A graphical language for quantum protocols based on the category of cobordisms

Abstract

As shown by Abramsky and Coecke, quantum mechanics can be studied in terms of dagger compact closed categories with biproducts. Within this structure, many well-known quantum protocols can be described and their validity can be shown by establishing the commutativity of certain diagrams in that category. In this paper, we propose an explicit realization of a category with enough structure to check the validity of a certain class of quantum protocols. To do this, we construct a category based on one-dimensional cobordisms with attached elements of a certain group freely generated by a finite set. We use this category as a graphical language, and we show that it is dagger compact closed with biproducts. Then relying on the coherence result for compact closed categories, proved by Kelly and Laplaza, we show the coherence result, which enables us to check the validity of quantum protocols just by drawing diagrams. In particular, we show the validity of quantum teleportation, entanglement swap** (as formulated in the work of Abramsky and Coecke) and superdense coding protocol.

Appendices

Appendix

The language and the equations for dagger compact closed categories with dagger biproducts

Our choice of a language for dagger compact closed categories with dagger biproducts is the one in which enrichment over $\textbf{Cmd}$ is primitive and not derived from the biproduct structure. Such a language is siutable for the proofs of our results. A dagger compact closed category with dagger biproducts $\mathcal {A}$ consists of a set of objects and a set of arrows. There are two functions (source and target) from the set of arrows to the set of objects of $\mathcal {A}$. For every object a of $\mathcal {A}$ there is the identity arrow ${\textbf {1}}_a:a\rightarrow a$. The set of objects includes two distinguished objects I and 0. Arrows $f:a\rightarrow b$ and $g:b\rightarrow c$ compose to give $g\circ f:a\rightarrow c$, and arrows $f_1,f_2:a\rightarrow b$ add to give $f_1+f_2:a\rightarrow b$. For every object a of $\mathcal {A}$, there is the object $a^*$, and for every pair of objects a and b of $\mathcal {A}$, there are the objects $a\otimes b$ and $a\oplus b$. Also, for every arrow $f:a\rightarrow b$, there is the arrow $f^\dagger :b\rightarrow a$, and for every pair of arrows $f:a\rightarrow a'$ and $g:b\rightarrow b'$ there are the arrows $f\otimes g:a\otimes b\rightarrow a'\otimes b'$ and $f\oplus g:a\oplus b\rightarrow a'\oplus b'$. In $\mathcal {A}$ we have the following families of arrows indexed by its objects.

$$\begin{aligned}&\alpha _{a,b,c} :a\otimes (b\otimes c)\rightarrow (a\otimes b)\otimes c, \quad{} & {} \alpha ^{-1}_{a,b,c}:(a\otimes b)\otimes c\rightarrow a\otimes (b\otimes c), \\&\lambda _a :I\otimes a\rightarrow a, \quad{} & {} \lambda ^{-1}_a:a\rightarrow I\otimes a, \\&\sigma _{a,b}:a\otimes b\rightarrow b\otimes a, \\&\eta _a:I\rightarrow a^*\otimes a, \quad{} & {} \varepsilon _a:a\otimes a^*\rightarrow I \\&\pi ^1_{a,b} :a\oplus b\rightarrow a,\quad{} & {} \iota ^1_{a,b}:a\rightarrow a\oplus b, \\&\pi ^2_{a,b}:a\oplus b\rightarrow b,\quad{} & {} \iota ^2_{a,b}:b\rightarrow a\oplus b, \\&0_{a,b} :a\rightarrow b. \end{aligned}$$

The arrows of $\mathcal {A}$ should satisfy the following equalities:

$$\begin{aligned}{} & {} f\circ {\textbf {1}}_a=f={\textbf {1}}_{a'}\circ f,\quad (h\circ g)\circ f=h\circ (g\circ f), \end{aligned}$$

(A.1)

$$\begin{aligned}{} & {} \quad {\textbf {1}}_a\otimes {\textbf {1}}_b={\textbf {1}}_{a\otimes b},\quad (f_2\otimes g_2)\circ (f_1\otimes g_1)= (f_2\circ f_1)\otimes (g_2\circ g_1), \end{aligned}$$

(A.2)

$$\begin{aligned}{} & {} \quad \begin{array}{c}((f\otimes g)\otimes h)\circ \alpha _{a,b,c}= \alpha _{a',b',c'}\circ (f\otimes (g\otimes h)),\\ \alpha ^{-1}_{a,b,c}\circ \alpha _{a,b,c}={\textbf {1}}_{a\otimes (b\otimes c)},\quad \alpha _{a,b,c}\circ \alpha ^{-1}_{a,b,c}={\textbf {1}}_{(a\otimes b)\otimes c}, \end{array} \end{aligned}$$

(A.3)

$$\begin{aligned}{} & {} \quad f\circ \lambda _a=\lambda _{a'}\circ (I\otimes f),\quad \lambda ^{-1}_a\circ \lambda _a={\textbf {1}}_{I\otimes a},\quad \lambda _a\circ \lambda ^{-1}_a={\textbf {1}}_a, \end{aligned}$$

