Quantum Channels and the Reversible/Irreversible Dichotomy

Introduction

The companion article Quantum Mechanics in Biquaternionic Form developed the qubit in the biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$: states and observables are elements of the Hermitian subspace $\mathbb{M}_+$; reversible evolution is rotor conjugation $\tilde{\rho} \mapsto \tilde{U}\tilde{\rho}\tilde{U}^{*}$ by a unitary biquaternion; measurement is the sandwich operation $\tilde{\rho} \mapsto \tilde{P}\tilde{\rho}\tilde{P}$. It listed the dynamics of open systems — dissipation, decoherence, and the Lindblad equation — among its open questions.

This article takes up that question. Its subject is the general state map on $\mathbb{M}_+$: the most general process carrying a state of the informational sector to a state of the informational sector — in quantum information theory, a quantum channel or quantum operation. We develop the requirements that define it — linearity, complete positivity, and trace preservation — in the biquaternion language, and then examine the reversible/irreversible dichotomy they produce.

The dichotomy has a sharp algebraic form here. A state map is reversible if and only if it is conjugation by a single unitary biquaternion, $\tilde{\rho} \mapsto \tilde{U}\tilde{\rho}\tilde{U}^{*}$ with $\tilde{U}\tilde{U}^{*} = e_0$. Each one-parameter subgroup is generated by a Hamiltonian $\tilde{H} \in \mathbb{M}_+$ through the exponential $\tilde{U}(t) = \exp(-i\tilde{H}t/\hbar)$. The irreversible maps are the rest: the completely positive, trace-preserving maps, which are sums of conjugations rather than a single conjugation. The canonical example is dephasing, worked out below in the $\mathbb{M}_+$ formulation.

The relation to the two subspaces is the conceptual thread. States live in $\mathbb{M}_+$; the reversible motion of those states is generated by elements of the complementary subspace $\mathbb{M}_-$; and the irreversible maps are exactly those that are not inner automorphisms of the algebra. Irreversibility is thus a statement about which kinds of map the algebra admits, not an extra postulate imposed on it.

Two cautionary notes, in the spirit of the companion articles. First, the mathematics is standard quantum information theory transcribed into biquaternion notation; no result below depends on the physical hypothesis that $\mathbb{M}_+$ is a distinct sector of the world. Second, the reformulation reframes the dichotomy without resolving the conceptual problems that attend it in standard quantum mechanics — the position taken in Entangled Subsystems in the Biquaternion Framework, that the framework changes the language and not the physics, applies here too.

The conventions are those of the companion articles: the quaternion basis $e_0 = 1, e_1, e_2, e_3$ with $e_k^2 = -e_0$; the central scalar imaginary $i$; the fixed-point subspaces $\mathbb{C}_{\mathbb{B}}$ (the complex subspace, the center), $\mathbb{H}_{\mathbb{B}}$ (the real-quaternion subspace), $\mathbb{M}_+$ (Hermitian), and $\mathbb{M}_-$ (anti-Hermitian); a general element $\tilde{Q} = Q_0 e_0 + Q_1 e_1 + Q_2 e_2 + Q_3 e_3$ with $Q_\mu \in \mathbb{C}$; and the trace $\mathrm{Tr}(\tilde{Q}) = 2\,\mathrm{Sc}(\tilde{Q})$.

The State Space and the Trace Pairing

A state of the informational sector is an element $\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$ of $\mathbb{M}_+$ with $\mathbf{r} \in \mathbb{R}^3$, positive if and only if $|\mathbf{r}| \leq 1$. The states form the Bloch ball, whose boundary is the set of pure states; these are exactly the idempotents $\tilde\Pi_\pm(\hat{\mu}) = \tfrac{1}{2}(e_0 \pm i\hat{\mu})$, with $\hat{\mu}$ a unit pure real quaternion. Every state has trace one, $\mathrm{Tr}(\tilde{\rho}) = 2\,\mathrm{Sc}(\tilde{\rho}) = 1$.

An observable is a general Hermitian element $\tilde{H} = h_0 e_0 + i\mathbf{h}$, and the trace pairing

$$ \mathrm{Tr}(\tilde{\rho}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{\rho}\tilde{H}) = h_0 + \mathbf{r}\cdot\mathbf{h} $$

is the Born rule. The trace pairing is a symmetric, positive-definite bilinear form, the Euclidean pairing $\mathrm{Tr}(\tilde{H}\tilde{K}) = 2(h_0k_0 + \mathbf{h}\cdot\mathbf{k})$ up to normalization. It plays a second, equally central role here: it is the operation that pairs a state map with its dual, and therefore the operation that makes the Heisenberg picture of a channel possible.

Two structural facts will be used repeatedly. First, $\mathbb{B}$ is the complexification of the real subspace $\mathbb{M}_+$, with the central scalar $i$ as the complex structure: $i\,\mathbb{M}_+ = \mathbb{M}_-$ and $\mathbb{B} = \mathbb{M}_+ \oplus \mathbb{M}_-$, so every element decomposes uniquely into a Hermitian and an anti-Hermitian part. Second, positivity is the biquaternion-norm condition $N(\tilde{\rho}) = \tfrac{1}{4}(1 - |\mathbf{r}|^2)e_0 \geq 0$; the state space is the trace-one slice of the future light cone of the biquaternion norm.

