QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

This review introduces a new development in theoretical quantum physics, the ``resource-theoretic'' point of view. The approach aims to be closely linked to experiment, and to state exactly what result you can hope to achieve for what expenditure of effort in the laboratory. This development is an extension of the principles of thermodynamics to quantum problems; but there are resources that would never have been considered previously in thermodynamics, such as shared knowledge of a frame of reference. Many additional examples and new quantifications of resources are provided.

Quantum resource theories

We study information retrieval from evaporating black holes, assuming that the internal dynamics of a black hole is unitary and rapidly mixing, and assuming that the retriever has unlimited control over the emitted Hawking radiation. If the evaporation of the black hole has already proceeded past the ``half-way'' point, where half of the initial entropy has been radiated away, then additional quantum information deposited in the black hole is revealed in the Hawking radiation very rapidly. Information deposited prior to the half-way point remains concealed until the half-way point, and then emerges quickly. These conclusions hold because typical local quantum circuits are efficient encoders for quantum error-correcting codes that nearly achieve the capacity of the quantum erasure channel. Our estimate of a black hole's information retention time, based on speculative dynamical assumptions, is just barely compatible with the black hole complementarity hypothesis.

http://absimage.aps.org/image/MAR08/MWS_MAR08-2007-003180.pdf

Black holes as mirrors: quantum information in random subsystems

We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numerical evidence that the corresponding inequality for the entanglement of purification may be true in general. Finally, we use intuition from bit threads to extend the conjecture to holographic duals of suboptimal purifications.

/pdf/holographic-inequalities-and-entanglement-of-purification-1q84b466ht.pdf

Holographic inequalities and entanglement of purification

A long-standing open problem whether a heat engine with finite power achieves the Carnot efficiency is investgated. We rigorously prove a general trade-off inequality on thermodynamic efficiency and time interval of a cyclic process with quantum heat engines. In a first step, employing the Lieb-Robinson bound we establish an inequality on the change in a local observable caused by an operation far from support of the local observable. This inequality provides a rigorous characterization of the following intuitive picture that most of the energy emitted from the engine to the cold bath remains near the engine when the cyclic process is finished. Using this description, we prove an upper bound on efficiency with the aid of quantum information geometry. Our result generally excludes the possibility of a process with finite speed at the Carnot efficiency in quantum heat engines. In particular, the obtained constraint covers engines evolving with non-Markovian dynamics, which almost all previous studies on this topic fail to address.

Efficiency versus speed in quantum heat engines: Rigorous constraint from Lieb-Robinson bound.

Suppose that Alice and Bob are located in distant laboratories, which are connected by an ideal quantum channel. Suppose further that they share many copies of a quantum state ${\ensuremath{\rho}}_{ABE}$, such that Alice possesses the $A$ systems and Bob the $BE$ systems. In our model, there is an identifiable part of Bob's laboratory that is insecure: a third party named Eve has infiltrated Bob's laboratory and gained control of the $E$ systems. Alice, knowing this, would like use their shared state and the ideal quantum channel to communicate a message in such a way that Bob, who has access to the whole of his laboratory ($BE$ systems), can decode it, while Eve, who has access only to a sector of Bob's laboratory ($E$ systems) and the ideal quantum channel connecting Alice to Bob, cannot learn anything about Alice's transmitted message. We call this task the conditional one-time pad, and in this Letter, we prove that the optimal rate of secret communication for this task is equal to the conditional quantum mutual information $I(A;B|E)$ of their shared state. We thus give the conditional quantum mutual information an operational meaning that is different from those given in prior works, via state redistribution, conditional erasure, or state deconstruction. We also generalize the model and method in several ways, one of which is a secret-sharing task, i.e., the case in which Alice's message should be secure from someone possessing only the $AB$ or $AE$ systems, but should be decodable by someone possessing all systems $A$, $B$, and $E$.

Conditional Quantum One-Time Pad.

We analyze implementations of bipartite unitaries by means of local operations and classical communication (LOCC) assisted by shared entanglement We employ concepts and techniques developed in quantum Shannon theory to study an asymptotic scenario, in which two distant parties perform the same bipartite unitary on infinitely many pairs of inputs We analyze minimum cost of entanglement and classical communication per copy For two-round LOCC protocols, we derive a single-letter formula for the minimum cost of entanglement and classical communication, under an additional requirement that the error converges to zero faster than $1/n^4$, where $n$ is the number of input pairs The formula is given by the "Markovianizing cost" of a tripartite state associated with the unitary, which can be computed by a finite-step algorithm We also derive a lower bound on the minimum cost of resources, which applies for protocols with arbitrary number of rounds

A Coding Theorem for Bipartite Unitaries in Distributed Quantum Computation

We prove a trade-off relation between the entanglement cost and classical communication round complexity of a protocol in implementing a class of two-qubit unitary gates by two distant parties, a key subroutine in distributed quantum information processing. The task is analyzed in an information theoretic scenario of asymptotically many input pairs with a small error that is required to vanish sufficiently quickly. The trade-off relation is shown by (i) one ebit of entanglement per pair is necessary for implementing the unitary by any two-round protocol, and (ii) the entanglement cost by a three-round protocol is strictly smaller than one ebit per pair. We also provide an example of bipartite unitary gates for which there is no such trade-off.

Complexity of Causal Order Structure in Distributed Quantum Information Processing: More Rounds of Classical Communication Reduce Entanglement Cost

We introduce and analyze a task that we call Markovianization , in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of $n$ copies of a tripartite state $\rho ^{ABC}$ , and is transformed to a quantum Markov chain conditioned by $B^{n}$ with a small error, using a random unitary operation on $A^{n}$ . In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost , that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a single-letter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is close to a quantum Markov chain. Our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication.

Markovianizing Cost of Tripartite Quantum States

In order to analyze non-Markovianity of tripartite quantum states from a resource theoretical viewpoint, we introduce a class of quantum operations performed by three distant parties, and investigate an operational resource theory (ORT) induced it. A tripartite state is a free state if and only if it is a quantum Markov chain. We prove monotonicity of functions such as the conditional quantum mutual information, intrinsic information, squashed entanglement, a generalization of entanglement of purification, and the relative entropy of recovery. The ORT has five bound sets, each of which corresponds to one of the monotone functions. We introduce a task of "non-Markovianity dilution", and prove that the optimal rate for the task, namely the "non-Markovianity cost", is bounded from above by entanglement of purification in the case of pure states. We also propose a classical resource theory of non-Markovianity.

Eyuri Wakakuwa

Papers

Conditional Quantum One-Time Pad.

A Coding Theorem for Bipartite Unitaries in Distributed Quantum Computation

Complexity of Causal Order Structure in Distributed Quantum Information Processing: More Rounds of Classical Communication Reduce Entanglement Cost

Markovianizing Cost of Tripartite Quantum States

Operational Resource Theory of Non-Markovianity