Entanglement is not only the resource that fuels many quantum technologies but also plays a key role for some of the most profound open questions of fundamental physics. Experiments controlling ...quantum systems at the single quantum level may shed light on these puzzles. However, measuring, or even bounding, entanglement experimentally has proven to be an outstanding challenge, especially when the prepared quantum states are mixed. We use entropic uncertainty relations for bipartite systems to derive measurable lower bounds on distillable entanglement. We showcase these bounds by applying them to physical models realizable in cold-atom experiments. The derived entanglement bounds rely on measurements in only two different bases and are generically applicable to any quantum simulation platform.
We investigate the performance of parallel and adaptive quantum channel discrimination strategies for a finite number of channel uses. It has recently been shown that, in the asymmetric setting with ...asymptotically vanishing type I error probability, adaptive strategies are asymptotically not more powerful than parallel ones. We extend this result to the non-asymptotic regime with finitely many channel uses, by explicitly constructing a parallel strategy for any given adaptive strategy, and bounding the difference in their performances, measured in terms of the decay rate of the type II error probability per channel use. We further show that all parallel strategies can be optimized over in time polynomial in the number of channel uses, and hence our result can also be used to obtain a poly-time-computable asymptotically tight upper bound on the performance of general adaptive strategies.
We study experimentally accessible lower bounds on entanglement measures based on entropic uncertainty relations. Experimentally quantifying entanglement is highly desired for applications of quantum ...simulation experiments to fundamental questions, e.g., in quantum statistical mechanics and condensed-matter physics. At the same time it poses a significant challenge because the evaluation of entanglement measures typically requires the full reconstruction of the quantum state, which is extremely costly in terms of measurement statistics. We derive an improved entanglement bound for bipartite systems, which requires measuring joint probability distributions in only two different measurement settings per subsystem, and demonstrate its power by applying it to currently operational experimental setups for quantum simulation with cold atoms. Examining the tightness of the derived entanglement bound, we find that the set of pure states for which our relation is tight is strongly restricted. We show that for measurements in mutually unbiased bases the only pure states that saturate the bound are maximally entangled states on a subspace of the bipartite Hilbert space (this includes product states). We further show that our relation can also be employed for entanglement detection using generalized measurements, i.e., when not all measurement outcomes can be resolved individually by the detector. In addition, the impact of local conserved quantities on the detectable entanglement is discussed.
Entanglement is not only the resource that fuels many quantum technologies but also plays a key role for some of the most profound open questions of fundamental physics. Experiments controlling ...quantum systems at the single quantum level may shed light on these puzzles. However, measuring, or even bounding, entanglement experimentally has proven to be an outstanding challenge, especially when the prepared quantum states are mixed. We use entropic uncertainty relations for bipartite systems to derive measurable lower bounds on distillable entanglement. We showcase these bounds by applying them to physical models realizable in cold-atom experiments. The derived entanglement bounds rely on measurements in only two different bases and are generically applicable to any quantum simulation platform.
Given a quantum channel and a state which satisfy a fixed point equation
approximately (say, up to an error $\varepsilon$), can one find a new channel
and a state, which are respectively close to the ...original ones, such that they
satisfy an exact fixed point equation? It is interesting to ask this question
for different choices of constraints on the structures of the original channel
and state, and requiring that these are also satisfied by the new channel and
state. We affirmatively answer the above question, under fairly general
assumptions on these structures, through a compactness argument. Additionally,
for channels and states satisfying certain specific structures, we find
explicit upper bounds on the distances between the pairs of channels (and
states) in question. When these distances decay quickly (in a particular,
desirable manner) as $\varepsilon\to 0$, we say that the original approximate
fixed point equation is rapidly fixable. We establish rapid fixability, not
only for general quantum channels, but also when the original and new channels
are both required to be unitary, mixed unitary or unital. In contrast, for the
case of bipartite quantum systems with channels acting trivially on one
subsystem, we prove that approximate fixed point equations are not rapidly
fixable. In this case, the distance to the closest channel (and state) which
satisfy an exact fixed point equation can depend on the dimension of the
quantum system in an undesirable way. We apply our results on approximate fixed
point equations to the question of robustness of quantum Markov chains (QMC)
and establish the following: For any tripartite quantum state, there exists a
dimension-dependent upper bound on its distance to the set of QMCs, which
decays to zero as the conditional mutual information of the state vanishes.
We study the problem of binary composite channel discrimination in the asymmetric setting, where the hypotheses are given by fairly arbitrary sets of channels, and samples do not have to be ...identically distributed. In the case of quantum channels we prove: (i) a characterization of the Stein exponent for parallel channel discrimination strategies and (ii) an upper bound on the Stein exponent for adaptive channel discrimination strategies. We further show that already for classical channels this upper bound can sometimes be achieved and be strictly larger than what is possible with parallel strategies. Hence, there can be an advantage of adaptive channel discrimination strategies with composite hypotheses for classical channels, unlike in the case of simple hypotheses. Moreover, we show that classically this advantage can only exist if the sets of channels corresponding to the hypotheses are non-convex. As a consequence of our more general treatment, which is not limited to the composite i.i.d. setting, we also obtain a generalization of previous composite state discrimination results.
We investigate the performance of parallel and adaptive quantum channel discrimination strategies for a finite number of channel uses. It has recently been shown that, in the asymmetric setting with ...asymptotically vanishing type I error probability, adaptive strategies are asymptotically not more powerful than parallel ones. We extend this result to the non-asymptotic regime with finitely many channel uses, by explicitly constructing a parallel strategy for any given adaptive strategy, and bounding the difference in their performances, measured in terms of the decay rate of the type II error probability per channel use. We further show that all parallel strategies can be optimized over in time polynomial in the number of channel uses, and hence our result can also be used to obtain a poly-time-computable asymptotically tight upper bound on the performance of general adaptive strategies.
We study asymmetric binary channel discrimination, for qantum channels acting on separable Hilbert spaces. We establish quantum Stein's lemma for channels for both adaptive and parallel strategies, ...and show that under finiteness of the geometric Rényi divergence between the two channels for some \(\alpha > 1\), adaptive strategies offer no asymptotic advantage over parallel ones. One major step in our argument is to demonstrate that the geometric Rényi divergence satisfies a chain rule and is additive for channels also in infinite dimensions. These results may be of independent interest. Furthermore, we not only show asymptotic equivalence of parallel and adaptive strategies, but explicitly construct a parallel strategy which approximates a given adaptive \(n\)-shot strategy, and give an explicit bound on the difference between the discrimination errors for these two strategies. This extends the finite dimensional result from B. Bergh et al., arxiv:2206.08350. Finally, this also allows us to conclude, that the chain rule for the Umegaki relative entropy in infinite dimensions, recently shown in O. Fawzi, L. Gao, and M. Rahaman, arxiv:2212.14700v2 given finiteness of the max divergence between the two channels, also holds under the weaker condition of finiteness of the geometric Rényi divergence. We give explicit examples of channels which show that these two finiteness conditions are not equivalent.
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