State Maps on $\mathbb{M}_+$

Linearity, Positivity, and Complete Positivity

A state map is a map $\Phi$ sending states to states. We require it to be linear. This is the standard framework of quantum operations and a genuine restriction: nonlinear modifications are excluded by the requirement that the map act consistently on mixtures, since a state prepared as a convex combination of ensembles must be transformed in a way that respects the preparation. In the biquaternion language, $\Phi$ is a real-linear map on $\mathbb{M}_+$,

$$ \Phi : \mathbb{M}_+ \longrightarrow \mathbb{M}_+, \qquad \Phi(a\tilde{\rho}_1 + b\tilde{\rho}_2) = a\,\Phi(\tilde{\rho}_1) + b\,\Phi(\tilde{\rho}_2), \qquad a, b \in \mathbb{R}. $$

Because $\mathbb{B} = \mathbb{M}_+ \oplus i\mathbb{M}_+$ is the complexification of $\mathbb{M}_+$, every real-linear map on $\mathbb{M}_+$ extends uniquely to a complex-linear map on $\mathbb{B}$, $\Phi(x + iy) = \Phi(x) + i\Phi(y)$, and it is standard to work with this extension, regarding a quantum operation as a map on $\mathbb{B} \cong M_2(\mathbb{C})$.

The first requirement is positivity: $\tilde{\rho} \geq 0$ must imply $\Phi(\tilde{\rho}) \geq 0$. Positivity alone is not enough. The system may be entangled with a reference system on which $\Phi$ does not act, and a map that is positive on the system alone can fail to be positive on the joint system. Complete positivity is the condition that the extended map

$$ \Phi \otimes \mathrm{id}_n : \mathbb{B}\otimes M_n(\mathbb{C}) \longrightarrow \mathbb{B}\otimes M_n(\mathbb{C}) $$

be positive for every $n$. A positive but not completely positive map is not a physical process, because acting on half of an entangled pair it can produce an operator with a negative eigenvalue.

The standard illustration is the transpose $\tilde{Q} \mapsto \tilde{Q}^{\mathsf{T}}$ in a fixed matrix representative. It is positive and complex-linear, but not completely positive: transposing one factor of a maximally entangled two-qubit state produces a non-positive operator. Complete positivity is strictly stronger than positivity, and it is the condition the algebra imposes on physical operations.

Trace Preservation and the Dual Map

Because a state map must send trace-one elements to trace-one elements, we require trace preservation, $\mathrm{Tr}(\Phi(\tilde{\rho})) = \mathrm{Tr}(\tilde{\rho})$ for all $\tilde{\rho} \in \mathbb{M}_+$, equivalently $\mathrm{Tr}(\Phi(\tilde{Q})) = \mathrm{Tr}(\tilde{Q})$ for all $\tilde{Q} \in \mathbb{B}$. The trace pairing defines a dual map on observables, the Heisenberg-picture representative of $\Phi$, by

$$ \mathrm{Tr}\bigl(\Phi(\tilde{\rho})\,\tilde{H}\bigr) = \mathrm{Tr}\bigl(\tilde{\rho}\,\Phi^{*}(\tilde{H})\bigr) \qquad \text{for all } \tilde{\rho}, \tilde{H}. $$

Trace preservation of $\Phi$ is equivalent to unitality of the dual, $\Phi^{*}(e_0) = e_0$: setting $\tilde{H} = e_0$ gives $\mathrm{Tr}(\Phi(\tilde{\rho})) = \mathrm{Tr}(\tilde{\rho}\,\Phi^{*}(e_0))$, which equals $\mathrm{Tr}(\tilde{\rho})$ for every $\tilde{\rho}$ precisely when $\Phi^{*}(e_0) = e_0$. The duality is mediated throughout by the trace formula $\mathrm{Tr}(\tilde{P}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{P}\tilde{H})$ that gives the Born rule.

The Kraus Representation

The structure theorem for completely positive maps is the Kraus representation (the Choi–Kraus theorem). A map $\Phi$ on $\mathbb{B}$ is completely positive if and only if there exist finitely many Kraus operators $\tilde{K}_l \in \mathbb{B}$ with

$$ \Phi(\tilde{Q}) = \sum_l \tilde{K}_l\,\tilde{Q}\,\tilde{K}_l^{*} , $$

and it is trace preserving if and only if $\sum_l \tilde{K}_l^{*} \tilde{K}_l = e_0$. A completely positive, trace-preserving map is a quantum channel; its dual is $\Phi^{*}(\tilde{H}) = \sum_l \tilde{K}_l^{*} \tilde{H}\tilde{K}_l$, and the normalization is exactly $\Phi^{*}(e_0) = e_0$. The representation is not unique: two Kraus sets define the same channel precisely when they differ by a partial isometry. The Kraus rank is the minimal number of Kraus operators, equal to the rank of the Choi matrix $J(\Phi) = \sum_{jk} \tilde{E}_{jk}\otimes\Phi(\tilde{E}_{jk})$, and complete positivity is the condition $J(\Phi) \geq 0$. For a qubit channel the Choi matrix is $4\times4$, so the Kraus rank is at most four.

