A new description of orthogonal bases

Introduction

Given any orthonormal basis $\{|\phi_{i}\rangle\}_{i}$ in a finite dimensional Hilbert space $H$ we can always define the linear maps

It was observed in that the triple $(H,\delta,\epsilon)$ is a so-called commutative special $\dagger$-Frobenius comonoid in the category $\mathbf{FdHilb}$ of finite dimensional Hilbert spaces and linear maps with the tensor product as monoidal structure. Meanwhile, this fact that orthonormal bases can be encoded as commutative special $\dagger$-Frobenius monoids has resulted in many important applications in the area of categorical quantum mechanics, for example, for describing the flow of classical information in quantum informatic protocols , for defining complementarity and special quantum logic gates in quantum computational schemes and for constructing discrete models for quantum reasoning . In this paper, we establish that every commutative special $\dagger$-Frobenius monoids arises from an orthonormal basis, and that dropping the specialty condition gives an orthogonal basis.

Section 2 provides category-theoretic preliminaries.

Section 3 spells out in more detail how a commutative $\dagger$-Frobenius monoid arises from an orthogonal basis.

Section 4 describes how to extract an orthogonal basis from any commutative $\dagger$-Frobenius monoid in $\mathbf{FdHilb}$.

Section 6 spells out that within this established bijective correspondence normalisation of basis vectors means speciality of the corresponding commutative $\dagger$-Frobenius monoid. We also compare this result to an already-known result which classifies arbitrary bases on a finite-dimensional complex vector space as special Frobenius algebras.

Section 7 describes some categorical statements that come out of these results; in particular, we describe how to obtain the category of finite sets as a category of commutative $\dagger$-Frobenius monoids in $\mathbf{FdHilb}$.

Preliminaries

The research area of categorical quantum mechanics emerged from the observation that the subtle details of important, experimentally-established quantum informatic protocols can already be specified at an abstract category-theoretic level . The background structure is that of a symmetric monoidal $\dagger$-category , a symmetric monoidal category together with a identity-on-objects involutive endofunctor which coherently preserves the symmetric monoidal structure. Within this context one then aims to maximise the

ratio. Additional structure on which we rely in this paper is that of internal Frobenius algebras , more specifically, internal commutative $\dagger$-Frobenius monoids . Relative to the quantum universe which is modelled by the symmetric monoidal $\dagger$-category these commutative $\dagger$-Frobenius monoids model the classical interfaces, and enable us to specify projector spectra, measurements, and classical data flows .

A Frobenius monoid in a symmetric monoidal category is a quintuple $(X,m,u,\delta,\epsilon)$ consisting of an internal monoid

which together satisfy the Frobenius condition

A (special) (commutative) $\dagger$-Frobenius monoid in a symmetric monoidal $\dagger$-category is a triple $(X,m,u)$ such that $(X,m,u,\delta=m^{\dagger},\epsilon=u^{\dagger})$ is a (special) (commutative) Frobenius algebra.

We will also use $\dagger$-Frobenius comonoid to refer to a $\dagger$-Frobenius monoid, depending on whether we want the emphasis to lie either on the monoid or the comonoid structure.

Recall that in a symmetric monoidal category a morphism $f:X\to Y$ is a monoid homomorphism for monoids $(X,m,u)$ and $(Y,m^{\prime},u^{\prime})$ if

and that it is a comonoid homomorphism for comonoids $(X,\delta,\epsilon)$ and $(Y,\delta^{\prime},\epsilon^{\prime})$ if

A copyable element of a $\dagger$-Frobenius monoid $(X,m_{X},u_{X})$ is a comonoid homomorphism $\alpha:I\to X$.

Turning an orthogonal basis into a commutative †\dagger-Frobenius monoid

Given an orthogonal basis $\{|\phi_{i}\rangle\}_{i}$, the maps defined in (1) and (2) are the linear extensions of copying and uniformly deleting these basis vectors.

