Quantum Physics

The variance of an observable in a pre-selected quantum system, which is always real and non-negative, appears as an increase in the probe wave packet width in indirect measurements. Extending this framework to pre- and post-selected systems, we formulate a complex-valued counterpart of the variance called "weak variance." In our formulation, the real and imaginary parts of the weak variance appear as changes in the probe wave packet width in the vertical-horizontal and diagonal-antidiagonal directions, respectively, on the quadrature phase plane. Using an optical system, we experimentally demonstrate these changes in the probe wave packet width caused by the real negative and purely imaginary weak variances. Furthermore, we show that the weak variance can be expressed as the variance of the weak-valued probability distribution in pre- and post-selected systems. These operational and statistical interpretations support the rationality of formulating the weak variance as a complex counterpart of the variance in pre- and post-selected systems.

We present methods to strictly calculate the finite-key effects in quantum key distribution (QKD) with error rejection through two-way classical communication (TWCC) for the sending-or-not-sending twin-field protocol. Unlike the normal QKD without TWCC, here the probability of tagging or untagging for each two-bit random group is not independent. We rigorously solve this problem by imagining a virtual set of bits where every bit is independent and identical. We show the relationship between the outcome starting from this imagined set containing independent and identical bits and the outcome starting with the real set of non-independent bits. With explicit formulas, we show that simply applying Chernoff bound in the calculation gives correct key rate, but the failure probability changes a little bit.

Conditional distributions, as defined by the Markov category framework, are studied in the setting of matrix algebras (quantum systems). Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such linear maps are positive are obtained. Several examples are provided, including the standard EPR scenario, where the EPR correlations are reproduced in a purely compositional (categorical) manner. A comparison between the Bayes map, the Petz recovery map, and the Leifer--Spekkens acausal belief propagation is provided, illustrating some similarities and key differences.

We consider fast high-fidelity quantum control by using a shortcut to adiabaticity (STA) technique and optimal control theory (OCT). Three specific examples, including expansion of cold atoms from the harmonic trap, atomic transport by moving harmonic trap, and spin dynamics in the presence of dissipation, are explicitly detailed. Using OCT as a qualitative guide, we demonstrate how STA protocols designed from inverse engineering method, can approach with very high precision optimal solutions built about physical constraints, by a proper choice of the interpolation function and with a very reduced number of adjustable parameters.

The surface Hamiltonian for a spin zero particle that is pinned to a surface by letting the thickness of a layer surrounding the surface go to zero -- assuming a strong normal force -- is constructed. The new approach we follow to achieve this is to start with an expression for the 3D momentum operators whose components along the surface and the normal to the surface are separately Hermitian. The normal part of the kinetic energy operator is a Hermitian operator in this case. When this operator is dropped and the thickness of the layer is set to zero, one automatically gets the Hermitian surface Hamiltonian that contains the geometric potential term as expected. Hamiltonians for both a neutral and a charged particle in an electromagnetic field are constructed. We show that a Hermitian surface and normal momenta emerge automatically once one symmetrizes the usual normal and surface momentum operators. The present approach makes it manifest that the geometrical potential originates from the term that is added to the surface momentum operator to render it Hermitian; this term itself emerges from symmetrization/ordering of differential momentum operators in curvilinear coordinates. We investigate the connection between this approach and the similar approach of Jenssen and Koppe and Costa ( the so called Thin-Layer Quantization (TLQ)). We note that the critical transformation of the wavefunction introduced there before taking the thickness of the layer to zero actually -- while not noted explicitly stated by the authors -- renders each of the surface and normal kinetic energy operators Hermitian by itself, which is just what our approach does from the onset.

We propose two systematic constructions of deletion-correcting codes for protecting quantum information. The first one works with qudits of any dimension, but only one deletion is corrected and the constructed codes are asymptotically bad. The second one corrects multiple deletions and can construct asymptotically good codes. The second one also allows conversion of stabilizer-based quantum codes to deletion-correcting codes, and entanglement assistance.

