nLab quantum state monad

Idea

The state monads and costate comonads (store comonads) are typically discussed on cartesian closed categories — but, of course, the analogous construction exists on any closed monoidal category.

An immediate non-cartesian example is a category of vector spaces. At least for a finite-dimensional vector space $\mathscr{H}$ (and generally on a compact closed category), the linear $\mathscr{H}$-state/store (co)monads merge into a single Frobenius monad (see below) which reflects some core structure of open quantum systems (mixed states, quantum channels and their quantum observables).

Due to this fact and the general broad identification of (multiplicative) linear logic with a form of quantum logic, the linear (co)state (co)monad may deserve to be called the quantum state monad (cf. the similar situation for the quantum reader monad).

For example, unitary quantum channels turn out to be embodied by monad transformations between such quantum state monads, so that with this terminology the plausible and desireable statement “quantum channels are quantum state transformations” becomes an actual theorem (see below).

Preliminaries

For definiteness (only), we consider the category $Mod_{\mathbb{C}}$ of complex vector spaces with, as usual:

• monoidal structure given by the tensor product of vector spaces,

• tensor unit$\;$ $\mathbb{1} \,\equiv\, \mathbb{C}$ the complex numbers themselves,

• internal hom$\;$ $\mathscr{H} \multimap \mathscr{H}'$ the vector space of linear maps

  (1)$$\mathscr{H} \otimes (\text{-}) \;\;\dashv\;\; \mathscr{H} \multimap (\text{-})$$

Throughout the following we fix

(2)$$\mathscr{H} \,\in\, FinDimMod_{\mathbb{C}} \hookrightarrow Mod_{\mathbb{C}}$$

a finite-dimensional complex vector space.

But most or all of the following discussion works for $Mod_{\mathbb{C}}$ replaced by any symmetric monoidal closed category and $\mathscr{H}$ taken from a compact closed subcategory.

In this vein, for notational purposes only (at this point) we choose on $\mathscr{H}$:

1. a Hermitian inner product $\left\langle\cdot \vert \cdot\right\rangle$, hence consider it as a finite-dimensional Hilbert space and use bra-ket notation to denote the generic elements of $\mathscr{H}$ by

   $$\left\vert \psi \right\rangle \,\equiv\, \psi \,\in\, \mathscr{H}$$

   and those of its linear dual space

   $$\mathscr{H}^\ast \,\equiv\, \mathscr{H} \multimap \mathbb{1}$$

   by

   $$\left\langle \phi \right\vert \,\equiv\, \left\langle \psi \vert (\text{-}) \right\rangle \,\in\, \mathscr{H}^\ast$$

2. an orthonormal linear basis $\big(\left\vert w \right\rangle \,\in\, \mathscr{H} \big)_{w \colon W}$, hence such that

   (3)$$w,w' \,\colon\, W \;\;\;\; \vdash \;\;\;\; \left\langle w \vert w' \right\rangle \,=\, \delta_{w, w'}$$

   (Kronecker delta).

Furthermore, for $\psi \in \mathscr{H}$, $\phi \in \mathscr{K}$ we leave the tensor product-symbol implicit in

$$\left\vert \psi \right\rangle \left\langle \phi \right\vert \;\; \equiv \;\; \psi \otimes \left\langle \phi \vert \text{-} \right\rangle \;\; \in \;\; \mathscr{H} \otimes \mathscr{K}^\ast \,.$$

In this bra-ket-notation, the compact closure for finite-dimensional $\mathscr{H}$ (see there) is witnessed by the following isomorphism

(4)$$\array{ \Big( \mathscr{H} \multimap \mathscr{H}' \Big) &\overset{\;\; \sim \;\;}{\longrightarrow}& \mathscr{H}' \otimes \mathscr{H}^\ast \\ \Big( \left\vert w \right\rangle \,\mapsto\, \underset{w'}{\sum} \left\vert w' \right\rangle \cdot A_{w', w} \Big) &\mapsto& \underset{w,w'}{\sum} \left\vert w' \right\rangle A_{w', w} \left\langle w \right\vert \mathrlap{\,.} }$$

Definition

For $\mathscr{H}$ a finite-dimensional Hilbert space — or generally an object in a compact closed category $QuTypes$ — its internal hom-adjunction is ambidextrous and associated to a pair of Frobenius monads, in linear version of the classical state monads/store comonad-construction:

Moreover, these (co)monads may equivalently be understood as the (co-)writer monads induced by the endomorphism algebra $End(\mathscr{H}) \,\simeq\, \mathscr{H}^\ast \otimes \mathscr{H}$, which is a Frobenius algebra (by general arguments as recalled in Lauda 2006, made explicit for instance in Vicary 2011 Lem 3.17) and as such makes its (co-)writer monad into a Frobenius monad.

