Pure-to-Mixed Evolution

A pure-to-mixed claim is meaningful only after the complete system and the retained observables have been fixed. For a genuinely closed system, mapping a rank-one density operator to a mixed operator is fundamental nonunitarity. For a subsystem, the same change is ordinary entanglement with degrees of freedom that were traced out. Hawking’s proposal concerns the former possibility, not merely the thermality of an exterior marginal.

Required background. Quantum Information supplies density operators, partial traces, completely positive trace-preserving maps, and entropy used below.

Helpful background. Semiclassical Gravity supplies the evaporation approximation and backreaction boundary; Quantum Fields in Curved Spacetime supplies the Hawking state on a prescribed background; Causality and Global Structure supplies the late-slice and asymptotic-region distinctions.

Closed evolution and reduced channels

Let $X$ be the declared complete quantum system. Closed unitary evolution has the form

$$\rho_X^{\mathrm{out}}=S\rho_X^{\mathrm{in}}S^\dagger, \qquad S^\dagger S=SS^\dagger=I.$$

Unitary conjugation preserves the eigenvalues of $\rho_X$. It therefore preserves purity $\operatorname{Tr}\rho_X^2$, von Neumann entropy $S(\rho_X)=-\operatorname{Tr}(\rho_X\ln\rho_X)$, trace, and positivity. If $\rho_X^{\mathrm{in}}=\lvert\psi\rangle\langle\psi\rvert$, then $\rho_X^{\mathrm{out}}$ is again rank one.

A general quantum channel $\Phi$ is linear, completely positive, and trace preserving (CPTP):

$$\rho_X^{\mathrm{out}}=\Phi(\rho_X^{\mathrm{in}}) =\sum_a K_a\rho_X^{\mathrm{in}}K_a^\dagger, \qquad \sum_aK_a^\dagger K_a=I.$$

Such a channel can send a pure state to a mixed state. That fact alone says nothing about whether a larger system evolved unitarily. By Stinespring dilation, the channel can arise from an environment $E$ in a fixed state,

$$\Phi(\rho_X)=\operatorname{Tr}_E\!\left[V(\rho_X\otimes\lvert0\rangle\langle0\rvert_E)V^\dagger\right].$$

The complete system $XE$ remains pure; only the retained system $X$ becomes mixed. Fundamental nonunitarity is the stronger assertion that no physically included environment or complementary algebra restores an invertible unitary description.

Hawking’s complete-system proposal

In the leading semiclassical picture, collapse produces Hawking quanta at future null infinity while their partners enter a region ending at a singularity. If the proposed final system contains only asymptotic exterior radiation, the interior partners are absent from the final Hilbert space. Hawking represented the resulting evolution by a “superscattering” map on density operators rather than by an ordinary S-matrix Hawking 1976, §§ II–IV. In modern notation one may write

$$\rho_{\mathrm{out}}=\mathcal S_{\mathrm{sup}}(\rho_{\mathrm{in}}),$$

with $\mathcal S_{\mathrm{sup}}$ not assumed to factor as $S(\cdot)S^\dagger$. The information problem is then: are the discarded interior degrees of freedom truly absent from the complete quantum theory, or has the semiclassical description omitted correlations carried by a microscopic sector?

This formulation needs three declarations. The initial algebra must include every degree of freedom whose purity is asserted. The final algebra must say whether a remnant, baby universe, long-lived interior, soft sector, or gravitational boundary degree of freedom is included. The approximation must say that local quantum fields are treated semiclassically and that corrections nonperturbative in the black-hole entropy are omitted. Harlow emphasizes that the Hawking calculation establishes a semiclassical reduced evolution, while the claim of exact information destruction adds assumptions about the complete theory Harlow 2016, § 4.6.

An explicit discriminator

Consider a system qubit $R$ and an environment qubit $I$. The isometry

$$\lvert0\rangle_R\mapsto\lvert0\rangle_R\lvert0\rangle_I, \qquad \lvert1\rangle_R\mapsto\lvert1\rangle_R\lvert1\rangle_I$$

sends $\lvert+\rangle_R$ to a Bell state. The complete density operator has purity one and entropy zero, whereas

$$\rho_R=\operatorname{Tr}_I\lvert\Phi^+\rangle\langle\Phi^+\rvert=\frac{I_R}{2}, \qquad S(\rho_R)=\ln2.$$

Trace and positivity are preserved in both descriptions. The decisive difference is reversibility: access to $RI$ permits the unitary to be inverted, while the reduced dephasing channel on $R$ has erased its off-diagonal entries and has no CPTP inverse on all states.

Two limits sharpen the point. If $R$ never entangles with $I$, the reduced state remains pure; thermality is therefore not automatic. If $I$ is declared part of the final system, the apparent pure-to-mixed transition disappears. Conversely, replacing every input by $I_R/2$ is trace preserving and positive but destroys even classical distinctions between orthogonal inputs. Interpreting that replacement as fundamental evolution is a much stronger claim than tracing over a specified partner.

What this formulation does not settle

The argument isolates the logical alternatives; it does not decide which degrees of freedom quantum gravity contains. Generic channel theory belongs to Quantum Information, and the fixed-background particle-production calculation belongs to Quantum Fields in Curved Spacetime. Stable thermodynamics is treated in Semiclassical Gravity, while explicit string or holographic microphysics is separate. Here the essential conclusion is conditional: once the same complete system is used at both ends, pure-to-mixed evolution rules out unitary conjugation; if the system has changed by a partial trace, it does not.

Exercise

Let $\Phi_p(\rho)=(1-p)\rho+pZ\rho Z$ with $0\leq p\leq1$. Determine when $\Phi_p$ sends $\lvert+\rangle$ to a mixed state, and explain whether that demonstrates fundamental nonunitarity.

Solution

In the $X$ basis, the output has eigenvalues $1-p$ and $p$, so it is mixed for $0\lt p\lt1$ and pure at the endpoints. Its trace is one and its eigenvalues are nonnegative. The channel can be implemented by coherently controlling $I$ and $Z$ with an environment and then tracing that environment, so the mixing does not demonstrate fundamental nonunitarity. It would do so only if the qubit were the declared complete system and no dilation sector were physically allowed.

References

• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.
• Hawking, Stephen W. 1976. “Breakdown of Predictability in Gravitational Collapse.” Physical Review D 14: 2460–2473. DOI.