Power Series & Quantum States

What is a formal power series? What has it to do with quantum states?

There is a vast literature about formal power series and their relation to automata (see  [1],  [2],  [3]), here I want to stress their similarities with ordinary vectors by abusing the ket notation¹.

So the plan is:

1. introduce the (free) vector space and the bra-ket notation
2. rework the vector space definition and discover that it is natural to look at power series as vectors whose components are indexed by words
3. the result is a richer algebraic structure equipped with a quotient operator that can be used to progressively disassemble a power series
4. look at a couple of examples and see how some sort of entanglement emerges

let's get started.

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(1) Free Vector Spaces

The free vector space on a set \(\Sigma\) is the space of functions

\[\mathbb{C}^\Sigma \equiv \{ \Sigma \to \mathbb{C} \}\]

any vector of \(\mathbb{C}^\Sigma\) will be denoted by a ket

\[\ket{\psi}: \Sigma \to \mathbb{C}\] \[\sigma \mapsto \psi_\sigma\]

it is a fancy way to encode the components of a vector, the advantage of writing vectors as functions is that they naturally inherit \(\mathbb{C}\text{'s}\) sum and product

\[(\ket{\psi} + \ket{\phi})(\sigma) \equiv \ket{\psi}(\sigma) + \ket{\phi}(\sigma) \hspace{1cm} (\alpha\ket{\psi})(\sigma) \equiv \alpha \ket{\psi}(\sigma)\]

Now introduce the computational basis \(\{ \ket{\sigma} \}_{\sigma \in \Sigma}\)

\[\ket{\sigma}: \Sigma \to \mathbb{C}\] \[\ket{\sigma}(\sigma') \equiv \delta_{\sigma,\sigma'}\]

any ket \(\ket{\psi} \in \mathbb{C}^\Sigma\) can be written as

\begin{align*} \ket{\psi}(\sigma') &= \sum_{\sigma \in \Sigma} \ket{\psi}(\sigma) \, \delta_{\sigma,\sigma'} \\ &= \sum_{\sigma \in \Sigma} \psi_\sigma \, \ket{\sigma}(\sigma') \\ \\ \ket{\psi} &= \sum_{\sigma \in \Sigma} \psi_\sigma \ket{\sigma} \\ \end{align*}

then \(\{ \ket{\sigma} \}_{\sigma \in \Sigma}\) spans the whole \(\mathbb{C}^\Sigma\) and if we order \((\sigma_1, \dots, \sigma_d)\) each \(\ket{\sigma_i}\) can be identified with a basis vector of \(\mathbb{C}^d\) where the \(i\text{-th}\) component is 1

\[\ket{i} = \ket{\sigma_i} = \begin{bmatrix} 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{bmatrix}\]

this defines an isomorphism \(\mathbb{C}^\Sigma \cong \mathbb{C}^d\) and any ket \(\ket{\psi}\) can be written as a column vector

\[\ket{\psi} = \sum_{i=1}^d \psi_i \ket{i} = \begin{bmatrix} \psi_1 \\ \vdots \\ \psi_d \end{bmatrix}\]

this is the form we are most familiar to. The inner product is defined as usual as

\[\langle \psi, \phi \rangle = \sum_{\sigma \in \Sigma} \psi^\star_\sigma \phi_\sigma\]

Finally introduce the bra as the dual

\[\bra{\psi}: \mathbb{C}^\Sigma \to \mathbb{C}\] \[\bra{\psi}\left( \ket{\phi} \right) \equiv \braket{\psi|\phi} = \langle \psi, \phi \rangle\]

then we can write the bra as a row vector

\[\bra{\psi} = \sum_{i=1}^d \psi^\star_i \bra{i} = \begin{bmatrix} \psi^\star_1 \dots \psi^\star_d \end{bmatrix}\]

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(2) Formal Power Series

If we look at the definition (see  [1] Ch.1.1 p.379) the formal power series over the free monoid \(\Sigma^\star\) with coefficients in \(\mathbb{C}\) form the space of functions

\[\mathbb{C}^{\Sigma^\star} \equiv \{ \Sigma^\star \to \mathbb{C} \}\]

you see, it's not very different from the definition above, let's repeat exactly what we have done.

