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

Fault tolerance is the study of reliable computation using unreliable components. With a given noise model, can one still reliably compute? For example, one can run many copies of a classical calculation in parallel, periodically using majority gates to catch and correct faults. Von Neumann showed in 1956 that if each gate fails independently with probability p, flipping its output bit 0 $ 1, then such a fault tolerance scheme still allows for arbitrarily reliable computation provided that p is below some constant threshold (whose value depends on the model details) [10]. In a quantum computer, the basic gates are much more vulnerable to noise than classical transistors – after all, depending on the implementation, they are manipulating single electron spins, photon polarizations, and similarly fragile subatomic particles. It might not be possible to engineer systems with noise rates less than 10 , or perhaps 10 , per gate. Additionally, the phenomenon of entanglement makes quantum systems inherently fragile. For example, in Schrödinger’s cat state – an equal superposition between a living cat and a dead cat, often idealized as 1= p 2.j0ni C j1ni/ – an interaction with just one quantum bit (“qubit”) can collapse, or decohere, the entire system. Fault tolerance techniques will therefore be essential for achieving the considerable potential of quantum computers. Practical fault tolerance techniques will need to control high noise rates and do so with low overhead, since qubits are expensive. Quantum systems are continuous, not discrete, so there are many possible noise models. However, the essential features of quantum noise for fault tolerance results can be captured by a simple discrete model similar to the one Von Neumann used. The main difference is that, in addition to bit-flip X errors which swap 0 and 1, there can also be phase-flip Z errors which swap jCi 1=p2.j0i C j1i/ and j i 1=p2.j0i j1i/ (Fig. 1). A noisy gate is modeled as a perfect gate followed by independent introduction of X, Z, or Y (which is both X and Z) errors with respective probabilities pX; pZ; pY. One popular model is independent depolarizing noise .pX D pZ D pY p=3/; a depolarized qubit is completely randomized. Faulty measurements and preparations of single-qubit states must additionally be modeled, and there can be memory noise on resting qubits. It is often assumed that measurement results can

Fault-Tolerant Quantum Computation.

Developments in the thermodynamics of small quantum systems envisage nonclassical thermal machines. In this scenario, energy fluctuations play a relevant role in the description of irreversibility. We experimentally implement a quantum heat engine based on a spin-1/2 system and nuclear magnetic resonance techniques. Irreversibility at a microscope scale is fully characterized by the assessment of energy fluctuations associated with the work and heat flows. We also investigate the efficiency lag related to the entropy production at finite time. The implemented heat engine operates in a regime where both thermal and quantum fluctuations (associated with transitions among the instantaneous energy eigenstates) are relevant to its description. Performing a quantum Otto cycle at maximum power, the proof-of-concept quantum heat engine is able to reach an efficiency for work extraction (η≈42%) very close to its thermodynamic limit (η=44%).

Experimental Characterization of a Spin Quantum Heat Engine

We consider the speed limit for classical stochastic Markov processes with and without the local detailed balance condition. We find that, for both cases, a trade-off inequality exists between the speed of the state transformation and the entropy production. The dynamical activity is related to a time scale and plays a crucial role in the inequality. For the dynamics without the local detailed balance condition, we use the Hatano-Sasa entropy production instead of the standard entropy production. Our inequalities consist of the quantities that are commonly used in stochastic thermodynamics and explicitly show underlying physical mechanisms.

Speed Limit for Classical Stochastic Processes.

According to the laws of thermodynamics, no heat engine can beat the efficiency of a Carnot cycle. This efficiency traditionally comes with vanishing power output and practical designs, optimized for power, generally achieve far less. Recently, various strategies to obtain Carnot's efficiency at large power were proposed. However, a thermodynamic uncertainty relation implies that steady-state heat engines can operate in this regime only at the cost of large fluctuations that render them immensely unreliable. Here, we demonstrate that this unfortunate trade-off can be overcome by designs operating cyclically under quasistatic conditions. The experimentally relevant yet exactly solvable model of an overdamped Brownian heat engine is used to illustrate the formal result. Our study highlights that work in cyclic heat engines and that in quasistatic ones are different stochastic processes.

