Unitarity and Quantum Channels

Black-hole evolution is often called “unitary” while the equation being discussed is actually a reduced quantum channel. The distinction is exact. A unitary maps one complete closed system reversibly to another of the same dimension; an isometry preserves the input inside a possibly larger output; a completely positive trace-preserving map describes the retained output after an environment or complementary algebra has been discarded.

Required background. Quantum Information supplies linear maps, complete positivity, Kraus operators, Stinespring dilation, and channel distance used in the proof.

Helpful background. Semiclassical Gravity supplies the evaporation regime; Quantum Fields in Curved Spacetime supplies outgoing and interior mode systems; Causality and Global Structure supplies the slices and asymptotic regions on which inputs and outputs are declared.

Three kinds of evolution

Let $X$ be the input system and $Y$ the complete output system. A unitary operator $U:\mathcal H_X\to\mathcal H_Y$ satisfies

$$U^\dagger U=I_X, \qquad UU^\dagger=I_Y.$$

Both conditions require equal dimensions and give a reversible channel $\mathcal U(\rho)=U\rho U^\dagger$. An isometry $V:\mathcal H_X\to\mathcal H_Y$ satisfies only $V^\dagger V=I_X$; it preserves inner products but may occupy a proper output subspace.

If an observer retains an output system $R$ and discards $E$, the reduced channel is

$$\mathcal N_{X\to R}(\rho) =\operatorname{Tr}_E(V\rho V^\dagger).$$

It is completely positive and trace preserving (CPTP). Equivalently,

$$\mathcal N(\rho)=\sum_aK_a\rho K_a^\dagger, \qquad \sum_aK_a^\dagger K_a=I_X.$$

Complete positivity ensures that $\mathcal N\otimes\mathrm{id}_Q$ remains positive when the input is entangled with any reference $Q$. Trace preservation is probability conservation for every input. Neither property implies reversibility.

Stinespring theorem and physical meaning

Every finite-dimensional CPTP channel has an isometric dilation: there exist $E$ and $V:X\to R\otimes E$ such that the partial-trace formula above holds. Conversely, tracing an isometry always produces a CPTP channel. Watrous develops these equivalences and the criteria for reversibility in Watrous 2018, Chapters 2–3.

One may further realize $V$ by adjoining an environment in a fixed state and applying a unitary on a larger space. The environment is mathematical until a physical theory identifies what it is. In black-hole applications it might denote interior partners, a remaining black hole, uncollected radiation, a remnant, a baby-universe sector, or auxiliary degrees introduced only by an effective description. A dilation does not prove that the corresponding sector survives in quantum gravity.

The input and output algebras must also be fixed. A channel from collapse data to all asymptotic states tests a complete S-matrix. A channel from collapse data to radiation collected before time $t$ is necessarily reduced. Calling the latter nonunitary is correct as a channel property but does not establish fundamental nonunitarity of the former.

Approximate statements need a channel norm, not just agreement on one state. The diamond distance

$$\lVert\mathcal N-\mathcal M\rVert_\diamond =\sup_{Q,\rho_{XQ}} \left\lVert (\mathcal N\otimes\mathrm{id}_Q)(\rho_{XQ}) -(\mathcal M\otimes\mathrm{id}_Q)(\rho_{XQ}) \right\rVert_1$$

tests the worst input including an entangled reference. A recovery map $\mathcal R$ supports approximate unitarity on a code space only when $\mathcal R\circ\mathcal N$ is close to the identity there in a stated norm. Agreement on thermal one-point functions can coexist with a large diamond distance because phases or high-order correlations are missing. Conversely, a channel can be reversible on a small code subspace while irreversible on the full microscopic Hilbert space. The domain is therefore part of every accuracy claim.

For evaporation this distinguishes three targets: preserving one semiclassical state, preserving every state in a low-energy code subspace, and preserving all black-hole microstates. Success at the first target does not imply the second or third. A reference-entangled test is especially discriminating because it detects information lost without changing the marginal probabilities of a chosen basis.

