Abstract
We purify the thermal density matrix of a free harmonic oscillator as a two-mode squeezed state, characterized by a squeezing parameter and squeezing angle. While the squeezing parameter is fixed by the temperature and frequency of the oscillator, the squeezing angle is otherwise undetermined, so that the complexity of purification is obtained by minimizing the complexity of the squeezed state over the squeezing angle. The resulting complexity of purification of the thermal state is minimized at non-zero values of the squeezing angle and saturates to an order one number at low frequencies, indicating that there is no additional operator cost required to build thermal mixed states when the oscillator probes length scales that are large compared to the thermal length scale. We also review applications in which thermal density matrices arise for quantum fields on curved spacetimes, including Hawking radiation and a simple model of decoherence of cosmological density perturbations in the early Universe. The complexity of purification for these mixed states also saturates as a function of the effective temperature, which may have interesting consequences for the quantum information stored in these systems.
Highlights
The resulting complexity of purification of the thermal state is minimized at non-zero values of the squeezing angle and saturates to an order one number at low frequencies, indicating that there is no additional operator cost required to build thermal mixed states when the oscillator probes length scales that are large compared to the thermal length scale
We will focus on the thermal density matrix of a free harmonic oscillator of frequency ω, both for its simplicity and because it can be directly related to several interesting applications
We constructed a purification of a thermal density matrix of a harmonic oscillator as a generic two-mode squeezed state, where the squeezing is fixed by the temperature and the squeezing angle is a free parameter
Summary
For any mixed state ρmix on the Hilbert space H, we can construct a purification of ρmix which consists of a pure state |Ψ in an enlarged Hilbert space Hpure = H ⊗ Hanc, where Hanc corresponds to an “ancillary” set of degrees of freedom. If the trace of the density matrix of |Ψ over the ancillary degrees of freedom gives the original mixed state Tranc (|Ψ Ψ|) = ρmix, we say that |Ψ is a “purification” of ρmix.
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