We next spell this out in more detail

(Beware that there is now a switch in conventions. Will streamline later.)

The defining adjunction

The internal hom-adjunction for $\mathscr{H}$ (2)

(5)$$Mod_{\mathbb{C}} \underoverset { \underset{ \mathscr{H} \multimap (\text{-}) }{\longrightarrow}} { \overset{ \mathscr{H} \otimes (\text{-}) }{\longleftarrow} } {\;\; \bot \;\;} Mod_{\mathbb{C}}$$

the $\big(\mathscr{H} \otimes (\text{-}) \,\dashv\, \mathscr{H} \multimap (\text{-})\big)$-unit is:

(6)$$\array{ (\text{-}) &\overset{ ret ^{ \mathscr{H}State } _{\mathbb{1},\mathbb{1}} }{\longrightarrow}& \mathscr{H} \multimap \big( \mathscr{H} \otimes (\text{-}) \big) \\ \left\vert \psi \right\rangle &\mapsto& \big( \left\vert \psi' \right\rangle \mapsto \left\vert \psi' \right\rangle \otimes \left\vert \psi \right\rangle \big) \mathrlap{\,.} }$$

Using the symmetric braiding, an orthonormal basis (3) and the compact closure (4) we may equivalently write this as the “insertion of an identity” in the form known from quantum mechanics textbooks:

$$\array{ (\text{-}) & \overset{ ret ^{ \mathscr{H}State } _{\mathbb{1},\mathbb{1}} }{\longrightarrow}& \mathscr{H}^\ast \otimes \mathscr{H} \otimes (\text{-}) \\ \vert \psi \rangle &\mapsto& \sum_w \left\vert w \right\rangle \left\langle w \right\vert \otimes \left\vert \psi \right\rangle \mathrlap{\,.} }$$

The adjunction counit is the evaluation map:

(7)$$\array{ \mathscr{H} \otimes \big( \mathscr{H} \multimap (\text{-}) \big) &\overset{ obt^{ \mathscr{H}Store }_{} }{\longrightarrow}& (\text{-}) \\ \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\vert \text{-} \right\rangle &\mapsto& \left\langle \phi \vert \psi \right\rangle \, \left\vert \text{-} \right\rangle \mathrlap{\,.} }$$

which — again using the symmetric braiding, an orthonormal basis (3) and the compact closure (4) — is equivalently the partial trace over $\mathscr{H}$:

(8)$$\array{ \mathscr{H} \otimes \mathscr{H}^\ast \otimes (\text{-}) &\overset{ obt^{ \mathscr{H}Store }_{} }{\longrightarrow}& (\text{-}) \\ \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\vert \text{-} \right\rangle &\mapsto& \underset{w}{\sum} \left\langle w \vert \psi \right\rangle \left\langle \phi \vert w \right\rangle \otimes \left\vert \text{-} \right\rangle \mathrlap{\,.} }$$

The linear state monad

The adjunction (5) induces a linear version of the state monad:

The underlying endofunctor is

(9)$$\array{ \mathllap{ \mathscr{H} State \;\colon\; } Mod_{\mathbb{C}} &\longrightarrow& Mod_{\mathbb{C}} \\ \mathscr{V} &\mapsto& \Big( \mathscr{H} \multimap \big( \mathscr{H} \otimes \mathscr{V} \big) \Big) \mathrlap{\,.} }$$

The join operation is induced from the adjunction counit (8) as:

$$\array{ \mathscr{H}^\ast \otimes \mathscr{H} \otimes \mathscr{H}^\ast \otimes \mathscr{H} & \overset{ join^{\mathscr{H}State}_{(\text{-})} }{\longrightarrow} & \mathscr{H}^\ast \otimes \mathscr{H} \\ \left\langle \text{-} \right\vert \otimes \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\vert \text{-} \right\rangle &\mapsto& \left\langle \phi \vert \psi \right\rangle \, \left\langle \text{-} \right\vert \otimes \left\vert \text{-} \right\rangle \mathrlap{\,.} }$$

