Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes

It is a standard result in the theory of quantum error-correcting codes that no code of length n can fix more than n/4 arbitrary errors, regardless of the dimension of the coding and encoded Hilbert spaces. However, this bound only applies to codes which recover the message exactly. Naively, one might expect that correcting errors to very high fidelity would only allow small violations of this bound. This intuition is incorrect: in this paper we describe quantum error-correcting codes capable of correcting up to (n-1)/2 arbitrary errors with fidelity exponentially close to 1, at the price of increasing the size of the registers (i.e., the coding alphabet). This demonstrates a sharp distinction between exact and approximate quantum error correction. The codes have the property that any $t$ components reveal no information about the message, and so they can also be viewed as error-tolerant secret sharing schemes. The construction has several interesting implications for cryptography and quantum information theory. First, it suggests that secret sharing is a better classical analogue to quantum error correction than is classical error correction. Second, it highlights an error in a purported proof that verifiable quantum secret sharing (VQSS) is impossible when the number of cheaters t is n/4. More generally, the construction illustrates a difference between exact and approximate requirements in quantum cryptography and (yet again) the delicacy of security proofs and impossibility results in the quantum model.Comment: 14 pages, no figure

Decision theory for agents with incomplete preferences

Orthodox decision theory gives no advice to agents who hold two goods to be incommensurate in value because such agents will have incomplete preferences. According to standard treatments, rationality requires complete preferences, so such agents are irrational. Experience shows, however, that incomplete preferences are ubiquitous in ordinary life. In this paper, we aim to do two things: (1) show that there is a good case for revising decision theory so as to allow it to apply non-vacuously to agents with incomplete preferences, and (2) to identify one substantive criterion that any such non-standard decision theory must obey. Our criterion, Competitiveness, is a weaker version of a dominance principle. Despite its modesty, Competitiveness is incompatible with prospectism, a recently developed decision theory for agents with incomplete preferences. We spend the final part of the paper showing why Competitiveness should be retained, and prospectism rejected

Atmospheric Heat Redistribution on Hot Jupiters

Infrared lightcurves of transiting hot Jupiters present a trend in which the atmospheres of the hottest planets are less efficient at redistributing the stellar energy absorbed on their daysides---and thus have a larger day-night temperature contrast---than colder planets. No predictive atmospheric model has been published that identifies which dynamical mechanisms determine the atmospheric heat redistribution efficiency on tidally locked exoplanets. Here we present a two-layer shallow water model of the atmospheric dynamics on synchronously rotating planets that explains the observed trend. Our model shows that planets with weak friction and weak irradiation exhibit a banded zonal flow with minimal day-night temperature differences, while models with strong irradiation and/or strong friction exhibit a day-night flow pattern with order-unity fractional day-night temperature differences. To interpret the model, we develop a scaling theory that shows that the timescale for gravity waves to propagate horizontally over planetary scales, t_wave, plays a dominant role in controlling the transition from small to large temperature contrasts. This implies that heat redistribution is governed by a wave-like process, similar to the one responsible for the weak temperature gradients in the Earth's tropics. When atmospheric drag can be neglected, the transition from small to large day-night temperature contrasts occurs when t_wave ~ sqrt(t_rad/Omega), where t_rad is the radiative relaxation time and Omega is the planetary rotation frequency. Alternatively, this transition criterion can be expressed as t_rad ~ t_vert, where t_vert is the timescale for a fluid parcel to move vertically over the difference in day-night thickness. These results subsume the commonly used timescale comparison for estimating heat redistribution efficiency between t_rad and the global horizontal advection timescale, t_adv.Comment: Accepted to ApJ with minor edits compared to version 1; 17 pages, 11 figure