The companion article already introduced this form for generalized measurements. The point here is that it describes every physical process, not only measurement.

The Reversible Case: Unitary Rotor Conjugation

Unitary Elements and Their Conjugation Action

The simplest channel is conjugation by a single unitary biquaternion:

$$ \Phi_{\tilde{U}}(\tilde{\rho}) = \tilde{U}\,\tilde{\rho}\,\tilde{U}^{*}, \qquad \tilde{U} \in \mathbb{B}, \quad \tilde{U}\tilde{U}^{*} = e_0 . $$

This is a channel with the single Kraus operator $\tilde{K} = \tilde{U}$, and it maps states to states because $\tilde{U}\tilde{\rho}\tilde{U}^{*}$ is Hermitian, positive, and of trace one.

Here "unitary" means unitary in the matrix sense, $\tilde{U}\tilde{U}^{*} = e_0$, and the unitary biquaternions form the group $U(2) \subset \mathbb{B}$. This must be distinguished from the condition $\tilde{U}\tilde{U}^{\natural} = e_0$ of unit biquaternion norm, which defines $SL(2,\mathbb{C})$ and describes the Lorentz rotors acting on the material sector $\mathbb{M}_-$. A Lorentz boost rotor is Hermitian and hence is not matrix-unitary: for a boost, $\tilde{\Lambda}\tilde{\Lambda}^{*} = \tilde{\Lambda}^2 \neq e_0$. Conjugation by a boost does not preserve the trace, so a boost is not a channel. The reversible processes of the informational sector are the matrix-unitary conjugations.

Under $\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$, the state $\Phi_{\tilde{U}}(\tilde{\rho})$ is again of the form $\tfrac{1}{2}(e_0 + i\mathbf{r}')$ with $|\mathbf{r}'| = |\mathbf{r}|$; the scalar part is fixed and the vector part is rotated. Purity, $\mathrm{Tr}(\tilde{\rho}^2) = \tfrac{1}{2}(1 + |\mathbf{r}|^2)$, and the spectrum $\lambda_\pm = (1\pm|\mathbf{r}|)/2$ are invariant. The unitary channels are exactly the rotations of the Bloch ball.

The One-Parameter Group Generated by a Hamiltonian

The unitary channels describing continuous evolution are the one-parameter subgroups generated by Hamiltonians. For Hermitian $\tilde{H} = h_0 e_0 + i\mathbf{h} \in \mathbb{M}_+$, the Schrödinger equation $i\hbar\,\tfrac{d}{dt}\tilde{U}(t) = \tilde{H}\tilde{U}(t)$ with $\tilde{U}(0) = e_0$ has the solution

$$ \tilde{U}(t) = \exp\!\left(-\frac{i\tilde{H}t}{\hbar}\right) = e^{-ih_0 t/\hbar}\left(\cos\!\left(\frac{|\mathbf{h}|t}{\hbar}\right)e_0 + \sin\!\left(\frac{|\mathbf{h}|t}{\hbar}\right)\hat{\mathbf{h}}\right), \qquad \hat{\mathbf{h}} = \frac{\mathbf{h}}{|\mathbf{h}|}, $$

the trigonometric form following from $\hat{\mathbf{h}}^2 = -e_0$. The elements satisfy $\tilde{U}(t)\tilde{U}(t)^{*} = e_0$ and form a one-parameter group, $\tilde{U}(t+s) = \tilde{U}(t)\tilde{U}(s)$, because they are functions of the single element $\tilde{H}$ and therefore commute. The state evolves by rotor conjugation $\tilde{\rho}(t) = \tilde{U}(t)\tilde{\rho}(0)\tilde{U}(t)^{*}$, or in differential form $i\hbar\,\tfrac{d}{dt}\tilde{\rho}(t) = [\tilde{H}, \tilde{\rho}(t)]$, the von Neumann equation.

The Generator Lies in $\mathbb{M}_-$

The generator of the group is $-\tfrac{i\tilde{H}}{\hbar} \in \mathbb{M}_-$, since $\tilde{H} \in \mathbb{M}_+$ implies $(-i\tilde{H})^{*} = i\tilde{H}^{*} = i\tilde{H} = -(-i\tilde{H})$. Because $i\mathbb{M}_+ = \mathbb{M}_-$, every element of $\mathbb{M}_-$ is of this form, and the exponential map sends $\mathbb{M}_-$ onto the unitary group. This is the cleanest expression of the complementarity that organizes the subject: states and observables live in $\mathbb{M}_+$, while the generators of their reversible evolution live in $\mathbb{M}_-$. The subspace $\mathbb{M}_-$ is the Lie algebra of the unitary group under the commutator, and reversible dynamics is its action on $\mathbb{M}_+$ by inner derivations.

