Quantum 6, 650 (2022).

https://doi.org/10.22331/q-2022-02-10-650

The thermal equilibrium properties of physical systems can be described using Gibbs states. It is therefore of great interest to know when such states allow for an easy description. In particular, this is the case if correlations between distant regions are small. In this work, we consider 1D quantum spin systems with local, finite-range, translation-invariant interactions at any temperature. In this setting, we show that Gibbs states satisfy uniform exponential decay of correlations and, moreover, the mutual information between two regions decays exponentially with their distance, irrespective of the temperature. In order to prove the latter, we show that exponential decay of correlations of the infinite-chain thermal states, exponential uniform clustering and exponential decay of the mutual information are equivalent for 1D quantum spin systems with local, finite-range interactions at any temperature. In particular, Araki's seminal results yields that the three conditions hold in the translation-invariant case. The methods we use are based on the Belavkin-Staszewski relative entropy and on techniques developed by Araki. Moreover, we find that the Gibbs states of the systems we consider are superexponentially close to saturating the data-processing inequality for the Belavkin-Staszewski relative entropy.

Quantum 6, 649 (2022).

https://doi.org/10.22331/q-2022-02-10-649

The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is $textit{general and model-independent}$, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization.

Quantum 6, 648 (2022).

https://doi.org/10.22331/q-2022-02-10-648

We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.

Quantum 6, 647 (2022).

https://doi.org/10.22331/q-2022-02-09-647

In this article we consider flagged extensions of convex combination of quantum channels, and find general sufficient conditions for the degradability of the flagged extension. An immediate application is a bound on the quantum $Q$ and private $P$ capacities of any channel being a mixture of a unitary map and another channel, with the probability associated to the unitary component being larger than $1/2$. We then specialize our sufficient conditions to flagged Pauli channels, obtaining a family of upper bounds on quantum and private capacities of Pauli channels. In particular, we establish new state-of-the-art upper bounds on the quantum and private capacities of the depolarizing channel, BB84 channel and generalized amplitude damping channel. Moreover, the flagged construction can be naturally applied to tensor powers of channels with less restricting degradability conditions, suggesting that better upper bounds could be found by considering a larger number of channel uses.

Quantum 6, 646 (2022).

https://doi.org/10.22331/q-2022-02-09-646

Quantum imaging with undetected photons is a recently introduced technique that goes significantly beyond what was previously possible. In this technique, images are formed without detecting the light that interacted with the object that is imaged. Given this unique advantage over the existing imaging schemes, it is now of utmost importance to understand its resolution limits, in particular what governs the maximal achievable spatial resolution. We show both theoretically and experimentally that the momentum correlation between the detected and undetected photons governs the spatial resolution — a stronger correlation results in a higher resolution. In our experiment, the momentum correlation plays the dominating role in determining the resolution compared to the effect of diffraction. We find that the resolution is determined by the wavelength of the undetected light rather than the wavelength of the detected light. Our results thus show that it is in principle possible to obtain resolution characterized by a wavelength much shorter than the detected wavelength.

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