Corrigendum: A system’s wave function is uniquely determined by its underlying physical state

ISSN:1367-263

Non-Shannon inequalities in the entropy vector approach to causal structures

A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.Comment: 23 pages + appendix; v2: minor changes to Section IV A; v3: paper has been significantly shortened, an expanded version of the removed review section can be found in arXiv:1709.08988; v4: version to be published, supplementary information available as ancillary file

Quantum Circuits for Quantum Channels

We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of Controlled-NOT (C-NOT) gates required (single-qubit gates are free). We consider three different models. In the first, the Quantum Circuit Model (QCM), we consider sequences of single-qubit and C-NOT gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of C-NOT gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from m qubits to n qubits that uses at most one ancilla and has a low C-NOT count. We give explicit examples for small numbers of qubits that provide the lowest known C-NOT counts.Comment: 6(+4) page