(A.4)

$$\begin{aligned}{} & {} \quad (g\otimes f)\circ \sigma _{a,b}=\sigma _{a',b'}\circ (f\otimes g),\quad \sigma _{b,a}\circ \sigma _{a,b}={\textbf {1}}_{a\otimes b}, \end{aligned}$$

(A.5)

$$\begin{aligned}{} & {} \quad \alpha _{a\otimes b,c,d}\circ \alpha _{a,b,c\otimes d}=(\alpha _{a,b,c}\otimes d)\circ \alpha _{a,b\otimes c,d}\circ (a\otimes \alpha _{b,c,d}), \end{aligned}$$

(A.6)

$$\begin{aligned}{} & {} \quad \lambda _{a\otimes b}=(\lambda _a\otimes b)\circ \alpha _{I,a,b}, \end{aligned}$$

(A.7)

$$\begin{aligned}{} & {} \quad \alpha _{c,a,b}\circ \sigma _{a\otimes b,c}\circ \alpha _{a,b,c}= (\sigma _{a,c}\otimes b)\circ \alpha _{a,c,b} \circ (a\otimes \sigma _{b,c}), \end{aligned}$$

(A.8)

$$\begin{aligned}{} & {} \quad (a^*\otimes \varepsilon )\circ \alpha ^{-1}_{a^*,a,a^*}\circ (\eta \otimes a^*)=\sigma _{I,a^*},\quad (\varepsilon \otimes a)\circ \alpha _{a,a^*,a}\circ (a\otimes \eta )=\sigma _{a,I}, \end{aligned}$$

(A.9)

$$\begin{aligned}{} & {} \quad f_1+(f_2+f_3)=(f_1+f_2)+f_3,\quad f_1+f_2=f_2+f_1,\quad f+0_{a,a'}=f, \end{aligned}$$

(A.10)

$$\begin{aligned}{} & {} \quad (g_1+g_2)\circ f=g_1\circ f + g_2\circ f,\quad g\circ (f_1+f_2)=g\circ f_1 + g\circ f_2, \end{aligned}$$

(A.11)

$$\begin{aligned}{} & {} \quad 0_{a',b}\circ f=0_{a,b},\quad f\circ 0_{b,a}=0_{b,a'}, \end{aligned}$$

(A.12)

$$\begin{aligned}{} & {} \quad {\textbf {1}}_a\oplus {\textbf {1}}_b={\textbf {1}}_{a\oplus b},\quad (f_2\oplus g_2)\circ (f_1\oplus g_1)= (f_2\circ f_1)\oplus (g_2\circ g_1), \end{aligned}$$

(A.13)

$$\begin{aligned}{} & {} \quad (f\oplus g)\circ \iota ^1_{a,b}=\iota ^1_{a',b'}\circ f,\quad (f\oplus g)\circ \iota ^2_{a,b}=\iota ^2_{a',b'}\circ g, \end{aligned}$$

(A.14)

$$\begin{aligned}{} & {} \quad f\circ \pi ^1_{a,b}=\pi ^1_{a',b'}\circ (f\oplus g),\quad g\circ \pi ^2_{a,b}=\pi ^2_{a',b'}\circ (f\oplus g), \end{aligned}$$

(A.15)

$$\begin{aligned}{} & {} \quad \pi ^1_{a,b}\circ \iota ^1_{a,b}={\textbf {1}}_a,\quad \pi ^2_{a,b}\circ \iota ^2_{a,b}={\textbf {1}}_b, \end{aligned}$$

(A.16)

$$\begin{aligned}{} & {} \quad \pi ^2_{a,b}\circ \iota ^1_{a,b}=0_{a,b},\quad \pi ^1_{a,b}\circ \iota ^2_{a,b}=0_{b,a}, \end{aligned}$$

(A.17)

$$\begin{aligned}{} & {} \quad \iota ^1_{a,b}\circ \pi ^1_{a,b} + \iota ^2_{a,b}\circ \pi ^2_{a,b}={\textbf {1}}_{a\oplus b}. \end{aligned}$$

(A.18)

$$\begin{aligned}{} & {} \quad 0_{0,0}={\textbf {1}}_0, \end{aligned}$$

(A.19)

$$\begin{aligned}{} & {} \quad {\textbf {1}}_a^\dagger ={\textbf {1}}_a,\quad (g\circ f)^\dagger = f^\dagger \circ g^\dagger ,\quad f^{\dagger \dagger }=f, \end{aligned}$$

(A.20)

$$\begin{aligned}{} & {} \quad (f\otimes g)^\dagger =f^\dagger \otimes g^\dagger , \end{aligned}$$

(A.21)

$$\begin{aligned}{} & {} \quad \alpha _{a,b,c}^\dagger =\alpha ^{-1}_{a,b,c},\quad \lambda _a^\dagger =\lambda ^{-1}_a,\quad \sigma _{a,b}^\dagger =\sigma _{b,a}, \end{aligned}$$

(A.22)

$$\begin{aligned}{} & {} \quad \varepsilon ^\dagger =\sigma _{a^*,a}\circ \eta , \end{aligned}$$

