

In finite-dimensional quantum theory, the adjoint channel is defined by

$$\operatorname{Tr}[E\mathcal E(\rho)] =\operatorname{Tr}[\mathcal E^\dagger(E)\rho].$$

It moves the final test back to an equivalent test on the input. Trace preservation of $\mathcal E$ gives $\mathcal E^\dagger(I)=I$: a certain final event remains certain. Positivity then ensures that $0\leq E\leq I$ implies $0\leq\mathcal E^\dagger(E)\leq I$. This is the Heisenberg description of the same experiment, as developed in Watrous’s channel formalism.

For a concrete example, let amplitude damping transfer $\lvert1\rangle$ to $\lvert0\rangle$ with probability $\gamma\in[0,1]$. Its adjoint sends the ground-state test to

$$\mathcal A_\gamma^\dagger(\lvert0\rangle\langle0\rvert) =\lvert0\rangle\langle0\rvert +\gamma\lvert1\rangle\langle1\rvert.$$

The resulting probability is $\rho_{00}+\gamma\rho_{11}$: the system either started in the ground state or decayed into it. The operator is a valid effect, but its trace is $1+\gamma$. For $\gamma>0$, interpreting the same operator as the output density matrix of a deterministic reversed process would violate normalization. The adjoint preserves the probability calculation; it does not undo the relaxation.

Exercises

Why does a preparation $I\to A$ followed by discarding yield the scalar one?

Solution

A normalized preparation occurs with certainty when all outcomes are ignored. In quantum theory this is $\operatorname{Tr}\rho=1$; in a classical theory it is $\sum_i p_i=1$. A probabilistic selected preparation instead yields its success probability.

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.
• Coecke, Bob, and Aleks Kissinger, Picturing Quantum Processes, Cambridge University Press, 2017 — compositional processes and diagrammatic reasoning. Publisher.
• Watrous, John, The Theory of Quantum Information, Cambridge University Press, 2018, Chapter 2 — quantum channels and their adjoints. Author’s text.