Purity-preservation proof

Statement. Under the hypotheses that the complete system evolves by an isometry $V$ and no output factor is discarded, every pure input remains pure and all input inner products are preserved.

Proof. For $\rho=\lvert\psi\rangle\langle\psi\rvert$,

$$V\rho V^\dagger =\lvert\phi\rangle\langle\phi\rvert, \qquad \lvert\phi\rangle=V\lvert\psi\rangle.$$

Normalization follows from $\langle\phi\vert\phi\rangle=\langle\psi\vert V^\dagger V\vert\psi\rangle=1$, and the output is rank one. More generally,

$$\langle V\psi\vert V\chi\rangle =\langle\psi\vert V^\dagger V\vert\chi\rangle =\langle\psi\vert\chi\rangle.$$

If input and complete output have the same dimension, the isometry is unitary. If $E$ is traced out, the reduced state is pure only when $R$ and $E$ are unentangled for that input. Hence global unitary evolution guarantees global purity but does not guarantee purity of a reduced subsystem.

The converse needs care. A constant channel can send every input to one fixed pure state while destroying all distinctions, so “pure outputs” alone do not prove unitarity. A CPTP map with a CPTP inverse on the full state space must preserve reference entanglement and distinguishability; for equal full matrix algebras this forces a unitary channel. Harlow applies this complete-versus-reduced distinction to black-hole evaporation in Harlow 2016, §§ 5.1–5.4.

Stinespring counterexample

Define the isometry

$$V\lvert0\rangle_X=\lvert0\rangle_R\lvert0\rangle_E, \qquad V\lvert1\rangle_X=\lvert1\rangle_R\lvert1\rangle_E.$$

For $\lvert+\rangle_X$, the complete output is the pure Bell state $\lvert\Phi^+\rangle_{RE}$. The reduced output is

$$\mathcal N(\lvert+\rangle\langle+\rvert)=\frac{I_R}{2}.$$

The channel preserves trace and positivity but removes the off-diagonal entries in the computational basis. Inputs $\lvert+\rangle$ and $\lvert-\rangle$ become the same reduced state, so no inverse on $R$ exists. Access to $RE$ restores the phase and permits reversal. This counterexample directly shows why enlarged unitarity does not imply reduced-channel invertibility.

Black-hole boundary and checks

A valid evaporation claim should state $X$, $R$, $E$, their observable algebras, and whether $E$ is physical or only a dilation. It should verify $\operatorname{Tr}\mathcal N(\rho)=1$, positivity on reference-entangled inputs, and recovery fidelity for a basis plus superpositions. Checking only thermal occupation numbers is insufficient.

The theorem is finite-dimensional. Continuum and gravitational algebras require regulators or algebraic channel formulations, and exact inside–outside factorization may fail. Generic channel theory remains with Quantum Information, mode construction with Quantum Fields in Curved Spacetime, and explicit string or holographic realizations elsewhere.

Exercise

The amplitude-damping channel has $K_0=\lvert0\rangle\langle0\rvert+\sqrt{1-\gamma}\lvert1\rangle\langle1\rvert$ and $K_1=\sqrt\gamma\lvert0\rangle\langle1\rvert$. Verify trace preservation and explain why it is not unitary when $0\lt\gamma\leq1$.

Solution

Direct multiplication gives $K_0^\dagger K_0+K_1^\dagger K_1=\lvert0\rangle\langle0\rvert+(1-\gamma+\gamma)\lvert1\rangle\langle1\rvert=I$, so the channel is trace preserving. For positive $\gamma$, the excited state becomes a mixture or decays to the ground state, and distinguishability from other inputs contracts. Its Stinespring environment records whether decay occurred. The enlarged evolution can be unitary, but the output system alone has no CPTP inverse.

References

• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.
• Watrous, John. 2018. The Theory of Quantum Information. Cambridge: Cambridge University Press. DOI.