The fixed-point subspaces $\mathbb{C}_{\mathbb{B}}$ and $\mathbb{H}_{\mathbb{B}}$ mark two extremes. The trace part $h_0 e_0$ lies in $\mathbb{C}_{\mathbb{B}}\cap\mathbb{M}_+ = \mathbb{R}e_0$, the center; it generates the central phase $e^{-ih_0t/\hbar}$, which commutes with everything and cancels in the conjugation, affecting only the unobservable global phase. The traceless part $i\mathbf{h}$ has $h_0 = 0$, and then

$$ \tilde{U}(t) = \cos\!\left(\frac{|\mathbf{h}|t}{\hbar}\right)e_0 + \sin\!\left(\frac{|\mathbf{h}|t}{\hbar}\right)\hat{\mathbf{h}} \in \mathbb{H}_{\mathbb{B}} $$

is a unit real quaternion, whose conjugation is an ordinary spatial rotation of the Bloch sphere, the $SU(2)$ subgroup of $U(2)$.

Reversibility

The map $\Phi_{\tilde{U}}$ is invertible, with inverse $\Phi_{\tilde{U}}^{-1} = \Phi_{\tilde{U}^{*}} = \Phi_{\tilde{U}(-t)}$, again a unitary channel: unitary evolution is reversible.

The converse is the precise statement of the dichotomy. A completely positive, trace-preserving map on $\mathbb{B} \cong M_2(\mathbb{C})$ is invertible, with an inverse that is again completely positive and trace preserving, if and only if it is conjugation by a unitary element; equivalently, a channel preserves the purity of every state if and only if it is a unitary conjugation. In the Kraus language the criterion is transparent: a channel with Kraus rank one has the form $\Phi(\tilde{Q}) = \tilde{K}\tilde{Q}\tilde{K}^{*}$ with $\tilde{K}^{*}\tilde{K} = e_0$, so $\tilde{K}$ is unitary; and a unitary conjugation has Kraus rank one. Hence

$$ \text{reversible} \quad\Longleftrightarrow\quad \text{Kraus rank one} \quad\Longleftrightarrow\quad \Phi = \Phi_{\tilde{U}} \text{ for some unitary } \tilde{U} . $$

The kernel of $\tilde{U} \mapsto \Phi_{\tilde{U}}$ on $\mathbb{M}_+$ is the group of central unitary elements, the unit circle $U(1) \subset \mathbb{C}_{\mathbb{B}}$; unitaries differing by a global phase define the same channel. The effective group is $PU(2) \cong SO(3)$, acting on the Bloch ball by rotations — the biquaternion form of the unobservability of global phase.

The Irreversible Case: General Completely Positive Maps

Sums of Conjugations and Stinespring Dilation

A general quantum channel is a sum of conjugations, $\Phi(\tilde{\rho}) = \sum_l \tilde{K}_l\tilde{\rho}\tilde{K}_l^{*}$ with $\sum_l \tilde{K}_l^{*}\tilde{K}_l = e_0$, involving two or more Kraus operators. But a trace-preserving channel with a single Kraus operator is necessarily unitary, since $\tilde{K}^{*}\tilde{K} = e_0$ is unitarity. It follows that a channel is irreversible precisely when its Kraus rank is at least two. Irreversibility is not a matter of degree in the action of a single element; it is the presence of a sum, and the choice among the terms is what the map forgets.

The information-theoretic picture is that the Kraus operators describe the interaction of the system with an environment that is not observed. This is made precise by the Stinespring dilation: every channel can be written

$$ \Phi(\tilde{\rho}) = \mathrm{Tr}_E\!\left(\tilde{V}\,\tilde{\rho}\,\tilde{V}^{*}\right) $$

for an isometry $\tilde{V}$ into the system tensor an environment, with $\mathrm{Tr}_E$ the partial trace of the companion article on entangled subsystems, and the Kraus operators the environment components of $\tilde{V}$. This is the deep structural statement of the dichotomy: every irreversible process on the system is the reduction of a reversible evolution on a larger system, followed by discarding the environment. The irreversibility is entirely the discarding.

Contraction of the Bloch Ball

Geometrically, a qubit channel is an affine map of the Bloch ball into itself, $\mathbf{r} \mapsto M\mathbf{r} + \mathbf{t}$. The unitary channels are the rotations $M \in SO(3)$, $\mathbf{t} = 0$. Two canonical types are distinguished by whether the origin is fixed:

Unital channels have $\mathbf{t} = 0$ and fix the maximally mixed state, contracting the ball toward the center. The depolarizing channel $\Delta_p(\tilde{\rho}) = (1-p)\tilde{\rho} + p\,e_0/2$, with Bloch map $\mathbf{r} \mapsto (1-p)\mathbf{r}$, is completely positive and trace preserving for $0 \leq p \leq 1$; in the biquaternion algebra it has Kraus operators $\{\sqrt{1-\tfrac{3p}{4}}\,e_0,\ \sqrt{\tfrac{p}{4}}\,e_1,\ \sqrt{\tfrac{p}{4}}\,e_2,\ \sqrt{\tfrac{p}{4}}\,e_3\}$, whose normalization uses $e_k^{*} e_k = e_0$.
Non-unital channels have $\mathbf{t} \neq 0$ and move the maximally mixed state off center. The amplitude-damping channel is the standard example, acting on the Bloch vector as $(r_1, r_2, r_3) \mapsto (\sqrt{1-\gamma}\,r_1,\ \sqrt{1-\gamma}\,r_2,\ \gamma + (1-\gamma)r_3)$, $0 \leq \gamma \leq 1$, contractive for $\gamma > 0$ and the identity only at $\gamma = 0$.