The linear store comonad

The adjunction (5) induces a linear version of the costate comonad:

The underlying endofunctor is

(10)$$\array{ \mathllap{ \mathscr{H} Store \;\colon\; } Mod_{\mathbb{C}} &\longrightarrow& Mod_{\mathbb{C}} \\ \mathscr{V} &\mapsto& \mathscr{H} \otimes \big( \mathscr{H} \multimap \mathscr{V} \big) \mathrlap{\,.} }$$

The duplication (cojoin) operation in the $\mathscr{H}$-CoState comonad (?) is induced from the adjunction unit (6) as follows:

(11)$$\array{ \mathscr{H} \otimes \big( \mathscr{H} \multimap (\text{-}) \big) & \overset{ dupl ^{ \mathscr{H}Store } _{ (\text{-}) } }{\longrightarrow} & \mathscr{H} \otimes \bigg( \mathscr{H} \multimap \Big( \mathscr{H} \otimes \big( \mathscr{H} \multimap (\text{-}) \big) \Big) \bigg) \\ \left\vert \psi \right \rangle \left\langle \phi \right\vert \otimes \left\vert \text{-} \right \rangle &\mapsto& \sum_w \left\vert \psi \right\rangle \left\langle w \right\vert \otimes \left\vert w \right\rangle \left\langle \phi \right\vert \otimes \left\vert \text{-} \right \rangle \mathrlap{\,.} }$$

Finally, the obtain-operation (the comonad counit) is the adjunction counit (8)

Kleisli structure

We assume an ambient compact closed category such as that of finite-dimensional vector spaces.

For the formula in Prop. to come out naturally, we take – without loss of generality – the generic object on which to apply the linear store comonad to be a dual object $\mathscr{K}^\ast \coloneqq (\mathscr{K} \multimap \mathbb{1})$.

Remark

(the coKleisli morphisms)
The coKleisli morphisms of the quantum store comonad (?)

(12)$$\mathcal{O}_A \;\colon\; \mathscr{H} \otimes \big( \mathscr{H} \multimap \mathscr{K}^\ast \big) \longrightarrow \mathscr{K}^\ast$$

are isomorphically of the form

(13)$$\mathcal{O}_A \;\colon\; \mathscr{H} \otimes \mathscr{H}^\ast \otimes \mathscr{K}^\ast \longrightarrow \mathscr{K}^\ast$$

and as such are equivalently (adjunct to) morphisms of the form

(14)$$A \;\colon\; \mathscr{H} \otimes \mathscr{K}^\ast \longrightarrow \mathscr{H} \otimes \mathscr{K}^\ast$$

which in turn are isomorphically “superoperators” of the form

(15)$$\big( \mathscr{K} \multimap \mathscr{H} \big) \longrightarrow \big( \mathscr{K} \multimap \mathscr{H} \big) \,.$$

Proposition

The Kleisli composition on Kleisli morphisms (12) of the quantum store comonad is given by the plain composition of the linear duals of the corresponding superoperators (14):

(16)$$\big( extend^{\mathscr{H}Store} \mathcal{O}_{A'} \big) \circ \big( extend^{\mathscr{H}Store} \mathcal{O}_{A} \big) \;\;=\;\; \Big( \left\vert \psi \right\rangle \left\langle \phi, \kappa \right\vert \,\mapsto\, \left\vert \psi \right\rangle \left\langle \phi, \kappa \right\vert A \circ A' \Big) \,.$$

Here for $\psi \in \mathscr{H}$ and $\kappa \in \mathscr{K}$ we denote their tensor product as

$$\left\vert \psi, \kappa \right\rangle \,\in\, \mathscr{H} \otimes \mathscr{K} \,.$$

Proof

The relation between a Kleisli morphism $\mathcal{O}_A$ (13) and the corresponding super-operator $A$ (14) on matrices is

$$\array{ \mathllap{ \mathcal{O}_A \;\colon\; } \mathscr{H} \otimes \mathscr{H}^\ast \otimes \mathscr{K}^\ast &\longrightarrow& \mathscr{K}^\ast \\ \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\langle \kappa \right\vert &\mapsto& \left\langle \phi, \kappa \right\vert A \left\vert \psi, - \right\rangle \mathrlap{\,.} }$$