It is easily seen that from $\delta$ alone we can recover the basis by solving

No other vectors besides those in $\{|\phi_{i}\rangle\}_{i}$ will satisfy this equation since for

with at least two non-zero scalars in $\{\langle\phi_{i}|\psi\rangle\}_{i}$ we have that

will always be entangled, i.e. cannot be written in the form $|\psi_{a}\rangle\otimes|\psi_{b}\rangle$ for some $|\psi_{a}\rangle$ and $|\psi_{b}\rangle$, and hence not equal to $|\psi\rangle\otimes|\psi\rangle$. So $\delta$ and hence also the triple $(H,\delta,\epsilon)$ faithfully encodes $\{|\phi_{i}\rangle\}_{i}$.

To see that $\delta^{\dagger}$ and $\delta$ obey the Frobenius condition it suffices to note that

Turning a commutative †\dagger-Frobenius monoid into an orthogonal basis

We will freely switch between denoting elements of $H$ as linear maps

and as kets $|\alpha\rangle=\alpha(1)\in H$. Taking the adjoint of $\alpha$ gives us

and hence $\langle\alpha|=\alpha^{\dagger}\in H^{*}$.

Let $(H,m,u)$ be a commutative $\dagger$-Frobenius monoid. Given such a commutative $\dagger$-Frobenius monoid any element $\alpha\in H$ induces a linear map

its its right action. We draw this right action in the following way:

The diagram is read from bottom to top. This is a direct representation of our definition of $R_{\alpha}$ as right-multiplication by the element $\alpha$: vertical lines represent the vector space $H$, the dot represents the element $\alpha$, and the merging of the two lines represents the multiplication operation $m$. Since $H$ is a Hilbert space, $R_{\alpha}$ has an adjoint

We draw the adjoint $R_{\alpha}{}^{\dagger}$ by flipping the diagram on a horizonal axis, but keeping the arrows pointing in their original direction:

We draw the Frobenius law of definition 2.1 in the following way:

Representing the unit $u:I\to X$ as a small horizontal bar we draw the unit law in the following way:

We can now use the unit law and the Frobenius law to redraw the graphical representation of $R_{\alpha}{}^{\dagger}$ in the following way:

So the adjoint of $R_{\alpha}$ is indeed itself a right action of $\alpha^{\prime}$, as defined above. ∎

Let ${\bf C}$ be a symmetric monoidal $\dagger$-category and $(X,m,u)$ a commutative $\dagger$-Frobenius monoid in ${\bf C}$. The right action mapping

is an involution preserving monoid embedding, when endowing ${\bf C}({\rm I},X)$ with the monoid structure of the internal monoid $(X,m,u)$. In the case that ${\bf C}=\mathbf{FdHilb}$ then this mapping also preserves the vector space structure.

Using the Frobenius and unit identities and the fact that the $\dagger$-functor is an involution we first show that $(-)^{\prime}$ as in Lemma 4.1 is involutive:

The adjoint is an involution on ${\bf C}(X,X)$ so since $R_{\alpha^{\prime}}=R_{\alpha}{}^{\dagger}$ involution is indeed preserved. The mapping is moreover injective, since by the unit equation we have $R_{\alpha}\circ u=\alpha$. It is straightforward to show that multiplication and unit are preserved. Preservation of the vector space structure in the case that ${\bf C}=\mathbf{FdHilb}$ follows by linearity of $m$. ∎

Any $\dagger$-Frobenius monoid in $\mathbf{FdHilb}$ is a C*-algebra.

The endomorphism monoid $\mathbf{FdHilb}(H,H)$ is a C*-algebra. By the above Lemma we know that

inherits algebra structure from $\mathbf{FdHilb}(H,H)$. Now, since any finite-dimensional involution-closed subalgebra of a C*-algebra is also a C*-algebra, it follows that any $\dagger$-Frobenius monoid in $\mathbf{FdHilb}$ is a C*-algebra, in particular, it can be given a C*-algebra norm. ∎

Note that in the above we did not assume the $\dagger$-Frobenius monoid to be commutative. More on this is in .

The copyable elements for any commutative $\dagger$-Frobenius monoid on $H$ in $\mathbf{FdHilb}$ form a basis for $H$.