Dephasing, treated next, is unital: it contracts the ball toward an axis rather than its center.

The Lindblad Form

The infinitesimal version of the dichotomy is the Lindblad equation (the Gorini–Kossakowski–Sudarshan–Lindblad form), governing a one-parameter semigroup $\Phi_t$ of channels:

$$ \frac{d\tilde{\rho}}{dt} = -\frac{i}{\hbar}\,[\tilde{H}, \tilde{\rho}] + \sum_k \left(\tilde{L}_k\,\tilde{\rho}\,\tilde{L}_k^{*} - \tfrac{1}{2}\left\{\tilde{L}_k^{*}\tilde{L}_k, \tilde{\rho}\right\}\right), $$

with $\tilde{H} \in \mathbb{M}_+$ Hermitian, $\tilde{L}_k \in \mathbb{B}$ arbitrary, and $\{A, B\} = AB + BA$. The two terms separate exactly along the reversible/irreversible split. The Hamiltonian term is generated by $-i\tilde{H}/\hbar \in \mathbb{M}_-$ and alone would give a reversible group. The dissipator involves arbitrary jump operators $\tilde{L}_k \in \mathbb{B}$; each contributes a completely positive term, and the anticommutator keeps the trace equal to one, since $\mathrm{Tr}(\tilde{L}_k\tilde{\rho}\tilde{L}_k^{*} - \tfrac{1}{2}\{\tilde{L}_k^{*}\tilde{L}_k,\tilde{\rho}\}) = 0$ by cyclicity. The GKSL theorem states that every semigroup of completely positive, trace-preserving maps with a bounded generator has this form. The reversible part of the generator is an inner derivation by an element of $\mathbb{M}_-$ (equivalently, by a Hamiltonian in $\mathbb{M}_+$); the irreversible part is a sum of completely positive dissipators that are not derivations at all.

Dephasing in $\mathbb{M}_+$

Definition and Kraus Form

Dephasing destroys the phase coherence between the two eigenstates of a chosen direction without changing their populations. Let $\hat{\mathbf{n}}$ be a unit pure real quaternion and $\tilde\Pi_+(\hat{\mathbf{n}}) = \tfrac{1}{2}(e_0 + i\hat{\mathbf{n}})$, $\tilde\Pi_-(\hat{\mathbf{n}}) = \tfrac{1}{2}(e_0 - i\hat{\mathbf{n}})$ the complementary idempotents along $\hat{\mathbf{n}}$. For $p \in [0,1]$, the dephasing channel along $\hat{\mathbf{n}}$ is

$$ \Phi^{\mathrm{deph}}_p(\tilde{\rho}) = (1-p)\,\tilde{\rho} + p\left(\tilde\Pi_+\,\tilde{\rho}\,\tilde\Pi_+ + \tilde\Pi_-\,\tilde{\rho}\,\tilde\Pi_-\right). $$

It is completely positive and trace preserving with the three Kraus operators $\tilde{K}_0 = \sqrt{1-p}\,e_0$, $\tilde{K}_1 = \sqrt{p}\,\tilde\Pi_+(\hat{\mathbf{n}})$, $\tilde{K}_2 = \sqrt{p}\,\tilde\Pi_-(\hat{\mathbf{n}})$: because the idempotents are Hermitian, $\tilde{K}_1^{*}\tilde{K}_1 + \tilde{K}_2^{*}\tilde{K}_2 = p(\tilde\Pi_+ + \tilde\Pi_-) = p\,e_0$, and the normalization sums to $(1-p)e_0 + p\,e_0 = e_0$.

The Bloch Vector Formula

Put $\alpha = i\hat{\mathbf{n}}$, so that $\tilde\Pi_\pm = \tfrac{1}{2}(e_0 \pm \alpha)$ with $\alpha^\dagger = \alpha$ and $\alpha^2 = e_0$. A direct expansion gives

$$ \tilde\Pi_+\,\tilde{Q}\,\tilde\Pi_+ + \tilde\Pi_-\,\tilde{Q}\,\tilde\Pi_- = \tfrac{1}{2}\left(\tilde{Q} + \alpha\,\tilde{Q}\,\alpha\right), $$