With this and the cojoin operation (11) it follows that the extend-operation turns $\mathcal{O}_A$ into

$$\array{ \mathscr{H} \otimes \big( \mathscr{H} \multimap \mathscr{K}^\ast \big) & \overset{ dupl^{ \mathscr{H}Store }_{\mathscr{K}^\ast,\mathscr{K}^\ast} }{\longrightarrow} & \mathscr{H} \otimes \bigg( \mathscr{H} \multimap \Big( \mathscr{H} \otimes \big( \mathscr{H} \multimap \mathscr{K}^\ast \big) \Big) \bigg) &\overset{ \mathscr{H} \otimes \big( \mathscr{H} \multimap \mathcal{O}_A \big) }{\longrightarrow}& \mathscr{H} \otimes \big( \mathscr{H} \multimap (\mathscr{K}')^\ast \big) \\ \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\langle \kappa \right\vert &\mapsto& \sum_w \left\vert \psi \right\rangle \left\langle w \right\vert \otimes \left\vert w \right\rangle \left\langle \phi \right\vert \otimes \left\langle \kappa \right\vert &\mapsto& \;\; \underset{ = \left\vert \psi \right\rangle \left\langle \phi, \kappa \right\vert A }{ \underbrace{ \sum_{w, k'} \left\vert \psi \right\rangle \left\langle \phi, \kappa \right\vert A \left\vert w, k' \right\rangle \left\langle w, k' \right\vert } } }$$

hence

(17)$$\array{ \mathllap{ exend^{\mathscr{H} Store} \mathcal{O}_A \;\colon\; } \mathscr{H} \otimes \mathscr{H}^\ast \otimes \mathscr{K}^\ast &\longrightarrow& \mathscr{H} \otimes \mathscr{H}^\ast \otimes \mathscr{K}^\ast \\ \left\vert \psi \right\rangle \left\langle \phi, \kappa \right\vert &\mapsto& \left\vert \psi \right\rangle \left\langle \phi, \kappa \right\vert A }$$

This implies the claim (16).

The same argument applies with the second copy of $\mathscr{K}$ replaced by some $\mathscr{K}'$: The Kleisli category is the opposite of that of superoperators of the form

$$\mathscr{H}\otimes\mathscr{K} \longrightarrow \mathscr{H}\otimes\mathscr{K}' \longrightarrow \mathscr{H}\otimes\mathscr{K}'' \,.$$

Examples

Quantum observables are the Quantum state contextful scalars

(The following now starts out a little repetitive, recalling a special case of the general statement above. Will streamline…)

For the classical costate comonad on a cartesian closed category its value and its operations on the tensor unit (the terminal object in this case) are vacuous. Quite in contrast, the linear CoState comonad on the tensor unit encodes core structure of quantum physics:

Example

(quantum observables as quantum state contextful scalars)

First notice that the quantum store on the tensor unit

$$\mathscr{H}Store(\mathbb{1}) \;\equiv\; \mathscr{H} \otimes \big( \mathscr{H} \multimap \mathbb{1} \big) \;=\; \mathscr{H} \otimes \mathscr{H}^\ast$$

is the home of density matrices/mixed quantum states generalizing the pure states in $\mathscr{H}$.

Moreover, an $\mathscr{H}$-CoState coKleisli morphism on the tensor unit is equivalently a linear operator

$$A \,\colon\, \mathscr{H} \to \mathscr{H} \,,$$

hence a quantum observables – incarnated via its expectation values on quantum states

(18)$$\array{ \mathscr{H} \otimes \mathscr{H}^\ast &\overset{\mathcal{O}_A}{\longrightarrow}& \mathbb{1} \\ \left\vert \psi \right\rangle \left\langle \phi \right\vert &\mapsto& \left\langle \phi \right\vert A \left\vert \psi \right\rangle \mathrlap{\,.} }$$

Proposition

Composition of quantum observables $\mathcal{O}_A$, $\mathcal{O}_{A'}$ (18) in the coKleisli category is given by ordinary composition $A \circ A'$ of the corresponding linear operators:

$$\mathcal{O}_{A'} \circ \big( ext^{ \mathscr{H}Store }_{\mathbb{1},\mathbb{1}} \mathcal{O}_A \big) \;\; = \;\; \mathcal{O}_{ A' \circ A } \;\colon\; \left\vert \psi \right\rangle \left\langle \phi \right\vert \,\mapsto\, \left\langle \phi \right\vert A A' \left\vert \psi \right\rangle \,.$$