We propose and theoretically analyze the use of coherent population trapping of a single diamond nitrogen vacancy (NV) center for continuous real-time sensing. The formation of the dark state in coherent population trapping prevents optical emissions from the NV center. Fluctuating magnetic fields, however, can kick the NV center out of the dark state, leading to a sequence of single-photon emissions. A time series of the photon counts detected can be used for magnetic field estimations, even when the average photon count per update time interval is much smaller than 1. For a theoretical demonstration, the nuclear spin bath in a diamond lattice is used as a model fluctuating magnetic environment. For fluctuations with known statistical properties, such as an Ornstein-Uhlenbeck process, Bayesian inference-based estimators can lead to an estimation variance that approaches the classical Cramer-Rao lower bound and can provide dynamical information on a timescale that is comparable to the inverse of the average photon counting rate. Real-time sensing using coherent population trapping adds a new and powerful tool to the emerging technology of quantum sensing.

This thesis focuses on three main questions in the continuous variable and optical settings: where does a quantum advantage, that is, the ability of quantum machines to outperform classical machines, come from? How to ensure the proper functioning of a quantum machine? What advantages can be gained in practice from the use of quantum information? Quantum advantage in continuous variable comes in particular from the use of so-called non-Gaussian quantum states. We introduce the stellar formalism to characterize these states. We then study the transition from classically simulable models to models which are universal for quantum computing. We show that quantum computational supremacy, the dramatic speedup of quantum computers over their classical counterparts, may be realised with non-Gaussian states and Gaussian measurements. Quantum certification denotes the methods seeking to verify the correct functioning of a quantum machine. We consider certification of quantum states in continuous variable, introducing several protocols according to the assumptions made on the tested state. We develop efficient methods for the verification of a large class of multimode quantum states, including the output states of the Boson Sampling model, enabling the experimental verification of quantum supremacy with photonic quantum computing. We give several new examples of practical applications of quantum information in linear quantum optics. Generalising the swap test, we highlight a connection between the ability to distinguish two quantum states and the ability to perform universal programmable quantum measurements, for which we give various implementations in linear optics, based on the use of single photons or coherent states. Finally, we obtain, thanks to linear optics, the first implementation of a quantum protocol for weak coin flipping, a building block for many cryptographic applications.

A compact scheme for the preparation of macroscopic multipartite entanglement is proposed and analyzed. In this scheme the vibrational modes of a mechanical resonator constitute continuous variable (CV) subsystems that entangle to each other as a result of their interaction with a two-level system (TLS). By properly driving the TLS, we show that a selected set of modes can be activated and prepared in a multipartite entangled state. We first study entanglement properties of a three-mode system by evaluating the genuine multipartite entanglement. And investigate its usefulness as a quantum resource by computing the quantum Fisher information. Moreover, the robustness of the state against the qubit and thermal noises is studied, proving a long-lived entanglement. To examine the scalability and structural properties of the scheme, we derive an effective model for the multimode system through elimination of the TLS dynamics. This work provides a step towards a compact and versatile device for creating multipartite noise-resilient entangled state in vibrational modes as a resource for CV quantum metrology, quantum communication, and quantum computation.

We investigate the continuous-time dynamics of highly-entangling intermediate-scale quantum circuits in the presence of dissipation and decoherence. By compressing the Hilbert space to a time-dependent ``corner" subspace that supports faithful representations of the density matrix, we simulate a noisy quantum Fourier transform processor with up to 21 qubits. Our method is efficient to compute with a controllable accuracy the time evolution of intermediate-scale open quantum systems with moderate entropy, while taking into account microscopic dissipative processes rather than relying on digital error models. The circuit size reached in our simulations allows to extract the scaling behaviour of error propagation with the dissipation rates and the number of qubits. Moreover, we show that depending on the dissipative mechanisms at play, the choice of input state has a strong impact on the performance of the quantum algorithm.