We again denote power series by kets

\[\ket{\psi}: \Sigma^\star \to \mathbb{C}\] \[w \mapsto \psi_w\]

now the "components" \(\psi_w\) are indexed by words and as before they naturally inherit sum and product from \(\mathbb{C}\)

sum

take any \(\ket{\psi}, \ket{\phi} \in \mathbb{C}^{\Sigma^\star}\)

\[(\ket{\psi} + \ket{\phi}): \Sigma^\star \to \mathbb{C}\] \[(\ket{\psi} + \ket{\phi})(w) \equiv \ket{\psi}(w) + \ket{\phi}(w)\]

the sum is associative, symmetric and has the \(\ket{\emptyset} \equiv 0\) as zero element

\[\ket{\emptyset} + \ket{\psi} = \ket{\psi} + \ket{\emptyset} = \ket{\psi}\]

exterior product

take any \(\alpha \in \mathbb{C}, \ket{\psi} \in \mathbb{C}^{\Sigma^\star}\)

\[(\alpha \ket{\psi}): \Sigma^\star \to \mathbb{C}\] \[(\alpha \ket{\psi})(w) \equiv \alpha \ket{\psi}(w)\]

Now introduce the computational basis \(\{ \ket{w} \}_{w \in \Sigma^\star}\)

\[\ket{w}: \Sigma^\star \to \mathbb{C}\] \[\ket{w}(w') \equiv \delta_{w, w'}\]

then we can write

\begin{align*} \ket{\psi}(w') &= \sum_{w \in \Sigma^\star} \ket{\psi}(w) \, \delta_{w, w'} \\ &= \sum_{w \in \Sigma^\star} \ket{\psi}(w) \, \ket{w}(w') \\ &= \sum_{w \in \Sigma^\star} \psi_w \, \ket{w}(w') \\ \\ \ket{\psi} &= \sum_{w \in \Sigma^\star} \psi_w \ket{w} \\ \end{align*}

exactly as before (but we avoid introducing the column vector notation that would be too cumbersome).

The inner product is²

\[\langle \psi, \phi \rangle = \sum_{w \in \Sigma^\star} \psi^\star_w \phi_w\]

and the bra is defined as

\[\bra{\psi} = \sum_{w \in \Sigma^\star} \psi^\star_w \bra{w}\]

Power series form an infinite dimensional inner product space, let's denote it by \(\mathcal{H}^\Sigma\)³.

Now we turn to features that are unique to power series.

First note that \(\mathbb{C}\) is included into \(\mathcal{H}^\Sigma\): any \(\alpha \in \mathbb{C}\) is mapped to the ket \(\alpha \ket{\epsilon}\) where \(\epsilon\) is the empty word.

Next since \(\Sigma^\star\) is a free monoid, we can lift the string concatenation to kets, for example

• take just two words \(ab\) and \(bc\) then we can define naturally

  \[\ket{ab} \otimes \ket{bc} \equiv \ket{abbc}\]

• take two slightly more complicated power series

  \[\ket{\psi} = \ket{a} + 2\ket{ab} \hspace{1cm} \ket{\phi} = \ket{ba} + 2\ket{c}\]

  then we can define their concatenation by

  \begin{align*} \ket{\psi} \otimes \ket{\phi} &= (\ket{a} + 2\ket{ab}) \otimes (\ket{ba} + 2\ket{c}) \\ &= \ket{aba} + 2\ket{abba} + 2\ket{ac} + 4\ket{abc} \end{align*}

this gives the

Cauchy product

take any \(\ket{\psi}, \ket{\phi} \in \mathcal{H}^\Sigma\) their Cauchy product is defined as

\[\ket{\psi} \otimes \ket{\phi} \equiv \sum_{w \in \Sigma^\star} \left( \sum_{w_1 w_2 = w} \psi_{w_1} \phi_{w_2} \right) \ket{w}\]

it is associative, non-symmetric and has the empty ket \(\ket{\epsilon}\) as unit

\[\ket{\epsilon} \otimes \ket{\psi} = \ket{\psi} \otimes \ket{\epsilon} = \ket{\psi}\]

moreover it (left and right) distributes over the sum

\[\ket{\chi} \otimes (\ket{\psi} + \ket{\phi}) = \ket{\chi} \otimes \ket{\psi} + \ket{\chi} \otimes \ket{\phi}\] \[(\ket{\psi} + \ket{\phi}) \otimes \ket{\chi} = \ket{\psi} \otimes \ket{\chi} + \ket{\phi} \otimes \ket{\chi}\]

also the zero kills

\[\ket{\emptyset} \otimes \ket{\psi} = \ket{\psi} \otimes \ket{\emptyset} = \ket{\emptyset}\]

with the addition of the Cauchy product \(\mathcal{H}^\Sigma\) becomes a semiring.

