Unextendible Product Basis for Fermionic Systems

We discuss the concept of unextendible product basis (UPB) and generalized UPB for fermionic systems, using Slater determinants as an analogue of product states, in the antisymmetric subspace \wedge^ N \bC^M. We construct an explicit example of generalized fermionic unextendible product basis (FUPB) of minimum cardinality $N(M-N)+1$ for any $N\ge2,M\ge4$. We also show that any bipartite antisymmetric space \wedge^ 2 \bC^M of codimension two is spanned by Slater determinants, and the spaces of higher codimension may not be spanned by Slater determinants. Furthermore, we construct an example of complex FUPB of $N=2,M=4$ with minimum cardinality $5$. In contrast, we show that a real FUPB does not exist for $N=2,M=4$ . Finally we provide a systematic construction for FUPBs of higher dimensions using FUPBs and UPBs of lower dimensions.Comment: 17 pages, no figure. Comments are welcom

Essays of Asset Pricing

This dissertation explores issues related to asset pricing anomalies by focusing on options market. It consists of 3 chapters. In Chapter 1, we find that firms’ left-tail risk is a strong positive predictor of future bear spread returns, suggesting that the options market underreacts to firms’ left-tail risk and the downside protection provided by bear spreads is not adequately priced. We provide a behavioral explanation for this phenomenon. We find that the underreaction to firms’ left-tail risk is stronger when the underlying stocks experience larger recent losses, are closer to their 52-week lows, and have higher information uncertainty. The options market’s underreaction to firms’ left-tail risk mainly happens in high investor sentiment periods. In Chapter 2, we decompose and analyze the straddle returns around firms' earnings announcements. Previous study shows that delta-neutral straddles earn positive returns around earnings announcements, indicating an underpricing of earnings-induced risk. This study uses a volatility-jump decomposition to analyze the driving components of the delta-neutral straddle returns. We find that the volatility component consistently generates positive returns. The jump component’s return is positive over the pre-announcement period and becomes negative after announcement. Our findings suggest that options market anticipates earnings-induced jumps. The overall pattern of delta-neutral straddle’s cumulative return is mainly driven by its jump component but the positive cumulative return after announcement is mainly driven by its volatility component. In Chapter 3, we propose a gambling activity measure by jointly considering open interest and moneyness of out-of-the-money individual equity call options. The new measure, CallMoney, captures excessive optimism during the dot-com bubble, the oil price bubble, and the pre-GFC stock market bubble. CallMoney robustly and negatively predicts both out-of-the-money and at-the-money call option returns cross-sectionally. The option return predictability of CallMoney is stronger when stock price is further from its 52-week high, capital gains overhang is lower, and when information uncertainty of the underlying stock is higher. CallMoney also robustly and negatively predicts cross-sectional stock returns. Comparing to lottery-like-payoffs based (indirect) gambling measures, CallMoney performs better at predicting both option and stock returns

Stochastic gravitational-wave background from spin loss of black holes

Although spinning black holes are shown to be stable in vacuum in general relativity, there exists exotic mechanisms that can convert the spin energy of black holes into gravitational waves. Such waves may be very weak in amplitude, since the spin-down could take a long time, and a direct search may not be feasible. We propose to search for the stochastic background associated with the spin-down, and we relate the level of this background to the formation rate of spinning black holes from the merger of binary black holes, as well as the energy spectrum of waves emitted by the spin-down process. We argue that current LIGO-Virgo observations are not inconsistent with the existence of a spin-down process, as long as it is slow enough. On the other hand, the background may still exist as long as a moderate fraction of spin energy is emitted within Hubble time. This stochastic background could be one interesting target of next generation GW detector network, such as LIGO Voyager, and could be extracted from total stochastic background

Universal Entanglers for Bosonic and Fermionic Systems

A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles $1,2$ with a Hilbert space $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$, a UE exists when $\min{(d_1,d_2)}\geq 3$ and $(d_1,d_2)\neq (3,3)$. It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ if and only if $d\geq 3$ (resp. $d\geq 8$). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.Comment: 15 pages, comments welcome, TQC2013 Accepted Tal

Nasal Bacterial Microbiome: Probing a Healthy Porcine Family

Upper respiratory tract (URT) infection caused the leading and devastating diseases in pigs. It was believed that the normal microbiome of URT plays a vital role in health and disease development. As the entry point of the URT, little knowledge of bacterial microbiome in porcine nasal was known. A cultivation-independent approach directly to 16s ribosomal RNA genes enabled us to reveal the nasal bacterial community, structure and diversity. Here, we found that an unprecedented 207 phylotypes were characterized from 933 qualified clones, indicating the variable, species richness but particularly dominant bacterial microbiome. The dominant species were from genus Comamonas and Acinetobacter, which constitute core normal bacterial microbiome in porcine nasal. Moreover, a set of swine specific pathogens and zoonotic agents were detected in the swine nasal microbiome. Collectively, we provided a snapshot of our current knowledge of the community structure of porcine nasal bacterial ecosystem in a healthy family that will further enhance our view to understand URT infection and public health