Operator Entanglement and Channel–State Maps

Operator entanglement treats a time-evolution map as a state on doubled input and output degrees of freedom. In finite dimension this is a standard channel–state construction. In QFT, an unnormalizable maximally entangled vector and type-III local algebras force an energy-constrained or regulated replacement.

Required background. Tensor products and index structure supply the doubled-space partition, and channel–state methods in infinite dimensions supply the domain restrictions.

Helpful background. Entanglement growth supplies the state-entanglement comparison.

Finite-dimensional construction

For a $d$-dimensional system, let

$$|\Omega\rangle=\frac{1}{\sqrt d} \sum_{j=1}^{d}|j\rangle_{\rm out}|j\rangle_{\rm in}.$$

The channel state of a unitary $U(t)$ is $|U(t)\rangle=(U(t)\otimes\mathbf1)|\Omega\rangle$. Partition both input and output into spatial regions and compute the entropy of the chosen doubled subsystem. This operator entanglement measures how nonfactorizable the map is across that partition; it is not the entanglement generated from every physical input state. Zanardi 2001, Eqs. (2)–(6) makes this distinction explicit.

For a noisy channel $\mathcal N$, the Choi state is mixed, so mutual information, logarithmic negativity, or operator-space entropies answer different questions. State the normalization and diagnostic.

Energy-constrained QFT replacement

No normalized equal-weight vector exists over an infinite oscillator basis. Replace it by a finite-energy reference, for example a two-mode squeezed state

$$|\Omega_r\rangle= \sqrt{1-\lambda^2}\sum_{n=0}^{\infty} \lambda^n|n\rangle_{\rm out}|n\rangle_{\rm in}, \qquad \lambda=\tanh r.$$

Apply $U(t)$ to the output half, choose a spatial or mode partition, and report the reference energy $\sinh^2r$. Convergence means that the claim is stable over a stated energy-constrained class as $r$ increases, not merely that one large squeezing value looks settled.

For Gaussian evolution, propagate the doubled covariance matrix and compute symplectic entropies. Compare with a Fock cutoff at matched reference energy. A spatial partition must be defined before the mode basis; a momentum-mode operator entropy is not automatically a local-space quantity.

Operator entanglement is a direct channel diagnostic built from a regulated doubled state. It should not be substituted for the entanglement of the physical post-quench state. The map is schematic.

Reference and partition checks

Vary squeezing, Fock cutoff, and the input–output partition separately. Test a factorized evolution $U_A\otimes U_B$, whose spatial operator entanglement should vanish for the matching partition. Test a swap or beam splitter, which creates a known nonfactorizable channel state. If the answer changes qualitatively with the arbitrary squeezing profile, report reference dependence rather than a QFT invariant.

An unnormalized Choi vector, unconverged squeezing, or a changed doubled-space partition can manufacture operator entanglement. Energy and partition data are part of the observable. The map is schematic.

Exercises

Show that $|U_A\otimes U_B\rangle=|U_A\rangle\otimes|U_B\rangle$ under the doubled spatial partition $(A_{\rm out}A_{\rm in})|(B_{\rm out}B_{\rm in})$.

Solution

The maximally entangled reference factorizes as $|\Omega_A\rangle|\Omega_B\rangle$, and each local unitary acts only on its own output half. The resulting channel state factorizes, so its entropy across the stated cut is zero.

References

• Zanardi, Paolo. “Entanglement of Quantum Evolutions.” Physical Review A 63 (2001): 040304(R). DOI.