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(3) The Quotient

Looking at the literature on formal power series let's review the concept of quotient (see  [1] Ch.4.1, p.438)

the quotient

consider a ket \(\ket{\psi} = \sum_{w \in \Sigma^\star} \psi_w \ket{w}\) and a word \(m \in \Sigma^\star\), define \(\partial_m\) as

\[\partial_m: \mathcal{H}^\Sigma \to \mathcal{H}^\Sigma\] \[\partial_m \ket{\psi} \equiv \sum_{w \in \Sigma^\star} \psi_{m w} \ket{w}\]

it operates on any power series in this way:

1. remove all the words that do not start with \(m\)

   \[\sum_{w \in \Sigma^\star} \psi_{mw} \ket{mw}\]

2. erase \(m\) from the beginning of the remaining words

   \[\sum_{w \in \Sigma^\star} \psi_{mw} \ket{w}\]

let's see a few examples⁴:

• \(\partial_{abc}\ket{ab} = 0\) strings do not match
• \(\partial_{abc}\ket{xyz} = 0\) strings do not match
• \(\partial_{abc}\ket{abc} = \ket{\epsilon}\) erases 'abc' we are left with the empty string
• \(\partial_{abc}\ket{abcd} = \ket{d}\) erases 'abc' we are left with 'd'
• \(\partial_{abc} \left( \ket{a} + 2\ket{b} + 3\ket{abc} + 4\ket{abcd} + 5\ket{defg} \right) = 3\ket{\epsilon} + 4\ket{d}\) it's a linear operator: so it applies to each term in the sum

the quotient then is a kind of (left) inverse of the Cauchy product⁵

\[\partial_m \left( \ket{m} \otimes \ket{\psi} \right) = \ket{\psi}\]

Note that there are two different linearities at work:

• \(\partial_m\) is a linear operator⁶

  \[\partial_m \left( \alpha \ket{\phi} + \beta \ket{\phi} \right) = \alpha \partial_m \ket{\phi} + \beta \partial_m \ket{\phi}\]

• but you can also linearly combine \(\partial_m\) with different \(m\text{s}\)

  \[\partial_{\alpha u + \beta v} \equiv \alpha \partial_u + \beta \partial_v\]

  and still get a linear operator

Let's see what are the consequences, take the last example

\[\ket{\psi} = \ket{a} + 2\ket{b} + 3\ket{abc} + 4\ket{abcd} + 5\ket{defg}\]

as we have seen it is just a fancy way to write a function \(\psi(w)\)⁷

\(w\)	\(\psi(w)\)
\(a\)	1
\(b\)	2
\(abc\)	3
\(abcd\)	4
\(defg\)	5
otherwise	0

usually we think of functions as black boxes that take one thing as input and deliver another thing (always the same) as output in one shot.

But here we can somehow open the black box and play out evaluation in slow-motion it works like this

• I want to evaluate \(\psi(b)\), let's try to erase \(b\) from \(\ket{\psi}\)

  \[\partial_b \ket{\psi} = 2\ket{\epsilon}\]

  you see: the coefficient of the empty word is just the value of \(\psi(b)\)

• now I want to evaluate \(\psi(a)\), let's do as before and erase \(a\)

  \[\partial_a \ket{\psi} = \ket{\epsilon} + 3\ket{bc} + 4\ket{bcd}\]

  the coefficient of the empty word is still \(\psi(a)\) but now there is more, the remaining

  \[3\ket{bc} + 4\ket{bcd}\]

  is another function that allows to continue the computation of \(\psi(a \cdots)\) if we had more of the argument

• so we started with \(a\) let's evaluate \(ab\) and \(abc\)

  \[\partial_{ab} \ket{\psi} = 3\ket{c} + 4\ket{cd}\]

  you see that at this stage we don't have the empty string this means that \(\psi(ab) = 0\)

  \[\partial_{abc} \ket{\psi} = 3\ket{\epsilon} + 4\ket{d}\]

  and that's it \(\psi(abc) = 3\)

Ok, so \(\partial_m\) can be used to evaluate \(\psi(\cdot)\) (we apply it to the ket and retain the \(\epsilon\) term).

But there is more: thanks to the second linearity we can evaluate \(\psi(\cdot)\) over a superposition of its argument, it works like this: take again \(\ket{\psi}\)

\begin{align*} \partial_{\alpha a + \beta b} \ket{\psi} &= \alpha \left( \ket{\epsilon} + 3\ket{bc} + 4\ket{abcd} \right) + 2 \beta \ket{\epsilon} \\ &= \left( \alpha + 2\beta \right) \ket{\epsilon} + \cdots \\ \end{align*}

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(4) The Strange Entanglement

Ok, but we said that the quotient was somehow related to entanglement, how is it so?