Proof

From (11) we find that the extension operation is given as follows:

$$\array{ \mathscr{H} \otimes \mathscr{H}^\ast & \overset{ dupl^{ \mathscr{H}Store }_{\mathbb{1},\mathbb{1}} }{\longrightarrow} & \mathscr{H} \otimes \big( \mathscr{H} \multimap \mathscr{H} \otimes \mathscr{H}^\ast \big) &\overset{ \mathscr{H} \otimes \big( \mathscr{H} \multimap \mathcal{O}_A \big) }{\longrightarrow}& \mathscr{H} \otimes \mathscr{H}^\ast \\ \left\vert \psi \right\rangle \left\langle \phi \right\vert &\mapsto& \sum_w \left\vert \psi \right\rangle \left\langle w \right\vert \otimes \left\vert w \right\rangle \left\langle \phi \right\vert &\mapsto& \;\; \mathclap{ \underset{ { =\, \left\vert \psi \right\rangle \left\langle \phi \right\vert A } } { \underbrace{ \sum_w \left\vert \psi \right\rangle \left\langle w \right\vert \otimes \left\langle \phi \right\vert A \left\vert w \right\rangle } } } }$$

hence

$$ext^{ \mathscr{H}Store }_{\mathbb{1},\mathbb{1}} \,\colon\, \Big( \left\vert \psi \right\rangle \left\langle \phi \right\vert \,\mapsto\, \left\langle \phi \right\vert A \left\vert \psi \right\rangle \Big) \,\mapsto\, \Big( \left\vert \psi \right\rangle \left\langle \phi \right\vert \,\mapsto\, \left\vert \psi \right\rangle \left\langle \phi \right\vert A \Big)$$

Remark

Understanding $\mathscr{H}Store$ as a Frobenius monad as above makes it natural to combine the monad unit (6) with the Kleisli morphisms of Rem. .

This gives the trace of linear operators, as seen via (17):

$$\array{ \mathbb{1} &\overset{ret^{\mathscr{H}^\ast State}_{\mathbb{1}}}{\longrightarrow}& \mathscr{H} \otimes \mathscr{H}^\ast &\overset{ ext^{\mathscr{H}Store}_{\mathbb{1}, \mathbb{1}} \mathcal{O}_{A} }{\longrightarrow}& \mathscr{H} \otimes \mathscr{H}^\ast &\overset{obt^{\mathscr{H}Store}_{\mathbb{1}}}{\longrightarrow}& \mathbb{1} \\ 1 &\mapsto& \underset{w}{\sum} \left\vert w \right\rangle \left\langle w \right\vert &\mapsto& \underset{w}{\sum} \left\vert w \right\rangle \left\langle w \right\vert A &\mapsto& tr\big( A \big) }$$

and generally the partial trace:

$$\array{ \mathscr{K}^\ast &\overset{ret^{\mathscr{H}^\ast State}_{\mathscr{K}^\ast}}{\longrightarrow}& \mathscr{H} \otimes \mathscr{H}^\ast \otimes \mathscr{K}^\ast &\overset{ ext^{\mathscr{H}Store}_{\mathscr{K}^\ast, \mathscr{K}^\ast} \mathcal{O}_{A} }{\longrightarrow}& \mathscr{H} \otimes \mathscr{H}^\ast \otimes \mathscr{K}^\ast &\overset{obt^{\mathscr{H}Store}_{\mathscr{K}^\ast}}{\longrightarrow}& \mathscr{K}^\ast \\ \left\langle \kappa \right\vert &\mapsto& &\mapsto& &\mapsto& \left\langle \kappa \right\vert tr_{\mathscr{H}}\big(A \big) }$$

Quantum channels that are Quantum state transformations

Example

(unitary quantum channels are quantum state transformations)
If $U \,\colon\, \mathscr{H}_1 \to \mathscr{H}_2$ is an invertible map with inverse $U^\dagger \,\colon\, \mathscr{H}_2 \to \mathscr{H}_1$, then the “unitary quantum channel”

$$\array{ \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \otimes (\text{-}) & \overset{ U \otimes U^{\dagger \ast} }{\longrightarrow} & \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \otimes (\text{-}) }$$

is a transformation of quantum state monads.

