Imaginary pseudo entropy encodes temporal orientation

Technical Summary: Imaginary Pseudo Entropy Encodes Temporal Orientation

Problem Statement
Pseudo entropy, defined via the transition matrix between two non-orthogonal quantum states, is generally a complex quantity. While its real part has been studied in various contexts (holographic, thermal, spectral), the operational meaning of its imaginary part has remained elusive. Specifically, the imaginary part lacks a bounded physical interpretation. This is particularly problematic near small state overlaps, where normalization factors can parametrically amplify the pseudo entropy (analogous to weak-value amplification), potentially obscuring the physical significance of the phase. Furthermore, while exchanging initial and final states reverses the sign of the imaginary part (a "reversal of transition orientation"), it was unclear how much information regarding this orientation could be physically extracted and how much is lost under coarse graining, distinct from thermodynamic entropy production.

Methodology
The authors propose a calibrated replica interferometer to operationalize the imaginary part of the pseudo-Rényi entropy. The protocol involves:

1. Replica Construction: Preparing $n$ identical copies of a bipartite system.
2. Cyclic Permutation: Applying a unitary operator $V^{(n)}_A$ that cyclically permutes the subsystem $A$ across the $n$ copies.
3. Interferometric Setup: Using an ancilla qubit to control a superposition of two branches: one containing the initial state $|\Psi_0^{\otimes n}\rangle$ and the other containing the time-evolved, permuted state $V^{(n)}_A |\Psi_t^{\otimes n}\rangle$.
4. Calibration: Performing a separate one-copy measurement to determine the phase of the total overlap $\chi(t) = \langle \Psi_0 | \Psi_t \rangle$. This allows for a compensating rotation of the ancilla by $-n\alpha$ (where $\alpha = \arg \chi$), removing the normalization phase.
5. Readout: Measuring the ancilla in the $Y$-basis. The forward and backward transition orientations (swapping bra and ket) result in ancilla density matrices with opposite $Y$-components.

Key Contributions and Results

• Operational Meaning of the Imaginary Part: The paper establishes that the imaginary part of the pseudo-Rényi entropy, combined with the replica visibility, exactly determines the trace distance ($D_n$) between the forward and backward ancilla states.

  • The calibrated amplitude is $A_n^F = V_n e^{i\theta_n}$, where $V_n$ is the visibility (magnitude of the overlap) and $\theta_n = (1-n)\text{Im} S_{A,n}^F$ is the pseudo-Rényi phase.
  • The trace distance is given by $D_n = V_n |\sin \theta_n|$.
  • This distance directly yields the Helstrom-optimal single-shot success probability ($P^*_{succ} = \frac{1}{2}(1 + D_n)$) for distinguishing the forward from the backward transition orientation via an ancilla $Y$-measurement.

• Boundedness and Amplification: The authors resolve the tension between pseudo-entropy amplification and probability bounds. While the normalized pseudo entropy $S_{A,n}^F$ can become large due to small overlaps ($\chi \to 0$), the physical coherence $V_n$ (the visibility) vanishes in this limit. Consequently, the distinguishability $D_n$ remains bounded between 0 and 1, ensuring the imaginary part represents a measurable, bounded record of orientation rather than an unphysical divergence.

• Microscopic Origin (Short-Time Limit): For short times, the initial rate of the distinguishability signal is derived.

  • For integer $n$, the signal is a replica covariance between a subsystem swap operator and the Hamiltonian.
  • In the von Neumann limit ($n \to 1$), the initial rate of the imaginary pseudo entropy is shown to be proportional to the symmetrized covariance between the modular Hamiltonian ($K_A = -\log \rho_A$) and the physical Hamiltonian ($H$). This reproduces and extends previous findings [27], identifying this covariance as the normalized initial rate of optimal orientation distinguishability.

• Coarse Graining and Irreversibility: The paper analyzes the loss of orientation information under a common quantum channel (coarse graining).

  • The quantum relative entropy between the forward and backward ancilla states quantifies the orientation information.
  • By the monotonicity of relative entropy, coarse graining can only decrease this information.
  • The loss is zero if and only if the Petz recovery map can perfectly reconstruct the original ancilla states from the coarse-grained outputs.
  • Crucially, the authors distinguish this from thermodynamic irreversibility: a closed unitary system can possess a non-zero imaginary pseudo entropy (a reversible distinction between orientations) without dissipative entropy production. Irreversibility only enters when a physical coarse graining destroys the record.

• Specific Models:

  • Open Transverse-Field Ising Chain: Numerical results show that while the phase response remains finite near the critical point, the visibility-dressed discrimination signal is weakly reduced, scaling with the half-chain visibility.
  • Two-Qubit System: The signal vanishes for product states (no coherence between Schmidt sectors) and maximally entangled states (flat modular spectrum), requiring intermediate entanglement and non-trivial modular spectra for optimal sensitivity.

Significance and Claims
The paper claims to provide the first bounded operational meaning for the imaginary part of real-time pseudo entropy. It demonstrates that this quantity is not merely a mathematical artifact or a proxy for thermodynamic entropy production, but a coherent record of temporal orientation.

The significance lies in the ability to:

1. Measure the directionality of a quantum transition (forward vs. backward) with optimal efficiency using a specific interferometric protocol.
2. Quantify the loss of this directional information under coarse graining, linking the recoverability of the record to the Petz recovery map.
3. Clarify that a non-zero imaginary pseudo entropy in a closed system represents a reversible distinction, distinct from dissipative irreversibility.

The authors conclude that the protocol offers three direct experimental checks: the sign flip of the ancilla $Y$-signal upon orientation reversal, the trace distance relation in Eq. (8), and the specific information loss curve under dephasing in Eq. (19). The method relies on subsystem permutations rather than tomography of non-Hermitian operators, making it applicable beyond integrable systems.