I want to partially evaluate two special power series over the input \(\ket{\chi} = \alpha \ket{0} + \beta \ket{1}\):⁸

the "tensor product" power series

take \(\ket{\psi}\) as the Cauchy product of two factors \(\ket{\phi_1}, \ket{\phi_2}\)

\begin{align*} \ket{\psi} &= \ket{\phi_1} \otimes \ket{\phi_2} \\ &= \left( a_1 \ket{0} + b_1 \ket{1} \right) \otimes \left( a_2 \ket{0} + b_2 \ket{1} \right) \\ &= a_1 a_2 \ket{00} + a_1 b_2 \ket{01} + b_1 a_2 \ket{10} + b_1 b_2 \ket{11} \end{align*}

then

\begin{align*} \partial_\chi \ket{\psi} = \partial_{\alpha 0 + \beta 1} \ket{\psi} &= \alpha \partial_0 \left( a_1 a_2 \ket{00} + a_1 b_2 \ket{01} + b_1 a_2 \ket{10} + b_1 b_2 \ket{11} \right) \\ &+ \beta \partial_1 \left( a_1 a_2 \ket{00} + a_1 b_2 \ket{01} + b_1 a_2 \ket{10} + b_1 b_2 \ket{11} \right) \\ &= (\alpha a_1 + \beta b_1) a_2 \ket{0} + (\alpha a_1 + \beta b_1) b_2 \ket{1} \\ &= (\alpha a_1 + \beta b_1) (a_2 \ket{0} + b_2 \ket{1}) \\ &= \braket{\chi|\phi_1} \ket{\phi_2} \end{align*}

when we perform the partial evaluation of the whole \(\ket{\psi}\) we completely evaluate \(\ket{\phi_1}\) over \(\chi\) and leave \(\ket{\phi_2}\) unevaluated as the continuation⁹

this means that any subsequent evaluation will involve \(\ket{\phi_2}\) alone, the final result will be the product of two independent evaluations

\[\partial_\eta \partial_\chi \ket{\psi} = \braket{\chi|\phi_1} \, \partial_\eta \ket{\phi_2} = \braket{\chi|\phi_1} \, \braket{\eta|\phi_2} \, \ket{\epsilon}\]

the "entangled" power series

now take \(\ket{\psi} = \ket{01} + \ket{10}\), it cannot be decomposed into the Cauchy product of two power series

\begin{align*} \partial_\chi \ket{\psi} = \partial_{\alpha 0 + \beta 1} \ket{\psi} &= \alpha \partial_0 \left( \ket{01} + \ket{10} \right) \\ &+ \beta \partial_1 \left( \ket{01} + \ket{10} \right) \\ &= \beta \ket{0} + \alpha \ket{1} \end{align*}

different story! now the coefficients of \(\chi\) carry over to the continuation and will affect any subsequent evaluation

\[\partial_\eta \partial_\chi \ket{\psi} = \partial_{\gamma 0 + \delta 1} \partial_{\alpha 0 + \beta 1} \ket{\psi} = \partial_{\gamma 0 + \delta 1} \left( \beta \ket{0} + \alpha \ket{1} \right) = (\gamma \beta + \delta \alpha) \ket{\epsilon}\]

Now you may ask: how is all of this related to Quantum Mechanics?

And the answer is: it is not

• we are not assuming any of the quantum postulates
• we don't have any notion of quantum systems (simple or composite) or of measurements (yet)
• power series are definitely not quantum states

What we have done is just abuse the ket notation to make it easier to transplant superposition and entanglement into an algebraically richer humus.

The objective is to gain a new perspective on entanglement: less about unusual correlations between separate systems, more about progressive computations.

This is exactly what we have been doing in the last few posts but focusing on weighted automata instead of power series.

In this post for example we saw how to evaluate a function over superpositions of argument and we have defined a derivative operator that works very much like the quotient.

But this is to be expected, after all power series and automata are closely related: a recognizable power series

\[\ket{\psi} = \sum_{w \in \Sigma^\star} \psi_w \ket{w}\]

is one where each term can be computed by a weighted automaton \(\psi = (\boldsymbol{\alpha}, \mathbf{A}^{(\sigma)}, \boldsymbol{\omega})\)

\[\ket{\psi} = \sum_{w \in \Sigma^\star} \boldsymbol{\alpha}^\top \cdot \mathbf{A}^{(w)} \cdot \boldsymbol{\omega} \, \ket{w}\]

now take the quotient¹⁰

\begin{align*} \partial_m \ket{\psi} &= \sum_{w \in \Sigma^\star} \psi_{mw} \ket{w} \\ &= \sum_{w \in \Sigma^\star} \boldsymbol{\alpha}^\top \cdot \mathbf{A}^{(mw)} \cdot \boldsymbol{\omega} \, \ket{w} \\ &= \sum_{w \in \Sigma^\star} \boldsymbol{\alpha}^\top \cdot \mathbf{A}^{(m)} \cdot \mathbf{A}^{(w)} \cdot \boldsymbol{\omega} \, \ket{w} \\ \end{align*}

it is still a recognizable power series with the automaton

\[\partial_m \psi = (\boldsymbol{\alpha}^\top \cdot \mathbf{A}^{(m)}, \mathbf{A}^{(\sigma)}, \boldsymbol{\omega})\]

the effect of the quotient on the automaton is to pick up a specific state (or a combination of states) and use that as the new starting state.

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