Proof

By direct unwinding of the definitions, we find that the counit is respected

$$\array{ \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \otimes (\text{-}) & \overset{ U \otimes U^{\dagger \ast} \, \otimes (\text{-}) }{\longrightarrow} & \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \otimes (\text{-}) & \overset{ obt^{\mathscr{H}_2Store}_{\mathscr{K}_2} }{\longrightarrow} & (\text{-}) \\ \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\langle \text{-} \right\vert &\mapsto& U \left\vert \psi \right\rangle \left\langle \phi \right\vert U^\dagger \otimes \left\langle \text{-} \right\vert &\mapsto& \left\langle \phi \right\vert U^\dagger U \left\vert \psi \right\rangle \otimes \left\langle \text{-} \right\vert \\ && &=& \left\langle \phi \vert \psi \right\rangle \otimes \left\langle \text{-} \right\vert \\ && &=& obt^{\mathscr{H}_1 Store}_{\mathscr{K}^\ast} \big( \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\langle \text{-} \right\vert \big) }$$

by the property that $U^\dagger$ is a left inverse to $U$,

and the unit is respected

$$\array{ (\text{-}) & \overset{ ret^{ \mathscr{H}_1State }_{(\text{-})} }{\longrightarrow} & \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \otimes (\text{-}) & \overset{ U \otimes U^{\dagger\ast} \otimes (\text{-}) }{\longrightarrow} & \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \otimes (\text{-}) \\ \underset{w}{\sum} \left\vert w \right\rangle \left\langle w \right\vert \otimes \left\vert \text{-} \right\rangle &\mapsto& \underset{w_1}{\sum} U \left\vert w_1 \right\rangle \left\langle w_1 \right\vert U^{\dagger} \otimes \left\vert \text{-} \right\rangle \\ && &=& \underset{w_2}{\sum} \left\vert w_2 \right\rangle \left\langle w_2 \right\vert \otimes \left\vert \text{-} \right\rangle \mathrlap{\,,} }$$

because $U^\dagger$ is also a right inverse to $U$, using here the compact closure identification

(19)$$\array{ \mathscr{H}_2 \otimes \mathscr{H}_2^\ast &\overset{\sim}{\longrightarrow}& \mathscr{H}_2 \multimap \mathscr{H}_2 &\overset{\sim}{\longrightarrow}& \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \\ \underset{w_1}{\sum} U \left\vert w_1 \right\rangle \left\langle w_1 \right\vert U^{\dagger} &\mapsto& \underset{ \mathrm{Id} }{ \underbrace{ U \circ Id \circ U^{\dagger} } } &\mapsto& \underset{w_2}{\sum} \left\vert w_2 \right\rangle \left\langle w_2 \right\vert \mathrlap{\,.} }$$

Therefore also the join and cojoin are preserved, eg.:

$$\array{ \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \otimes (\text{-}) & \overset{ dupl^{\mathscr{H}_1 Store}_{\mathscr{K}^\ast} }{\longrightarrow} & \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \otimes \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \otimes (\text{-}) & \overset{ U \otimes U^{\dagger\ast} \otimes U \otimes U^{\dagger\ast} \, \otimes \mathrm{id} }{\longrightarrow} & \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \otimes \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \otimes (\text{-}) \\ \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\langle \text{-} \right\vert &\mapsto& \mathclap{ \underset{w_1}{\sum} \, \left\vert \psi \right\rangle \left\langle w_1 \right\vert \otimes \left\vert w_1 \right\rangle \left\langle \phi \right\vert U^\dagger \otimes \left\langle \text{-} \right\vert } &\mapsto& \mathclap{ U \left\vert \psi \right\rangle \Big( \underset{ = \underset{w_2}{\sum} \left\langle w_2 \right\vert \otimes \left\vert w_2 \right\rangle } { \underbrace{ \underset{w_1}{\sum} \left\langle w_1 \right\vert U^\dagger \otimes U \left\vert w_1 \right\rangle } } \Big) \left\langle \phi \right\vert U^\dagger \otimes \left\langle \text{-} \right\vert } \\ && &=& \mathclap{ dupl^{\mathscr{H}_2 Store}_{\mathscr{K}^\ast} \circ \big( U \otimes U^{\dagger \ast} \otimes id \big) \big( \left\vert \psi \right\rangle \left\langle \phi \right\vert \otimes \left\langle \text{-} \right\vert \big) } }$$