since the cross terms cancel between the two projectors. For a state $\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$, the elementary products are $\alpha\,e_0\,\alpha = \alpha^2 = e_0$ and $\alpha\,(i\mathbf{r})\,\alpha = -i\,\hat{\mathbf{n}}\,\mathbf{r}\,\hat{\mathbf{n}} = -i(\mathbf{r} - 2(\hat{\mathbf{n}}\cdot\mathbf{r})\hat{\mathbf{n}})$, using $\hat{\mathbf{n}}\,\mathbf{r}\,\hat{\mathbf{n}} = \mathbf{r} - 2(\hat{\mathbf{n}}\cdot\mathbf{r})\hat{\mathbf{n}}$. Combining,

$$ \tilde\Pi_+\tilde{\rho}\tilde\Pi_+ + \tilde\Pi_-\tilde{\rho}\tilde\Pi_- = \tfrac{1}{2}\left(e_0 + i(\hat{\mathbf{n}}\cdot\mathbf{r})\,\hat{\mathbf{n}}\right), $$

so the fully dephasing map projects the Bloch vector onto the $\hat{\mathbf{n}}$ axis. Including the mixture with weight $1-p$,

$$ \boxed{\;\Phi^{\mathrm{deph}}_p(\tilde{\rho}) = \tfrac{1}{2}\left(e_0 + i\,\mathbf{r}'\right), \qquad \mathbf{r}' = (1-p)\,\mathbf{r} + p\,(\hat{\mathbf{n}}\cdot\mathbf{r})\,\hat{\mathbf{n}}.\;} $$

The transverse components of the Bloch vector are multiplied by $1-p$, while the longitudinal component is untouched. In the eigenbasis of $\hat{\mathbf{n}}$, the off-diagonal entries of the density matrix — the coherences — are scaled by $1-p$, while the diagonal entries — the populations — are unchanged. This is the standard phase-damping channel, expressed entirely in the objects of $\mathbb{M}_+$.

Limiting Cases, Measurement, and Irreversibility

At $p = 0$ the channel is the identity, a reversible unitary evolution. At $p = 1$ it is full dephasing, $\Phi^{\mathrm{deph}}_1(\tilde{\rho}) = \tfrac{1}{2}(e_0 + i(\hat{\mathbf{n}}\cdot\mathbf{r})\hat{\mathbf{n}})$, whose image is the diameter of the Bloch ball along $\hat{\mathbf{n}}$; it is idempotent, $(\Phi^{\mathrm{deph}}_1)^2 = \Phi^{\mathrm{deph}}_1$, and non-unitary. Full dephasing is measure-and-forget: measuring $\alpha = i\hat{\mathbf{n}}$ and discarding the outcome leaves the average state

$$ p_+\,\tilde\Pi_+ + p_-\,\tilde\Pi_- = \Phi^{\mathrm{deph}}_1(\tilde{\rho}), \qquad p_\pm = \mathrm{Tr}(\tilde\Pi_\pm\tilde{\rho}) = \tfrac{1}{2}\left(1 \pm \hat{\mathbf{n}}\cdot\mathbf{r}\right), $$

as a direct computation confirms. The three-Kraus form shows that $\Phi^{\mathrm{deph}}_p$ is a convex mixture, with weight $p$, of this measurement channel and the identity; since the identity is reversible and full dephasing is not, the channel set is convex with its reversible elements among its extreme points.

Write $r_\parallel = \hat{\mathbf{n}}\cdot\mathbf{r}$ and $r_\perp^2 = |\mathbf{r}|^2 - r_\parallel^2$. Under dephasing,

$$ |\mathbf{r}'|^2 = (1-p)^2\,r_\perp^2 + r_\parallel^2 \leq |\mathbf{r}|^2 , $$

with equality only if $p = 0$, or the state is already on the axis, or $r_\perp = 0$. The purity $\mathrm{Tr}(\tilde{\rho}'^2) = \tfrac{1}{2}(1 + |\mathbf{r}'|^2)$ therefore strictly decreases for any off-axis state when $0 < p < 1$, and the von Neumann entropy

$$ S(\tilde{\rho}) = -\lambda_+\log\lambda_+ - \lambda_-\log\lambda_-, \qquad \lambda_\pm = \tfrac{1}{2}\left(1 \pm |\mathbf{r}|\right), $$

strictly increases, since $S$ is a decreasing function of $|\mathbf{r}|$. A pure state on the equator is driven into the interior of the Bloch ball; only the two pure states on the $\hat{\mathbf{n}}$ axis are fixed. No channel can restore the lost coherence, because the inverse map is not completely positive. Unitary evolution, by contrast, preserves $|\mathbf{r}|$ exactly, and with it the purity and the spectrum of every state.

The Dichotomy, the Two Subspaces, and the Informational Sector

The Algebraic Statement of the Split

Collecting the results, the dichotomy has the following exact algebraic form.

