Error probability analysis in quantum tomography: a tool for evaluating experiments

We expand the scope of the statistical notion of error probability, i.e., how often large deviations are observed in an experiment, in order to make it directly applicable to quantum tomography. We verify that the error probability can decrease at most exponentially in the number of trials, derive the explicit rate that bounds this decrease, and show that a maximum likelihood estimator achieves this bound. We also show that the statistical notion of identifiability coincides with the tomographic notion of informational completeness. Our result implies that two quantum tomographic apparatuses that have the same risk function, (e.g. variance), can have different error probability, and we give an example in one qubit state tomography. Thus by combining these two approaches we can evaluate, in a reconstruction independent way, the performance of such experiments more discerningly.Comment: 14pages, 2 figures (an analysis of an example is added, and the proof of Lemma 2 is corrected

Quantum computation over the butterfly network

In order to investigate distributed quantum computation under restricted network resources, we introduce a quantum computation task over the butterfly network where both quantum and classical communications are limited. We consider deterministically performing a two-qubit global unitary operation on two unknown inputs given at different nodes, with outputs at two distinct nodes. By using a particular resource setting introduced by M. Hayashi [Phys. Rev. A \textbf{76}, 040301(R) (2007)], which is capable of performing a swap operation by adding two maximally entangled qubits (ebits) between the two input nodes, we show that unitary operations can be performed without adding any entanglement resource, if and only if the unitary operations are locally unitary equivalent to controlled unitary operations. Our protocol is optimal in the sense that the unitary operations cannot be implemented if we relax the specifications of any of the channels. We also construct protocols for performing controlled traceless unitary operations with a 1-ebit resource and for performing global Clifford operations with a 2-ebit resource.Comment: 12 pages, 12 figures, the second version has been significantly expanded, and author ordering changed and the third version is a minor revision of the previous versio

Classification of delocalization power of global unitary operations in terms of LOCC one-piece relocalization

We study how two pieces of localized quantum information can be delocalized across a composite Hilbert space when a global unitary operation is applied. We classify the delocalization power of global unitary operations on quantum information by investigating the possibility of relocalizing one piece of the quantum information without using any global quantum resource. We show that one-piece relocalization is possible if and only if the global unitary operation is local unitary equivalent of a controlled-unitary operation. The delocalization power turns out to reveal different aspect of the non-local properties of global unitary operations characterized by their entangling power

Phase-random states: ensembles of states with fixed amplitudes and uniformly distributed phases in a fixed basis

Motivated by studies of typical properties of quantum states in statistical mechanics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We first show that canonical states typically appear in subsystems of phase-random states. We then investigate the simulatability of phase-random states, which is directly related to that of time evolution in closed systems, by studying their entanglement properties. We find that starting from a separable state, time evolutions under Hamiltonians composed of only separable eigenstates generate extremely high entanglement and are difficult to simulate with matrix product states. We also show that random quantum circuits consisting of only two-qubit diagonal unitaries can generate an ensemble with the same average entanglement as phase-random states.Comment: Revised, 12 pages, 4 figur

Frequency noise and intensity noise of next-generation gravitational-wave detectors with RF/DC readout schemes

The sensitivity of next-generation gravitational-wave detectors such as Advanced LIGO and LCGT should be limited mostly by quantum noise with an expected technical progress to reduce seismic noise and thermal noise. Those detectors will employ the optical configuration of resonant-sideband-extraction that can be realized with a signal-recycling mirror added to the Fabry-Perot Michelson interferometer. While this configuration can reduce quantum noise of the detector, it can possibly increase laser frequency noise and intensity noise. The analysis of laser noise in the interferometer with the conventional configuration has been done in several papers, and we shall extend the analysis to the resonant-sideband-extraction configuration with the radiation pressure effect included. We shall also refer to laser noise in the case we employ the so-called DC readout scheme.Comment: An error in Fig. 10 in the published version in PRD has been corrected in this version; an erratum has been submitted to PRD. After correction, this figure reflects a significant difference in the ways RF and DC readout schemes are susceptible to laser noise. In addition, the levels of mirror loss imbalances and input laser amplitude noise have also been updated to be more realistic for Advanced LIG

Effect of nonnegativity on estimation errors in one-qubit state tomography with finite data

We analyze the behavior of estimation errors evaluated by two loss functions, the Hilbert-Schmidt distance and infidelity, in one-qubit state tomography with finite data. We show numerically that there can be a large gap between the estimation errors and those predicted by an asymptotic analysis. The origin of this discrepancy is the existence of the boundary in the state space imposed by the requirement that density matrices be nonnegative (positive semidefinite). We derive an explicit form of a function reproducing the behavior of the estimation errors with high accuracy by introducing two approximations: a Gaussian approximation of the multinomial distributions of outcomes, and linearizing the boundary. This function gives us an intuition for the behavior of the expected losses for finite data sets. We show that this function can be used to determine the amount of data necessary for the estimation to be treated reliably with the asymptotic theory. We give an explicit expression for this amount, which exhibits strong sensitivity to the true quantum state as well as the choice of measurement.Comment: 9 pages, 4 figures, One figure (FIG. 1) is added to the previous version, and some typos are correcte

The chain rule implies Tsirelson's bound: an approach from generalized mutual information

In order to analyze an information theoretical derivation of Tsirelson's bound based on information causality, we introduce a generalized mutual information (GMI), defined as the optimal coding rate of a channel with classical inputs and general probabilistic outputs. In the case where the outputs are quantum, the GMI coincides with the quantum mutual information. In general, the GMI does not necessarily satisfy the chain rule. We prove that Tsirelson's bound can be derived by imposing the chain rule on the GMI. We formulate a principle, which we call the no-supersignalling condition, which states that the assistance of nonlocal correlations does not increase the capability of classical communication. We prove that this condition is equivalent to the no-signalling condition. As a result, we show that Tsirelson's bound is implied by the nonpositivity of the quantitative difference between information causality and no-supersignalling.Comment: 23 pages, 8 figures, Added Section 2 and Appendix B, result unchanged, Added reference