Example

A partial trace quantum channel

$$\array{ (\text{-}) \otimes \mathscr{H} \otimes \mathscr{B} \otimes \mathscr{B}^\ast \otimes \mathscr{H}^\ast & \xrightarrow{ id \otimes trace^{\mathscr{B}} } & \mathscr{H} \otimes \mathscr{H}^\ast \\ \left\vert \psi, \beta \right\rangle \left\langle \beta', \psi' \right\vert &\mapsto& \left\vert \psi \right\rangle \left\langle \beta' \vert \beta \right\rangle \left\langle \psi' \right\vert \mathrlap{\,.} }$$

is a comonadic quantum state transformation

$$trace^{\mathscr{B}} \,\colon\, (\mathscr{H} \otimes \mathscr{B})State \underoverset{ comon\;transf }{}{\longrightarrow} \mathscr{H}State \,.$$

Proof

Since the structure maps of the $(\mathscr{H} \otimes \mathscr{B})\mathrm{State}$-comonad are tensor products of structure maps of $\mathscr{H}\mathrm{State}$ and $\mathscr{B}\mathrm{State}$, it is sufficient to show this for $\mathscr{H} = \mathbb{1}$ the tensor unit, hence for the case that $\mathscr{H}\mathrm{State} = \mathrm{Id}$. But in this case $\mathrm{trace}^{\mathscr{B}} \,=\, obt^{ \mathscr{H}^\ast\mathrm{Store} }{}$, which is a comonadic transformation (in fact the terminal one) by this example).

Alternatively, onee readily checks the required conditions explicitly:

Heisenberg evolution is action of Quantum state transformations

namely on the quantum-state contextful scalars:

Example

The quantum state transformation induced by a unitary quantum channel $U$, according to Exp. , is invertible (its inverse given by the inverse unitary quantum channel) and hence it induces a functor between the Kleisli categories (by the discussion here) which on the quantum-state contextful scalars $\mathcal{O}_A$ (Exp. ) is given by conjugation with $U$:

$$\array{ (\text{-}) \otimes \mathscr{H}_1 \otimes \mathscr{H}_1^\ast & \overset{ (\text{-}) \otimes U \otimes U^{\dagger\ast} }{\longleftarrow} & (\text{-}) \otimes \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \\ Kl(\mathscr{H}Store) &\longrightarrow& Kl(\mathscr{H}Store) \\ \left[ \array{ \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \\ \Big\downarrow\mathrlap{ \mathcal{O}_A } \\ \mathbb{1} } \right] &\mapsto& \left[ \array{ \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \\ \Big\downarrow\mathrlap{ \mathcal{O}_{ U \cdot A \cdot U^\dagger } } \\ \mathbb{1} } \;\;\;\;\;\; \right] }$$

This transformation

$$A \;\mapsto\; U \cdot A \cdot U^\dagger$$

is of course the Heisenberg picture-evolution of quantum observables $A$ under unitary transformations $U$ of the quantum states.

Properties

Interaction with QuantumEnvironment monad

Proposition

For

• $\mathscr{H}$ a dualizable linear type

• $W$ a finite classical type,

we have:

1. the $\mathscr{H}$QuantumState comonad $\mathscr{H}^\ast Store$ distributes over the $W$QuantumEnvironment monad $\bigcirc_W$,

2. the $W$-QuantumEnvironment comonad $\bigstar_W$ distributes over over the $\mathscr{H}$QuantumState monad $\mathscr{H}^\ast Store$.

Proof

The relevant conditions (from Brookes & Van Stone 1993 Def. 3, see there) all follows immediately from the distributivity of the tensor product (being a left adjoint) over the direct sum (being a coproduct).

For definiteness, we spell it out. In the first direction we have:

and in the converse direction we have:

Example

The two-sided Kleisli category induced by Prop. (via Brookes & Van Stone 1993 Thm. 2) equivalently has (by arguing $W$-wise just as in Prop. ):

• as morphisms $W$-indexed sets of linear operators

• with composition given by their $W$-wise operator products

Related concepts

• quantum reader monad