What is the difference between quantum gates and quantum channels?

Question

I'm not sure if this is a dumb question, since they seem to be very basic building blocks of quantum information theory; however, I can't seem to wrap my head around the difference between the two. As I understand it, both quantum gates and quantum channels are operations that you can pass a quantum state into. Additionally, a quantum gate is represented by a unitary operator, whereas quantum channels are sums of probabilistic distributions of unitary operators and their inverses... Could anyone help me understand the intuitive distinction between the two, and their relation, if any?

Answer 1

A quantum gate is a unitary operator on a Hilbert space. Typically this Hilbert space is associated with a system of qubits. In the case of a single qubit a quantum gate is a $2\times 2$ unitary matrix, and the unit vectors $|0\rangle$ and $|1\rangle$ are the computational basis states. For example, the quantum NOT gate is the unitary matrix $X$ that maps $|0\rangle\mapsto|1\rangle$ and $|1\rangle \to |0\rangle$. Because of linearity, unitary operators $U$ can act on superpositions (i.e. linear combinations) of basis states, called pure states, transforming them via $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle \mapsto U|\psi\rangle=\alpha U|0\rangle +\beta U|1\rangle$. For systems of $n$ qubits, quantum gates are unitary matrices that act on the Hilbert space $\mathbb{C}^{2^n}\cong \underbrace{\mathbb{C}^2\otimes \cdots \otimes \mathbb{C}^2}_{\text{$n$ times}}$. Often when someones mentiones quantum circuits, one is referring to a collection of gates acting on 1 or 2 qubits arranged in a manner similar to that of a classical circuit acting on $n$ wires.

Quantum channels are a little bit more complicated mathematically. To explain it's convenient to recall the definition of a density matrix. A density matrix is a positive semi-definite matrix $\rho\geq 0$ with $Tr(\rho)=1$. Density matrices represent the information about a more general class of quantum states (not just the pure states), i.e. the probabilities and outcomes of the state that can arise upon measurement. Mathematically, a quantum channel is a linear map that sends density matrices to density matrices by preserving the fundamental properties of these matrices; such as the positivity, and trace condition ($=1$). However, any quantum channel must also have an additional property known as \emph{complete positivity} to ensure that it is compatible with the other axioms of quantum mechanics. Most importantly, the density matrix could be entangled with some auxiliary quantum system. By adding the word \emph{completeley} we guarantee compatibility with this possibility. Hence, a quantum channel is a completely positive trace-preserving (CPTP) map.

Distinctions aside, in the context of physics, quantum gates and quantum channels arise from the unitary evolution of (closed) quantum system according to Schrodinger's equation. The Hamiltonian (a mathematical object governing these dynamics) evolves in time, and the result is a unitary transformation of the quantum state of the system. Thus to physically implement a gate on a quantum computer, we procure a Hamiltonian and evolve it (in time) to produce the desired effect on our system of qubits. Although the entire quantum system evolves unitarily, if we only have access to a part of the system, the transformation of the subsystem is not unitary, nevertheless, it is described by a quantum channel (i.e. a CPTP map). This illustrates on a theoretical level the connection between unitary transformations and quantum channels.

Answer 2

When you model your quantum system as closed and look at transformations the whole system can go through, those are described by unitary evolutions.

But, what if you want to describe local transformations? That's where the notion of channels are introduced; They describe local transformations (There's a dilation theorem stating that given any channel you can always find extended systems and unitary evolutions of those systems such that the channel you have in your hand describes the local evolution). As such, channels describe the most general transformations where states are preserved.

Unitary evolutions are then dubbed unitary channels. To wind up, quantum channels are the most general evolutions, unitary channels being a special case. And, you just call unitary channels as gates when you see them in circuits. From an information-theoretic point of view, you can see unitary channels as describing noiseless evolutions where (non-unitary) channels as describing noisy ones (the loss happening due to the appended system).