To make use of the spectral theorem, we also need to show that these homomorphisms are involution-preserving. In fact, this is automatic: if a map between two finite-dimensional commutative C*-algebras preserves the algebra multiplication and unit, then it necessarily preserves the involution. ∎

It remains to be shown that this basis is orthogonal.

If $\phi_{i},\phi_{j}:I\to X$ are comonoid homomorphisms for a commutative $\dagger$-Frobenius comonoid and if $\langle\phi_{i}|\phi_{j}\rangle:=\phi_{i}^{\dagger}\circ\phi_{j}$, $\langle\phi_{i}|\phi_{i}\rangle$ and $\langle\phi_{j}|\phi_{j}\rangle$ are all cancellable scalars then they must be real and equal.

Given that $\phi_{i}$ and $\phi_{j}$ are comonoid homomorphisms, and making use of one of the Frobenius identities, we can derive the following equation:

Switching the roles of $\phi_{i}$ and $\phi_{j}$ we can obtain another similar equation, and writing both equations in bra-ket notation we obtain

If we cancel $\langle\phi_{i}|\phi_{i}\rangle$, $\langle\phi_{j}|\phi_{j}\rangle$ and $\langle\phi_{i}|\phi_{j}\rangle$ we obtain

It follows that these inner products are real, and that they are all equal:

The copyable elements for any commutative $\dagger$-Frobenius monoid on $H$ in $\mathbf{FdHilb}$ form an orthogonal basis for $H$.

we have $|\phi_{i}\rangle-|\phi_{j}\rangle=0$, so $|\phi_{i}\rangle$ and $|\phi_{j}\rangle$ are the same. This contradicts our assumption that ${\phi_{i}}$ and ${\phi_{j}}$ were different, and so the basis $|\phi_{i}\rangle$ is orthogonal. ∎

Statement of the main result

Every commutative $\dagger$-Frobenius monoid in $\mathbf{FdHilb}$ determines an orthogonal basis, consisting of its copyable elements, and every orthogonal basis determines a commutative $\dagger$-Frobenius monoid in $\mathbf{FdHilb}$ via prescriptions (1) and (2). These constructions are inverse to each other.

It only remains to be explained why the construction described in section 4, which obtains an orthogonal basis from a commutative $\dagger$-Frobenius algebra, is inverse to the construction described in section 3, which obtains a commutative $\dagger$-Frobenius algebra from an orthogonal basis.

Assume we begin with an orthogonal basis. We construct the $\dagger$-Frobenius monoid as the unique one with a comultiplication which perfectly copies our original basis, and we construct a new orthogonal basis consisting of those elements which are perfectly copied by the comultiplication. This new basis must be at least as large as our original basis. However, since any two bases for a finite-dimensional Hilbert space must have the same number of elements, the new basis is actually the same as the original basis.

Now assume we begin with a commutative $\dagger$-Frobenius monoid. We construct our orthogonal basis as those elements which are perfectly copied, and construct our new monoid as the unique $\dagger$-Frobenius monoid which perfectly copies this basis. However, since the original monoid also perfectly copies this basis, by uniqueness the two monoids are the same.

In other papers on this subject abstract basis vectors are required to be self-conjugate comonoid homomorphisms, where the conjugate of a morphism $f:X\to Y$ relative to $\dagger$-Frobenius monoids $(X,m_{X},u_{X})$ and $(Y,m_{Y},u_{Y})$ is defined to be

with $\eta_{Z}=m_{Z}{}^{\dagger}\circ u_{Z}:{\rm I}\to Z\otimes Z$. In $\mathbf{FdHilb}$ a morphism is self-conjugate if in its matrix representation in the corresponding bases all entries are self-conjugate, a fact which follows in $\mathbf{FdHilb}$ automatically from the fact of being a comonoid homomorphism. In other categories the additional constraint of the basis vectors being self-conjugate guarantees that they are involution preserving – cf. the last part of the proof of corollary 4.5.

Other types of basis

and it is clear that this is the identity if and only if the vectors $|\phi_{i}\rangle$ are normalised.

