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arXiv:2301.01053 · Physical Review Research

Sequences of resource monotones from modular Hamiltonian polynomials

Raúl Arias, Jan De Boer A, Giuseppe Di Giulio, Esko Keski-Vakkuri, Erik Tonni C

Published Oct 24, 2023• 4 citations
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Abstract

We introduce two infinite sequences of entanglement monotones, which are constructed from expectation values of polynomials in the modular Hamiltonian. These monotones yield infinite sequences of inequalities that must be satisfied in majorizing state transitions. We demonstrate this for information erasure, deriving an infinite sequence of ``Landauer inequalities'' for the work cost, bounded by linear combinations of expectation values of powers of the modular Hamiltonian. These inequalities give improved lower bounds for the work cost in finite-dimensional systems, and depend on more details of the erased state than just on its entropy and variance of modular Hamiltonian. Similarly one can derive lower bounds for marginal entropy production for a system coupled to an environment. These infinite sequences of entanglement monotones also give rise to relative quantifiers that are monotonic in more general processes, namely those involving so-called $\ensuremath{\sigma}$ majorization with respect to a fixed point full rank state $\ensuremath{\sigma}$; such quantifiers are called resource monotones. As an application to thermodynamics, one can use them to derive finite-dimension corrections to the Clausius inequality. Finally, in order to gain some intuition for what (if anything) plays the role of majorization in field theory, we compare pairs of states in discretized theories at criticality and study how majorization depends on the size of the bipartition with respect to the size of the entire chain.

DOI
10.1103/physrevresearch.5.043082

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We introduce two infinite sequences of entanglement monotones, which are constructed from expectation values of polynomials in the modular Hamiltonian. These monotones yield infinite sequences of inequalities that must be satisfied in majorizing state transitions. We demonstrate this for information erasure, deriving an infinite sequence of ``Landauer inequalities'' for the work cost, bounded by linear combinations of expectation values of powers of the modular Hamiltonian. These inequalities give improved lower bounds for the work cost in finite-dimensional systems, and depend on more details of the erased state than just on its entropy and variance of modular Hamiltonian. Similarly one can derive lower bounds for marginal entropy production for a system coupled to an environment. These infinite sequences of entanglement monotones also give rise to relative quantifiers that are monotonic in more general processes, namely those involving so-called \ensuremathσ\ensuremath{\sigma}\ensuremathσ majorization with respect to a fixed point full rank state \ensuremathσ\ensuremath{\sigma}\ensuremathσ; such quantifiers are called resource monotones. As an application to thermodynamics, one can use them to derive finite-dimension corrections to the Clausius inequality. Finally, in order to gain some intuition for what (if anything) plays the role of majorization in field theory, we compare pairs of states in discretized theories at criticality and study how majorization depends on the size of the bipartition with respect to the size of the entire chain.
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Entanglement has been studied extensively for understanding the mysteries of non-classical correlations between quantum systems. In the bipartite case, there are well known monotones for quantifying entanglement such as concurrence, relative entropy of entanglement (REE) and negativity, which cannot be increased via local operations. The study on these monotones has been a hot topic in quantum information [1-7] in order to understand the role of entanglement in this discipline. It can be observed that from any arbitrary quantum pure state a mixed state can obtained. A natural generalization of this observation would be to consider local operations classical communication (LOCC) transformations between general pure states of two parties. Although this question is a little more difficult, a complete solution has been developed using the mathematical framework of the majorization theory [8]. In this work, we analyze the relation between entanglement monotones concurrence and negativity with respect to majorization for general two-level quantum systems of two particles.
We derive a monogamy inequality for entanglement and local contextuality, for any finite bipartite system. It essentially results from the relations between the entropy of a local state and the entanglement of the global state, and between the purity of a state, in the sense of majorization, and its ability to violate a given state-dependent noncontextuality inequality. We build an explicit entanglement monotone that satisfies the found monogamy inequality. An important consequence of this inequality is that there are global states too entangled to violate the local noncontextuality inequality.
We introduce and analyse the problem of encoding classical information into different resources of a quantum state. More precisely, we consider a general class of communication scenarios characterised by encoding operations that commute with a unique resource destroying map and leave free states invariant. Our motivating example is given by encoding information into coherences of a quantum system with respect to a fixed basis (with unitaries diagonal in that basis as encodings and the decoherence channel as a resource destroying map), but the generality of the framework allows us to explore applications ranging from super-dense coding to thermodynamics. For any state, we find that the number of messages that can be encoded into it using such operations in a one-shot scenario is upper bounded in terms of the information spectrum relative entropy between the given state and its version with erased resources. Furthermore, if the resource destroying map is the twirling channel over some unitary group, we find matching one-shot lower bounds as well. In the asymptotic setting where we encode into many copies of the resource state, our bounds yield an operational interpretation of resource monotones such as the relative entropy of coherence and its corresponding relative entropy variance.
The Rains relative entropy of a bipartite quantum state is the tightest known upper bound on its distillable entanglement – which has a crisp physical interpretation of entanglement as a resource – and it is efficiently computable by convex programming. It has not been known to be a selective entanglement monotone in its own right. In this work, we strengthen the interpretation of the Rains relative entropy by showing that it is monotone under the action of selective operations that completely preserve the positivity of the partial transpose, reasonably quantifying entanglement. That is, we prove that Rains relative entropy of an ensemble generated by such an operation does not exceed the Rains relative entropy of the initial state in expectation, giving rise to the smallest, most conservative known computable selective entanglement monotone. Additionally, we show that this is true not only for the original Rains relative entropy, but also for Rains relative entropies derived from various Rényi relative entropies. As an application of these findings, we prove, in both the non-asymptotic and asymptotic settings, that the probabilistic approximate distillable entanglement of a state is bounded from above by various Rains relative entropies.Full version available at https://arxiv.org/abs/2201.00835
We obtain new constraints for the modular energy of general states by using the monotonicity property of relative entropy. In some cases, modular energy can be related to the energy density of states and these constraints lead to interesting relations between energy and entropy. In particular, we derive new quantum energy inequalities that improve some previous bounds for the energy density of states in a conformal field theory. Additionally, the inequalities derived in this manner also lead us to conclude that the entropy of the state further restricts the possible amount of negative energy allowed by the theory.