Quantum channels as a categorical completion

TL;DR

This paper establishes a categorical framework connecting pure and mixed quantum states using distributive monoidal categories, showing that quantum channels form a canonical completion of pure quantum operations, and extends this to topological settings.

Contribution

It introduces a categorical foundation for quantum channels as a completion of pure operations and generalizes it to topological contexts.

Findings

Quantum channels form a canonical completion of pure quantum operations.

The category of completely positive trace-preserving maps is a completion of vector spaces and isometries.

The operator norm topology on quantum channels is induced by the norm topology on isometries.

Abstract

We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.

Equations

$$\otimes ,\oplus ,\circ$$

$$\mathbf{Unitary}$$

$$\mathbb{Q}$$

$$\mathbb{E}:\mathbf{Isometry}\to \mathbf{CPTP}$$

$$\begin{gathered} \mathbb{E}(I\oplus I{)}^{\otimes 2}\stackrel{\text{preparation}}{\to }\mathbb{E}(I\oplus I{)}^{\otimes 5}\stackrel{\text{unitary}}{\to }\mathbb{E}(I\oplus I{)}^{\otimes 5} \\ \stackrel{\text{ measurement}}{\to }(\mathbb{E}(I)\oplus \mathbb{E}(I){)}^{\otimes 3}\otimes \mathbb{E}(I\oplus I{)}^{\otimes 2} \end{gathered}$$

$$\mathbf{Isometry}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbf{Isometry}(m,n)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\to A\otimes 0(A\otimes B)+(A\otimes C)\to A\otimes (B+C)$$

$$A=A\oplus (B\otimes N)\stackrel{A\oplus (B\otimes !)}{\to }A\oplus (B\otimes I)=A\oplus B$$

$$A\oplus B\stackrel{f\oplus g}{\to }C\oplus C\stackrel{=}{\to }(I\oplus I)\otimes C\stackrel{!\otimes C}{\to }I\otimes C\stackrel{=}{\to }C\text{.}$$

$$F(A\oplus B)\otimes FC\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$FN\otimes FA\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$I=U_i^{\ast }{U}_i=\left(\begin{array}{cc}{A}^{\ast }A+B^{\ast }B& A^{\ast }{C}_i+B^{\ast }{D}_i\\ C_i^{\ast }A+D_i^{\ast }B& C_i^{\ast }{C}_i+D_i^{\ast }{D}_i\end{array}\right)$$

$$U_1^{\ast }{U}_2$$

$$=\left(\begin{array}{cc}I& 0\\ 0& C_1^{\ast }{C}_2+D_1^{\ast }{D}_2\end{array}\right)$$

$$\mathbf{Unitary}$$

$$f(A_1,\dots ,A_k)=U(A_1\otimes {\mathrm{Id}}_{s_1}\oplus \dots \oplus A_k\otimes {\mathrm{Id}}_{s_k})U^{\ast }$$

$${\mathrm{Ad}}_U\circ {\mathrm{Ad}}_{V\otimes {\mathrm{Id}}_n}\circ \phi \circ {\Delta }_{s,n}$$

$$={\mathrm{Ad}}_U\circ ({\mathrm{Ad}}_V\otimes {\mathrm{Id}}_n)\circ (\phi \otimes {\mathrm{Id}}_n)\circ ({\Delta }_{s,1}\otimes {\mathrm{Id}}_n)$$

$$\displaystyle=\mathrm{Ad}_{U}\circ\big{(}(\mathrm{Ad}_{V}\circ\varphi\circ\Delta_{s,1})\otimes\mathrm{Id}_{n}\big{)}$$

$$\displaystyle=\mathrm{Ad}_{U}\circ\big{(}(\varphi\circ\Delta_{s,1})\otimes\mathrm{Id}_{n}\big{)}$$

$$={\mathrm{Ad}}_U\circ \phi \circ {\Delta }_{s,n}$$

$$0=M_{k,k}=(UM{U}^{\ast }{)}_{k,k}=\sum _{0\le i<n}{a}_{k,j+im}{m}_{j,j}{\bar{a}}_{k,j+im}$$

$$m_{i,j}=\sum _{0\le k<n}{a}_{i,i+km}{m}_{i,j}{\bar{a}}_{j,j+km}$$

$$1$$

$$=⟨(a_{i,i},\dots ,a_{i,i+(n-1)m}),(a_{j,j},\dots ,a_{j,j+(n-1)m})⟩$$

$$\le \text{norm}{(a_{i,i},\dots ,a_{i,i+(n-1)m})}^2.\text{norm}{(a_{j,j},\dots ,a_{j,j+(n-1)m})}^2$$

$$=1$$

$$(a_{i,i},\dots ,a_{i,i+(n-1)m})=\lambda (a_{j,j},\dots ,a_{j,j+(n-1)m})$$

$$0=M_{k,k}=\sum _{s_1{n}_1<j\le p}{a}_{k,j}{M}_{j,j}\overline{a_{k,j}}$$

$$\mathbf{Unitary}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries.

Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.

I Introduction

A popular explanation of quantum theory says that, in reality, everything is reversible (“pure quantum”), but conceptually we can hide and prepare things, and this is what leads to classical data, randomness and perceived irreversibility (“full quantum”). In this paper we explain the passage from theories of pure quantum to theories of full quantum in terms of categorical completions.

We test this passage in several ways:

• •

Starting from pure quantum with preparations (isometries), we recover quantum channels (completely positive maps between C*-algebras) as a completion with hiding — this is our main result (Thm. V.6);

• •

Starting from pure quantum (unitaries), we recover preparation of ancillas (isometries) as a completion with preparations (Thm. III.3);

• •

Also starting from pure quantum (unitaries), we recover finite non-commutative geometry (finite-dimensional C*-algebras and $*$-homomorphisms) as a different completion (Thm. IV.10);

• •

Starting from topologies on the isometries, we recover topologies on quantum channels as a completion (Thm. VI.8).

All these require slightly different kinds of completion, and in this introduction we discuss the kinds of categories and completion at hand. First we consider the pure situation (§I-A), then preparation of states (§I-B), and finally hiding of states (§I-C) and topology (§I-D). In what follows we use categorical terminology, but the casual reader may prefer the following informal picture of our main result.

admit **hidingPure quantum + preparationsQuantum channels\forall$$\exists!

Informally, the outer ellipse contains all the possible theories, including pure quantum theory with preparations. The inner circle contains the theories that admit hiding. Our main result is that of all the theories that admit hiding, quantum channels are the ‘closest’ to pure quantum with preparations. This notion of ‘closeness’ will be made precise using category theory.

In [21] we presented a similar paradigm for the restricted version of quantum channels between matrix algebras. We proved that those quantum channels are the affine completion of the category of isometries, both seen as monoidal categories. We go further here by considering all finite dimensional C*-algebras which amounts to handling classical data.

I-A Rudiments of pure / reversible computing

Before moving to categorical side, we recall some rudiments of reversible computing, which is one perspective on pure quantum theory. The basic idea is that a classical reversible operation on an $n$-level system is a bijection $n\to n$ on the natural number $n$ considered as a finite set. A quantum reversible operation is an $n\times n$ complex matrix that is unitary. But the reader unfamiliar with quantum theory can focus on the classical setting for now, because every bijection can be thought of as a unitary matrix valued in $\{0,1\}$. For example, there are two reversible classical operations on bits $2\to 2$, identity and negation, and a reversible 2-bit operation is a bijection $4\to 4$. The natural numbers form a rig (aka semiring) under addition and multiplication, and we find a simple calculus for building reversible operations by noticing that the bijections and unitaries can be composed but also they can be combined according to these rig operations. Here we write $(\oplus,N)$ and $(\otimes,I)$ instead of $+$ and $\times$ to emphasise their categorical nature.

• •

The multiplication of numbers corresponds to spatial juxtaposition of systems. For example, given two bijections on a bit, $f,g:2\to 2$, we have a bijection $(f\otimes g):2\otimes 2\to 2\otimes 2$ on $2\otimes 2\stackrel{{\scriptstyle\text{def}}}{{=}}4$ on two bits. In terms of matrices, this is the Kronecker product.

• •

The addition of numbers allows for conditional operations. Recall that most of the traditional logical operations are not reversible, however, it is possible to perform reversible controlled operations if the condition is kept. In terms of matrices, this is the block diagonal matrix. For example, the controlled-not gate is a bijection ${(\mathrm{id}\oplus\neg)}:{2\oplus 2}\to{2\oplus 2}$. Generally, given two unitaries ${f,g:n\to n}$, we can build a unitary ${(f\oplus g)}:{n\oplus n}\to{n\oplus n}$. Since $n\oplus n=2\otimes n$, we can think of $(f\oplus g):2\otimes n\to 2\otimes n$ as an operation that will either apply $f$ or $g$ to $n$ depending on the state of the first qubit, which is retained.

The unit $I=1$ represents a system with no levels. There is only one classical bijection $1\to 1$, but in quantum computation, the unitaries $1\to 1$ correspond to angles in the interval $[0,2\pi)$, known as ‘global phase’. Here the addition plays a further role, since the unitary ${(0\oplus\frac{\pi}{4})}:2\to 2$ is called the $T$-gate. The controlled gates can be used to induce quantum entanglement in the product $2\otimes 2$. For example, consider the following circuit, which is a quantum Fourier transform on three qubits. It is a graphical notation for a unitary $2\otimes 2\otimes 2\to 2\otimes 2\otimes 2$. Vertical juxtaposition is $\otimes$; horizontal juxtaposition is composition of unitaries. The first gate $H:2\to 2$ is the Hadamard unitary gate, the next is a controlled $T$ gate ${(\mathrm{id}\oplus T)}:2\otimes 2\to 2\otimes 2$; the final gate is the swap gate $2\otimes 2\to 2\otimes 2$ which amounts to the symmetry of $\otimes$.

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