Conditional mutual information of bipartite unitaries and scrambling

Abstract

One way to diagnose chaos in bipartite unitary channels is via the tripartite information of the corresponding Choi state, which for certain choices of the subsystems reduces to the negative conditional mutual information (CMI). We study this quantity from a quantum information-theoretic perspective to clarify its role in diagnosing scrambling. When the CMI is zero, we find that the channel has a special normal form consisting of local channels between individual inputs and outputs. However, we find that arbitrarily low CMI does not imply arbitrary proximity to a channel of this form, although it does imply a type of approximate recoverability of one of the inputs. When the CMI is maximal, we find that the residual channel from an individual input to an individual output is completely depolarizing when the other input is maximally mixed. However, we again find that this result is not robust. We also extend some of these results to the multipartite case and to the case of Haar-random pure input states. Finally, we look at the relationship between tripartite information and its Rényi-2 version which is directly related to out-of-time-order correlation functions. In particular, we demonstrate an arbitrarily large gap between the two quantities.

Entanglement, quantum randomness, and complexity beyond scrambling

“…To establish such separations, we need to show a large (Θ(log d)) gap between Rényi-α entropies and the maximum for some small D, as well as upper bound the difference between Rényiα entropies and the maximum by continuity. The gap side can work out by directly generalizing the corresponding calculation in [50]: Let β = log(…”

Entanglement Wedge Reconstruction via Universal Recovery Channels

“…The associated algebraic structure slightly generalizes operator algebra quantum error correction, precisely the structure relevant in [15]. Near-saturation of strong subadditivity, however, fails to imply proximity to a quantum Markov chain state for systems of large Hilbert space dimension [18,19].…”

Approximate Quantum Markov Chains

“…We further denote the limit cases 45) where in the final step we are allowed to interchange the infimum and the limit as the sequence {D α (ρ τ)} α is monotonically increasing (due to Proposition 2.40) and hence by Dini's theorem [130] it converges uniformly in τ. By the same arguments we also see that…”

“…To summarize, one-dimensional systems that satisfy the locality assumption (5.110) can be efficiently represented by a finite sequence of recovery maps given by Theorem 5.5. Theorem 5.5 has been successfully applied in other areas such as high energy physics [43,45,117], solid state physics [29,160,179], quantum error correction [66,118], quantum information theory [7,22,33,90,97,101], and foundations of quantum mechanics [109].…”