Cycling Tames Power Fluctuations near Optimum Efficiency.

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.

There exist two formulations for quantum heat engines that model energy transfer between two microscopic systems. One is the semiclassical scenario and the other is the full quantum scenario. The former is formulated as unitary evolution for the internal system and is adopted by the statistical mechanics community. In the latter, the whole process is formulated as unitary and is adopted by the quantum information community. This paper proposes a model for quantum heat engines that transfer energy from a collection of microscopic systems to a macroscopic system like a fuel cell. In such a situation, the amount of extracted work is visible for a human. For this purpose, we formulate a quantum heat engine as the measurement process whose measurement outcome is the amount of extracted work. Under this model, we derive a suitable energy-conservation law and propose a more concrete submodel. Then we derive a trade-off relation between the measurability of the amount of work extraction and the coherence of the internal system, which limits the applicability of the semiclassical scenario to a heat engine transferring energy from a collection of microscopic systems to a macroscopic system.

Measurement-based formulation of quantum heat engines

The optimal efficiency of quantum (or classical) heat engines whose heat baths are n-particle systems is given by the strong large deviation. We give the optimal work extraction process as a concrete energy-preserving unitary time evolution among the heat baths and the work storage. We show that our optimal work extraction turns the disordered energy of the heat baths to the ordered energy of the work storage, by evaluating the ratio of the entropy difference to the energy difference in the heat baths and the work storage, respectively. By comparing the statistical mechanical optimal efficiency with the macroscopic thermodynamic bound, we evaluate the accuracy of the macroscopic thermodynamics with finite-size heat baths from the statistical mechanical viewpoint. We also evaluate the quantum coherence effect on the optimal efficiency of the cycle processes without restricting their cycle time by comparing the classical and quantum optimal efficiencies.

Finite-size effect on optimal efficiency of heat engines.

Quantum coherence is a useful resource for increasing the speed and decreasing the irreversibility of quantum dynamics. Because of this feature, coherence is used to enhance the performance of various quantum information processing devices beyond the limitations set by classical mechanics. However, when we consider thermodynamic processes, such as energy conversion in nanoscale devices, it is still unclear whether coherence provides similar advantages. Here we establish a universal framework, clarifying how coherence affects the speed and irreversibility in thermodynamic processes described by the Lindblad master equation, and give general rules for when coherence enhances or reduces the performance of thermodynamic devices. Our results show that a proper use of coherence enhances the heat current without increasing dissipation; i.e., coherence can reduce friction. In particular, if the amount of coherence is large enough, this friction becomes virtually zero, realizing a superconducting-like "dissipation-less" heat current. Since our framework clarifies a general relation among coherence, energy flow, and dissipation, it can be applied to many branches of science from quantum information theory to biology. As an application to energy science, we construct a quantum heat engine cycle that exceeds the power-efficiency trade-off bound on classical engines and effectively attains the Carnot efficiency with finite power in fast cycles.

Superconducting-like Heat Current: Effective Cancellation of Current-Dissipation Trade-Off by Quantum Coherence.

The underlying mechanism in the implementation of unitary operation on a system with an external apparatus is studied. We implement the unitary time evolution in the system as a physical phenomenon that results from the interaction between the system and the apparatus. We investigate the fundamental limitation of an accurate implementation for the desired unitary time evolution. This limitation is manifested in the form of trade-off relations between the accuracy of the implementation and quantum fluctuation of energy in the external apparatus. Our relations clearly show that an accurate unitary operation requires a large energy fluctuation inside the apparatus originated from the quantum fluctuation.

Hiroyasu Tajima

Papers

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

Measurement-based formulation of quantum heat engines

Finite-size effect on optimal efficiency of heat engines.

Superconducting-like Heat Current: Effective Cancellation of Current-Dissipation Trade-Off by Quantum Coherence.

Uncertainty relations in implementation of unitary operations