States and observables live in $\mathbb{M}_+$, the Hermitian subspace, and are paired by $\mathrm{Tr}(\tilde{P}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{P}\tilde{H})$, the Born rule.
Reversible processes are the inner automorphisms $\tilde{\rho} \mapsto \tilde{U}\tilde{\rho}\tilde{U}^{*}$ by matrix-unitary biquaternions $\tilde{U} \in U(2)$. They are exactly the channels of Kraus rank one, and they preserve purity. Their one-parameter subgroups have generators $-i\tilde{H}/\hbar$ lying in the complementary subspace $\mathbb{M}_-$: the states live in $\mathbb{M}_+$, but their reversible motion is generated from $\mathbb{M}_-$.
Irreversible processes are the remaining channels: completely positive, trace-preserving maps of Kraus rank at least two. Their Kraus operators are arbitrary elements of $\mathbb{B} = \mathbb{M}_+ \oplus \mathbb{M}_-$, and their generators contain, besides a derivation by an $\mathbb{M}_-$ element, completely positive dissipators that are not derivations.

The dichotomy is therefore not a dichotomy between the two subspaces. It is a dichotomy within the set of maps on $\mathbb{B}$, and the two subspaces appear asymmetrically. The Hermitian subspace $\mathbb{M}_+$ is the arena: it contains the states acted upon and the observables labelling outcomes. The anti-Hermitian subspace $\mathbb{M}_-$ is the source of reversibility: it is the Lie algebra whose exponential is the group of reversible maps, and its elements are the Hamiltonians of motion. The irreversible maps are those needing more than a single element of this Lie algebra; by Stinespring dilation they are the reversible evolutions of a larger system with information discarded in the trace. Dephasing cannot be written as conjugation by any element at all — its three Kraus operators have no common unitary representative — and this is why its effect cannot be undone.

The Informational-Sector Reading

Under the interpretive hypothesis of the companion article, that $\mathbb{M}_+$ is the arena of an informational sector, this article is a theory of the operations of that sector. The states of $\mathbb{M}_+$ are informational configurations; a channel transforms them; the reversible channels are the deterministic, information-preserving transformations, and the irreversible channels are those in which information is lost to an environment the sector does not describe. Dephasing is the canonical such loss: it is decoherence, the destruction of phase relations, expressed as a map on $\mathbb{M}_+$.

In this reading, reversibility is the regime in which the informational sector is closed and its evolution is generated by the material Hamiltonian; irreversibility is the regime in which the sector is open and its states are entangled with degrees of freedom outside the description, with Stinespring dilation making the discarded environment precise. Nothing here requires the material sector $\mathbb{M}_-$ to be "accessed" by the channel: the generator of the reversible part of the motion is an $\mathbb{M}_-$ element, but irreversibility itself is a property of the map on $\mathbb{M}_+$. As elsewhere in this series, the interpretive reading is a hypothesis and not a result; the channel formalism above is standard quantum information theory, and whether the Hermitian subspace is a distinct physical sector is the open question identified in The Hermitian Subspace $\mathbb{M}_+$ as the Informational Sector.

What the Reformulation Does and Does Not Claim

The framework does provide a natural algebraic home for the standard theory of quantum operations. Complete positivity is a property of maps on $\mathbb{B} \cong M_2(\mathbb{C})$; trace preservation is the statement that the dual fixes $e_0$; the Kraus operators are elements of $\mathbb{B}$; and the reversible/irreversible split is the split between Kraus rank one and higher rank, equivalently between inner automorphisms and general completely positive maps. The trace formula $\mathrm{Tr}(\tilde{P}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{P}\tilde{H})$ makes the state/observable duality and the Schrödinger/Heisenberg duality of channels the same operation.

The framework does not claim to explain why a given physical process is irreversible, to derive the environment a channel traces over, or to resolve the measurement problem. The reformulation changes the vocabulary of the dichotomy, not its physics; the selection problem — why one outcome, or one channel, rather than another — is untouched.

Open Questions

1. Empirical content. As for the rest of the framework, the central question is whether the biquaternion formulation predicts anything standard quantum information theory does not. A reformulation of quantum channels is, by itself, empirically empty.

2. The geometry of the channel set. The general qubit channel is an affine contraction of the Bloch ball with a specific positivity structure. Which affine contractions are completely positive, and how does the biquaternion description of the Choi matrix read in terms of $\mathbb{C}_{\mathbb{B}}$, $\mathbb{H}_{\mathbb{B}}$, and $\mathbb{M}_\pm$? A complete classification in the biquaternion language has not been carried out here.

3. Departures from complete positivity. The framework is built on completely positive maps. Physically motivated modifications sometimes relax complete positivity to positivity while retaining trace preservation. How would such a relaxation read in the biquaternion algebra, and is there a natural algebraic home for it?

4. Entropy and the environment. The increase of entropy under dephasing is proved state by state; is there a monotonicity theorem for a general biquaternion channel, in the form of data-processing inequalities for a suitable entropy functional on $\mathbb{M}_+$? And is there a dynamics of the "informational environment" of Stinespring dilation, or is it purely formal?

5. The relation to measurement. Dephasing is measure-and-forget for the observable $i\hat{\mathbf{n}}$; more generally every channel is a generalized measurement followed by a state transformation. Whether this unification has consequences beyond the standard one is open.

