We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to the preservers of the product numerical range.

The accurate and reliable description of measurement devices is a central problem in both observing uniquely nonclassical behaviors and realizing quantum technologies from powerful computing to precision metrology. To date quantum tomography is the prevalent tool to characterize quantum detectors. However, such a characterization relies on accurately characterized probe states, rendering reliability of the characterization lost in circular argument. Here we report a self-characterization method of quantum measurements based on reconstructing the response range—the entirety of attainable measurement outcomes, eliminating the reliance on known states. We characterize two representative measurements implemented with photonic setups and obtain fidelities above 99.99% with the conventional tomographic reconstructions. This initiates range-based techniques in characterizing quantum systems and foreshadows novel device-independent protocols of quantum information applications.

Given commuting families of Hermitian matrices {A ₁, … , A _k } and {B ₁, … , B _k }, conditions for the existence of a completely positive map Φ, such that Φ(A _j ) = B _j for j = 1, … , k, are studied. Additional properties such as unital or/and trace preserving on the map Φ are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits and quantum information science are mentioned.

In quantum mechanics, geometry has been demonstrated as a useful tool for inferring nonclassical behaviors and exotic properties of quantum systems. One standard approach to illustrate the geometry of quantum systems is to project the quantum state space onto the Euclidean space via measurements of observables on the system. Despite the great success of this method in studying two-level quantum systems (qubits) with the celebrated Bloch sphere representation, it is still difficult to reveal the geometry of multidimensional quantum systems. Here we report the first experiment measuring the geometry of such projections beyond the qubit. Specifically, we observe the joint numerical ranges of a triple of observables in a three-level photonic system, providing a complete classification of these ranges. We further show that the geometry of different classes reveals ground-state degeneracies of a Hamiltonian as a linear combination of the observables, which is related to quantum phases in the thermodynamic limit. Our results offer a versatile geometric approach for exploring the properties of higher-dimensional quantum systems.

Although there is considerable interest in the advance bookings model as a forecasting method in the hotel industry, there has been little research analyzing the use of an advance booking curve in forecasting hotel reservations. The mainstream of advance booking models reviewed in the literature uses only the bookings-on-hand data on a certain day and ignores the previous booking data. This empirical study analyzes the entire booking data set for one year provided by the Hotel ICON in Hong Kong, and identifies the trends and patterns in the data. The analysis demonstrates the use of an advance booking curve in forecasting hotel reservations at property level.