Interestingly, it is already known that, for a finite-dimensional complex vector space, a basis exactly corresponds to a choice of special commutative Frobenius algebra Thanks to John Baez for pointing this out.. Of course, it does not make sense to ask whether such a Frobenius algebra is $\dagger$-Frobenius, or whether the elements of such a basis are normalised or orthogonal. This result follows from the fact that a special commutative Frobenius algebra is necessarily strongly separable, and since the ground field is of characteristic 0, it is therefore necessarily finite-dimensional and semisimple . Such an algebra is canonically isomorphic to a finite cartesian product of the complex numbers, up to permutation, and the basis elements are given by the number $1$ in each of the complex factors.

In summary, on a finite-dimensional complex Hilbert space, we can describe different types of basis very precisely with the following structures:

The inner product on the Hilbert space does not play a role for the case of the arbitrary basis. In every case, the basis is recovered from the Frobenius algebra as those vectors which are perfectly copied by the comultiplication, and the Frobenius algebra is recovered from the basis as the unique Frobenius algebra of the correct type with a comultiplication that perfectly copies the basis.

It is interesting to consider whether an arbitrary commutative Frobenius algebra on a complex vector space might also correspond to some type of basis structure. In fact, it does not, and such Frobenius algebras can be very wild indeed. It is perhaps surprising that the specialness axiom and the $\dagger$-Frobenius axiom can both serve independently to tame this wildness.

Categorical statements

We have shown that a commutative $\dagger$-Frobenius monoid in $\mathbf{FdHilb}$ corresponds to a basis of a finite-dimensional Hilbert space. Any such basis is determined up to unitary isomorphism by the norms of the basis elements, which constitute a list of positive real numbers.

If a homomorphism between two commutative $\dagger$-Frobenius monoids preserves all of the structure — the multiplication, unit, comultiplication and counit — then it is necessarily an isomorphism, and in $\mathbf{FdHilb}$, it is necessarily unitary. Such a homomorphism will map one basis onto another, taking basis elements onto basis elements of the same norm. This leads to the following result:

The category of commutative $\dagger$-Frobenius monoids in $\mathbf{FdHilb}$, with morphisms preserving all of the Frobenius structure, is equivalent to the groupoid of ‘finite lists of real numbers and isomorphisms that preserve the numbers’, which has objects given by finite sets equipped with functions into the positive real numbers, and morphisms given by isomorphisms of sets that preserve the functions into the real numbers.

This is interesting from the perspective of unitary 2-dimensional topological quantum field theory , since such things are given by commutative $\dagger$-Frobenius monoids in $\mathbf{FdHilb}$, and the natural notion of homomorphism is one that preserves all of the Frobenius structure.

If we only require that our homomorphisms preserve the comultiplication and counit then this gives arbitrary functions between bases, without the requirement of preserving the length of the basis vector. This follows from the spectral theorem for commutative C*-algebras; a comonoid homomorphism gives rise to an oppositely-directed monoid homomorphism by taking the adjoint, and any involution-preserving monoid homomorphism is equivalent to an oppositely-directed continuous function between the spectra of the C*-algebras.

The category of commutative $\dagger$-Frobenius monoids in $\mathbf{FdHilb}$, with morphisms preserving comultiplication and counit, is equivalent to the category $\mathbf{FinSet}$ of finite sets and functions.

This gives an interesting new perspective on the relationship between $\mathbf{FdHilb}$ and $\mathbf{FinSet}$. There is an obvious functor $F:\mathbf{FinSet}\to\mathbf{FdHilb}$ which takes a set with $n$ elements to a Hilbert space of dimension $n$ and chosen basis, and takes a function between sets to the induced linear map. This can be thought of as a ‘free’ functor, as it generates the free Hilbert space on a finite set. But using the equivalence between $\mathbf{FinSet}$ and the category of commutative $\dagger$-Frobenius monoids described in lemma 7.2, we see that this functor $F$ is equivalent to the forgetful functor which regards each finite set as a Hilbert space equipped with a commutative $\dagger$-Frobenius monoid structure, and forgets the monoid structure. So a finite set can be considered as a finite-dimensional Hilbert space with the extra structure of a commutative $\dagger$-Frobenius monoid, or a finite-dimensional Hilbert space can be considered as a finite set with the extra structure of a vector space and inner product.

References