Summary

A general state map on the informational sector is a linear map on $\mathbb{M}_+$, extended complex-linearly to $\mathbb{B} = \mathbb{M}_+ \oplus \mathbb{M}_-$ and required to be completely positive and trace preserving. Complete positivity guarantees that the map remains positive when the system is entangled with a reference system; trace preservation guarantees that it sends states to states. Every such map has a Kraus representation $\Phi(\tilde{\rho}) = \sum_l \tilde{K}_l\tilde{\rho}\tilde{K}_l^{*}$ with $\sum_l \tilde{K}_l^{*}\tilde{K}_l = e_0$, and the minimal number of Kraus operators is the Choi rank.

The reversible/irreversible dichotomy is the dichotomy of Kraus rank. A channel is reversible if and only if it has Kraus rank one, which happens if and only if it is conjugation by a unitary biquaternion, $\tilde{\rho} \mapsto \tilde{U}\tilde{\rho}\tilde{U}^{*}$ with $\tilde{U}\tilde{U}^{*} = e_0$. This is exactly the class of inner automorphisms of the algebra, and exactly the class of channels that preserve purity. The one-parameter subgroups are generated by Hamiltonians $\tilde{H} \in \mathbb{M}_+$ through $\tilde{U}(t) = \exp(-i\tilde{H}t/\hbar)$, whose generators $-i\tilde{H}/\hbar$ lie in the complementary subspace $\mathbb{M}_-$. A channel is irreversible if and only if its Kraus rank is at least two; the general irreversible channel is a sum of conjugations, and by Stinespring dilation it is the reduction of a reversible evolution on a larger system with the environment discarded.

Dephasing is the canonical irreversible channel. Along a direction $\hat{\mathbf{n}}$, it maps the Bloch vector by $\mathbf{r} \mapsto (1-p)\mathbf{r} + p(\hat{\mathbf{n}}\cdot\mathbf{r})\hat{\mathbf{n}}$, scaling the coherences by $1-p$ and leaving the populations fixed. At $p=1$ it is the idempotent "measure-and-forget" channel of the observable $i\hat{\mathbf{n}}$, whose image is the diameter of the Bloch ball. It strictly decreases purity and strictly increases entropy for off-axis states, and it cannot be inverted by any channel.

The relation of the split to the two subspaces is asymmetric. States and observables live in $\mathbb{M}_+$; the generators of reversible evolution live in $\mathbb{M}_-$; and the irreversible maps are those that are not inner automorphisms, requiring Kraus operators that are general elements of $\mathbb{B} = \mathbb{M}_+ \oplus \mathbb{M}_-$. The dichotomy is thus an algebraic fact about the space of maps on the biquaternion algebra rather than an additional postulate — but, as the companion articles emphasize, it is a reformulation of standard quantum theory, not a new physical theory.

Summary of Notation

Symbol	Meaning
$\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra, $\cong M_2(\mathbb{C})$
$\mathbb{M}_+$	Hermitian subspace (states, observables)
$\mathbb{M}_-$	Anti-Hermitian subspace (generators of reversible evolution)
$\mathbb{C}_{\mathbb{B}}$	Complex subspace (center); trace part of a Hamiltonian
$\mathbb{H}_{\mathbb{B}}$	Real-quaternion subspace (rotation rotors)
$e_0 = 1, e_1, e_2, e_3$	Quaternion basis, $e_k^2 = -e_0$
$i$	Scalar imaginary, $i^2 = -1$
$\tilde{\rho} = \tfrac{1}{2}(e_0 + i\mathbf{r})$	State of the informational sector
$\tilde{H} = h_0 e_0 + i\mathbf{h}$	Observable / Hamiltonian
$\tilde\Pi_\pm(\hat{\mathbf{n}}) = \tfrac{1}{2}(e_0 \pm i\hat{\mathbf{n}})$	Idempotents (pure states)
$\mathrm{Tr}(\tilde{P}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{P}\tilde{H})$	Trace formula (Born rule)
$\Phi(\tilde{\rho}) = \sum_l \tilde{K}_l\tilde{\rho}\tilde{K}_l^{*}$	Kraus representation of a channel
$\sum_l \tilde{K}_l^{*}\tilde{K}_l = e_0$	Trace-preservation condition
$\Phi^{}(\tilde{H}) = \sum_l \tilde{K}_l^{}\tilde{H}\tilde{K}_l$	Dual (Heisenberg-picture) map
$\tilde{U}(t) = \exp(-i\tilde{H}t/\hbar)$	One-parameter unitary group
$-i\tilde{H}/\hbar \in \mathbb{M}_-$	Generator of reversible evolution
$\Phi^{\mathrm{deph}}_p$	Dephasing channel along $\hat{\mathbf{n}}$
$\mathbf{r} \mapsto (1-p)\mathbf{r} + p(\hat{\mathbf{n}}\cdot\mathbf{r})\hat{\mathbf{n}}$	Dephasing action on the Bloch vector
$J(\Phi) = \sum_{jk}\tilde{E}_{jk}\otimes\Phi(\tilde{E}_{jk})$	Choi matrix; Kraus rank $=\mathrm{rank}\,J(\Phi)$