We consider games of chance played by someone with external capital that cannot be applied to the game and determine how this affects risk-adjusted optimal betting. Specifically, we focus on Kelly optimization as a metric, optimizing the expected log
arithm of total capital including both capital in play and the external capital. For games with multiple rounds, we determine the optimal strategy through dynamic programming and construct a close approximation through the WKB method. The strategy can be described in terms of short-term utility functions, with risk aversion depending on the ratio of the amount in the game to the external money. Thus, a rational players behavior varies between conservative play that approaches Kelly strategy as they are able to invest a larger fraction of total wealth and extremely aggressive play that maximizes linear expectation when a larger portion of their capital is locked away. Because you always have expected future productivity to account for as external resources, this goes counter to the conventional wisdom that super-Kelly betting is a ruinous proposition.
We argue that stringy effects in a putative gravity-dual picture for SYK-like models are related to the branching time, a kinetic coefficient defined in terms of the retarded kernel. A bound on the branching time is established assuming that the lead
ing diagrams are ladders with thin rungs. Thus, such models are unlikely candidates for sub-AdS holography. In the weak coupling limit, we derive a relation between the branching time, the Lyapunov exponent, and the quasiparticle lifetime using two different approximations.
We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with $Ngg 1$ flavors and a global U(1) charge. We provide a general definition of the charge in the $(G,Sigma)$ formalism, and compute its universal relation to the i
nfrared asymmetry of the Green function. The same relation is obtained by a renormalization theory. The conserved charge contributes a compact scalar field to the effective action, from which we derive the many-body density of states and extract the charge compressibility. We compute the latter via three distinct numerical methods and obtain consistent results. Finally, we present a two dimensional bulk picture with free Dirac fermions for the zero temperature entropy.
We present two techniques that can greatly reduce the number of gates required to realize an energy measurement, with application to ground state preparation in quantum simulations. The first technique realizes that to prepare the ground state of som
e Hamiltonian, it is not necessary to implement the time-evolution operator: any unitary operator which is a function of the Hamiltonian will do. We propose one such unitary operator which can be implemented exactly, circumventing any Taylor or Trotter approximation errors. The second technique is tailored to lattice models, and is targeted at reducing the use of generic single-qubit rotations, which are very expensive to produce by standard fault tolerant techniques. In particular, the number of generic single-qubit rotations used by our method scales with the number of parameters in the Hamiltonian, which contrasts with a growth proportional to the lattice size required by other techniques.
The boundary of a fractionalized topological phase can be gapped by condensing a proper set of bosonic quasiparticles. Interestingly, in the presence of a global symmetry, such a boundary can have different symmetry transformation properties. Here we
present an explicit example of this kind, in the double semion state with time reversal symmetry. We find two distinct cases where the semionic excitations on the boundary can transform either as time reversal singlets or as time reversal (Kramers) doublets, depending on the coherent phase factor of the Bose condensate. The existence of these two possibilities are demonstrated using both field theory argument and exactly solvable lattice models. Furthermore, we study the domain walls between these two types of gapped boundaries and find that the application of time reversal symmetry tunnels a semion between them.
Indistinguishable particles in two dimensions can be characterized by anyonic quantum statistics more general than those of bosons or fermions. Such anyons emerge as quasiparticles in fractional quantum Hall states and certain frustrated quantum magn
ets. Quantum liquids of anyons exhibit degenerate ground states where the degeneracy depends on the topology of the underlying surface. Here we present a novel type of continuous quantum phase transition in such anyonic quantum liquids that is driven by quantum fluctuations of topology. The critical state connecting two anyonic liquids on surfaces with different topologies is reminiscent of the notion of a `quantum foam with fluctuations on all length scales. This exotic quantum phase transition arises in a microscopic model of interacting anyons for which we present an exact solution in a linear geometry. We introduce an intuitive physical picture of this model that unifies string nets and loop gases, and provide a simple description of topological quantum phases and their phase transitions.
