An area law for the entropy of low-energy states

It is often observed in the ground state of spatially-extended quantum systems with local interactions that the entropy of a large region is proportional to its surface area. In some cases, this area law is corrected with a logarithmic factor. This contrasts with the fact that in almost all states of the Hilbert space, the entropy of a region is proportional to its volume. This paper shows that low-energy states have (at most) an area law with the logarithmic correction, provided two conditions hold: (i) the state has sufficient decay of correlations, (ii) the number of eigenstates with vanishing energy-density is not exponential in the volume. These two conditions are satisfied by many relevant systems. The central idea of the argument is that energy fluctuations inside a region can be observed by measuring the exterior and a superficial shell of the region.Comment: 6 pages + appendix, 1 figur

Certified randomness in quantum physics

The concept of randomness plays an important role in many disciplines. On one hand, the question of whether random processes exist is fundamental for our understanding of nature. On the other hand, randomness is a resource for cryptography, algorithms and simulations. Standard methods for generating randomness rely on assumptions on the devices that are difficult to meet in practice. However, quantum technologies allow for new methods for generating certified randomness. These methods are known as device-independent because do not rely on any modeling of the devices. Here we review the efforts and challenges to design device-independent randomness generators.Comment: 18 pages, 3 figure

Key Distillation and the Secret-Bit Fraction

We consider distillation of secret bits from partially secret noisy correlations P_ABE, shared between two honest parties and an eavesdropper. The most studied distillation scenario consists of joint operations on a large number of copies of the distribution (P_ABE)^N, assisted with public communication. Here we consider distillation with only one copy of the distribution, and instead of rates, the 'quality' of the distilled secret bits is optimized, where the 'quality' is quantified by the secret-bit fraction of the result. The secret-bit fraction of a binary distribution is the proportion which constitutes a secret bit between Alice and Bob. With local operations and public communication the maximal extractable secret-bit fraction from a distribution P_ABE is found, and is denoted by Lambda[P_ABE]. This quantity is shown to be nonincreasing under local operations and public communication, and nondecreasing under eavesdropper's local operations: it is a secrecy monotone. It is shown that if Lambda[P_ABE]>1/2 then P_ABE is distillable, thus providing a sufficient condition for distillability. A simple expression for Lambda[P_ABE] is found when the eavesdropper is decoupled, and when the honest parties' information is binary and the local operations are reversible. Intriguingly, for general distributions the (optimal) operation requires local degradation of the data.Comment: 12 page

Bell's inequalities detect efficient entanglement

We review the status of Bell's inequalities in quantum information, stressing mainly the links with quantum key distribution and distillation of entanglement. We also prove that for all the eavesdropping attacks using one qubit, and for a family of attacks of two qubits, acting on half of a maximally entangled state of two qubits, the violation of a Bell inequality implies the possibility of an efficient secret-key extraction.Comment: 9 pages, for the Proceedings of EQIS'03 (Kyoto, Sept. 2003

All bipartite entangled states display some hidden nonlocality

We show that a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality can be demonstrated in a certain kind of Bell experiment for all bipartite entangled states. Our protocol allows local filtering measurements and involves shared ancilla states that do not themselves violate CHSH. Our result follows from two main steps. We first provide a simple characterization of the states that violate the CHSH-inequality after local filtering operations in terms of witness-like operators. Second, we prove that for each entangled state $\sigma$, there exists another state $\rho$ not violating CHSH, such that $\rho\otimes\sigma$ violates CHSH. Hence, in this scenario, $\sigma$ cannot be substituted by classical correlations without changing the statistics of the experiment; we say that $\sigma$ is not simulable by classical correlations and our result is that entanglement is equivalent to non-simulability.Comment: 5 pages, 1 figur

Entanglement fluctuation theorems

Pure state entanglement transformations have been thought of as irreversible, with reversible transformations generally only possible in the limit of many copies. Here, we show that reversible entanglement transformations do not require processing on the many copy level, but can instead be undertaken on individual systems, provided the amount of entanglement which is produced or consumed is allowed to fluctuate. We derive necessary and sufficient conditions for entanglement manipulations in this case. As a corollary, we derive an equation which quantifies the fluctuations of entanglement, which is formally identical to the Jarzynski fluctuation equality found in thermodynamics. One can also relate a forward entanglement transformation to its reverse process in terms of the entanglement cost of such a transformation, in a manner equivalent to the Crooks relation. We show that a strong converse theorem for entanglement transformations is formally related to the second law of thermodynamics, while the fact that the Schmidt rank of an entangled state cannot increase is related to the third law of thermodynamics. Achievability of the protocols is done by introducing an entanglement battery, a device which stores entanglement and uses an amount of entanglement that is allowed to fluctuate but with an average cost which is still optimal. This allows us to also solve the problem of partial entanglement recovery, and in fact, we show that entanglement is fully recovered. Allowing the amount of consumed entanglement to fluctuate also leads to improved and optimal entanglement dilution protocols.Comment: comments welcome, v2 published versio

The second law of quantum thermodynamics as an equality

We investigate the connection between recent results in quantum thermodynamics and fluctuation relations by adopting a fully quantum mechanical description of thermodynamics. By including a work system whose energy is allowed to fluctuate, we derive a set of equalities which all thermodynamical transitions have to satisfy. This extends the condition for maps to be Gibbs-preserving to the case of fluctuating work, providing a more general characterisation of maps commonly used in the information theoretic approach to thermodynamics. For final states, block diagonal in the energy basis, this set of equalities are necessary and sufficient conditions for a thermodynamical state transition to be possible. The conditions serves as a parent equation which can be used to derive a number of results. These include writing the second law of thermodynamics as an equality featuring a fine-grained notion of the free energy. It also yields a generalisation of the Jarzynski fluctuation theorem which holds for arbitrary initial states, and under the most general manipulations allowed by the laws of quantum mechanics. Furthermore, we show that each of these relations can be seen as the quasi-classical limit of three fully quantum identities. This allows us to consider the free energy as an operator, and allows one to obtain more general and fully quantum fluctuation relations from the information theoretic approach to quantum thermodynamics.Comment: 11+3 pages. V4: Updated to match published version. Discussion of thermo-majorization and implementing arbitary unitaries added. V3: Added funding information. V2: Expanded discussion on relation to fluctuation theorem