(A.23)

$$\begin{aligned}{} & {} \quad (\pi ^1_{a,b})^\dagger =\iota ^1_{a,b},\quad (\pi ^2_{a,b})^\dagger =\iota ^2_{a,b}. \end{aligned}$$

(A.24)

The following equalities are derivable from A.1–A.24:

$$\begin{aligned}{} & {} (f\oplus g)^\dagger = f^\dagger \oplus g^\dagger , \end{aligned}$$

(A.25)

$$\begin{aligned}{} & {} \quad (f+ g)^\dagger = f^\dagger + g^\dagger ,\quad 0_{a,b}^\dagger =0_{b,a} \end{aligned}$$

(A.26)

$$\begin{aligned}{} & {} \quad f\otimes (g_1+g_2)=(f\otimes g_1)+(f\otimes g_2),\quad (f_1+f_2)\otimes g=(f_1\otimes g)+(f_2\otimes g), \end{aligned}$$

(A.27)

$$\begin{aligned}{} & {} \quad f\otimes 0_{b,b'}=0_{a\otimes b,a'\otimes b'}=0_{a,a'}\otimes g. \end{aligned}$$

(A.28)

Scalars and probability amplitudes

As firmly laid, quantum mechanics is based on complex vector spaces (Hilbert spaces, to be more precise). Implied in this structure is the notion of scalars, that correspond here to the field of complex numbers. In categorical language, one can define scalars more abstractly [1, 20]. A scalar is a morphism $s:\hspace{1mm}I\rightarrow I$. It can be proved that the hom-set $\textrm{Hom}\,(I,I)$, for a compact closed category, is a commutative monoid, therefore justifying further this structure’s name.

In $1\textbf{Cob}$, the scalars correspond to closed, one-dimensional manifolds, and the only candidate for such a structure is a finite collection of circles $S^1$ (as denoted on the left-hand side of the following picture). In $1\textbf{Cob}_\mathfrak {G}$, we have $\mathfrak {G}$-circles; topological circles dressed with group elements (right-hand side of the following picture). Due to the compact closed structure of this category, there is a natural interpretation of those circles. Namely, any compact closed category can be lifted to a traced category by a suitable definition of a categorical trace (see Sect. 8.3 for the definition).

That closed loops should be connected with traces is not limited to a categorical approach to quantum mechanics. Even when considering Feynman diagrams in quantum electrodynamics, fermions loops are accompanied by a trace in spinorial indices. Moreover, in TQFT, we are customed to the fact that closing manifold by gluing the outward future to inward past (if possible), results in a trace, that for a cylinder, i.e. the identity, simply gives the dimension of the respective Hilbert space.

Furthermore, as explained in [7], these traces correspond to the probability weights of different branches. This is further confirmed by a Hilbert-space picture computations. Recall that one reason we have scalars (different from the multiplicative unit) is normalization on states. In order to get the probabilistic interpretation, according to the Born rule, we must insist on normalized states. For a state $\beta _{00}| {0} \rangle \otimes | {0} \rangle +\beta _{01}| {0} \rangle \otimes | {1} \rangle +\beta _{10}| {1} \rangle \otimes | {0} \rangle +\beta _{11}| {1} \rangle \otimes | {1} \rangle$, we have that its norm squared is given by $|\beta _{00}|^2+|\beta _{01}|^2+|\beta _{10}|^2+|\beta _{11}|^2=\textrm{Tr}(\beta ^\dagger \beta )$, where $\beta$ is a $2\times 2$ matrix whose components are $\beta _{ij}$ constants. Therefore, we conclude that traces are as important in this set-up as in traditional Hilbert-space formulation.

When dealing with quantum protocols, one usually takes $\beta$ to be proportional to Pauli sigma matrices. (Extended) Pauli matrices are defined as

$$\begin{aligned} \sigma _0= \begin{pmatrix} 1&{}0\\ 0&{}1 \end{pmatrix}, \hspace{3mm} \sigma _1= \begin{pmatrix} 0&{}1\\ 1&{}0 \end{pmatrix},\hspace{3mm} \sigma _2= \begin{pmatrix} 0&{}-i\\ i&{}0 \end{pmatrix},\hspace{3mm} \sigma _3 =\begin{pmatrix} 1&{}0\\ 0&{}-1 \end{pmatrix}. \end{aligned}$$

We see that those matrices are unitary, self-adjoint and satisfy $\textrm{Tr}(\sigma _i\sigma _j)=2\delta _{ij}$, where $\delta _{ij}$ is a Kronecker delta symbol (equal to one if $i=j$ and zero otherwise). In order to make the connection with the Bell basis, introduced in Sect. 1, we take $\beta _1=\sigma _0$, $\beta _2=\sigma _1$, $\beta _3=\sigma _3$ and $\beta _4=-i\sigma _2$. This implies that we have $\textrm{Tr}(\beta _i\beta _j^\dagger )=2\delta _{ij}$, with the usual definition of matrix adjoint.

However, in order to check whether two diagrams commute, it is usually straightforward to include scalars into consideration. One can then just neglect this issue of scalars and work without explicitly using them (as done previously). They are, of course, needed if one is to obtain probabilities for different outcomes of a measurement, but in our work (and related work of [1, 7]) this is not a primary task.