Gravitational redshift as a quantum channel: modeling the effects of gravitational redshift in quantum optics

Technical Summary: Gravitational Redshift as a Quantum Channel

Problem Statement
The gravitational frequency shift of light is well-established in classical electromagnetism and general relativity. However, its description within quantum theory, particularly for single photons and multi-mode systems, has remained incomplete. Recent literature proposed modeling gravitational redshift as an effective linear multi-mode mixer (mmm) acting on a finite set of ladder operators associated with specific frequency spectra. While these models yielded plausible predictions for small redshifts, a subsequent study identified a critical inconsistency: the transformation matrices derived in the mmm model were found to be non-unitary, contradicting the expectation that the underlying physical evolution should be unitary. This lack of a firm mathematical basis hindered the rigorous application of standard quantum information-theoretic techniques to gravitational redshift.

Methodology and Theoretical Framework
The authors analyze the problem within the framework of quantum field theory in stationary space-times. They define the gravitational redshift as a unitary operator $U_\chi$ acting on the infinite-dimensional space of square-integrable spectral functions $L^2(\mathbb{R}^+)$. The parameter $\chi = \sqrt{\zeta_a/\zeta_b}$ relates the lapse functions of two stationary observers.

The core of the investigation involves a rigorous proof demonstrating that for any $\chi \neq 1$, the operator $U_\chi$ possesses no finite-dimensional invariant subspaces consisting of "well-behaved" functions (specifically, those with finite moments). This result establishes a "no-go theorem" for the original mmm model: it is mathematically impossible to represent the action of $U_\chi$ on a finite set of modes using a unitary matrix of the form $(f_I, U_\chi f_J)$, because the subspace spanned by these modes is not invariant under the redshift operation.

To resolve this, the authors propose two complementary approaches:

1. Compression Approach (Field-Theoretic): They demonstrate that the non-unitary matrix $u_\chi$ previously calculated in the literature is not an error, but rather a compression of the unitary operator $U_\chi$ onto a finite subspace. Specifically, $u_\chi = P_V U_\chi |_V$, where $P_V$ is the orthogonal projector onto the subspace $V$ spanned by the modes of interest. This approach avoids matrix representations entirely for the full evolution, relying instead on the direct action of $U_\chi$ on spectral functions to compute transition amplitudes.
2. Dilation Approach (Quantum Information): To utilize standard quantum information techniques which require unitary evolution, the authors construct a unitary dilation of the non-unitary compression $u_\chi$. Using a construction attributed to Halmos, they extend the system by adding auxiliary degrees of freedom (an enlarged space $V \oplus V$) to create a unitary operator $\Upsilon_\chi$. This allows the gravitational redshift to be modeled as a quantum channel acting on the original modes by "tracing out" the auxiliary modes.

Key Results

• Resolution of Inconsistency: The paper proves that the non-unitarity of the mmm model matrices is a necessary consequence of projecting an infinite-dimensional unitary transformation onto a finite-dimensional subspace. It does not imply photon loss or a violation of quantum mechanics; rather, it reflects the leakage of information into modes outside the chosen finite set.
• Gaussian Quantum Channel: By applying the unitary dilation $\Upsilon_\chi$ to Gaussian states, the authors derive an explicit form for the gravitational redshift as a Gaussian quantum channel. The transformation of the covariance matrix $\Sigma$ of the system modes is given by $\Sigma \mapsto X_\chi \Sigma X_\chi^\dagger + Y_\chi$, where $X_\chi$ is related to the compression matrix $u_\chi$ and $Y_\chi$ represents a noise term arising from the tracing out of auxiliary modes.
• Quantification of Non-Unitarity: The authors introduce a measure called "channel nonunitarity" ($\eta$), defined via the normalized trace of the noise matrix $Y_\chi$. They show that $\eta > 0$ for any non-trivial redshift ($\chi \neq 1$), confirming the channel is non-unitary.
• Fidelity and Squeezing: Analysis of the fidelity between initial and final states reveals that for initially pure Gaussian states, the fidelity remains unity ($F=1$) unless the state is squeezed. This indicates that the observational signatures of gravitational redshift on fidelity depend critically on the quantum properties (specifically squeezing) of the light.
• Roundtrip Protocols: The model is applied to a roundtrip communication protocol (Alice $\to$ Bob $\to$ Alice). The authors show that if Bob reflects the signal using a filter that only passes the agreed-upon modes (absorbing orthogonal components), the roundtrip channel is strictly more non-unitary than the one-way channel due to the cumulative effect of the projection.

Significance and Claims
The paper claims to provide a complete and consistent mathematical foundation for modeling gravitational redshift in quantum optics, correcting the inconsistencies of previous mmm models.

• Theoretical Correction: It clarifies that the original mmm predictions for transition amplitudes involving only the "main" modes were correct, but the assumption that these could be embedded in a larger unitary matrix using the same inner-product logic was flawed.
• Operational Utility: The dilation approach provides a practical recipe for characterizing the redshift as a quantum channel using standard covariance matrix techniques. This enables the calculation of channel capacities and the design of experimental protocols involving local measurements only, without needing to access the infinite set of "perpendicular" modes.
• Generality: Unlike previous works restricted to weak fields, this analysis holds for arbitrarily strong gravitational frequency shifts.
• Photon Number Conservation: The analysis confirms that gravitational redshift in stationary space-times preserves the photon number, resolving questions raised in prior literature about potential photon loss due to non-unitary matrices.

The authors conclude that their construction of finite-dimensional unitary matrices (via dilation) paves the way for further studies on relativistic effects in satellite-based quantum information protocols, specifically regarding channel capacity and its dependence on both gravitational and quantum features.