We present a method to estimate the probabilities of outcomes of a quantum observable, its mean value, and higher moments by measuring any other observable. This method is general and can be applied to any quantum system. In the case of estimating the mean energy of an isolated system, the estimate can be further improved by measuring the other observable at different times. Intuitively, this method uses interplay and correlations between the measured observable, the estimated observable, and the state of the system. We provide two bounds: one that is looser but analytically computable and one that is tighter but requires solving a nonconvex optimization problem. The method can be used to estimate expectation values and related quantities such as temperature and work in setups where performing measurements in a highly entangled basis is difficult, finding use in state-of-the-art quantum simulators. As a demonstration, we show that in Heisenberg and Ising models of ten sites in the localized phase, performing two-qubit measurements excludes 97.5% and 96.7% of the possible range of energies, respectively, when estimating the ground-state energy.

Dealing with a generic time-local non-Markovian master equation, we define current and power to be process-dependent as in classical thermodynamics. Each process is characterized by a symmetry transformation, a gauge of the master equation, and is associated with different amounts of heat and/or work. Once the symmetry requirement fixes the thermodynamical quantities, a consistent gauge interpretation of the laws of thermodynamics emerges. We also provide the necessary and sufficient conditions for a system to have a gauge-independent thermodynamical behavior and show that systems satisfying quantum detailed balance conditions are gauge-independent. Applying the theory to quantum thermal engines, we show that gauge transformations can change the machine efficiency, however, yet constrained by the classical Carnot bound.

The number partitioning problem (NPP) is one of the NP-complete (nondeterministic polynomial-time complete) computational problems. Its definite exact solution generally requires a check of all $N$ solution candidates, which is exponentially large. Here we describe a path to the fast solution of this problem in $\sqrt{N}$ quasi-adiabatic quantum annealing steps. We argue that the errors due to the finite duration of the quantum annealing can be suppressed if the annealing time scales with $N$ only logarithmically. Moreover, our adiabatic oracle is topologically protected, in the sense that it is robust against small uncertainty and slow time dependence of the physical parameters or the choice of the annealing protocol. We also argue that our approach can solve many other famous NP-complete computational problems in $\sqrt{N}$ steps.

Multiphoton emissions constitute a fundamental source of noise in quantum repeaters and other quantum communication protocols when probabilistic photon sources are employed. In this paper it is shown that by alternating the Bell state measurement basis in concatenated entanglement swapping links one can automatically identify and discard many errors from stimulated multiphoton emissions. The proposed protocol is shown to completely eliminate the dominant quadratic growth of multiphoton errors with the length of the repeater chain. Furthermore, it is shown that the protocol can be employed in satellite-assisted entanglement distribution links to enable links which are more robust in the presence of imbalanced channel losses. The analysis introduces a convenient calculus based on Clifford algebra for modeling concatenated entanglement swapping links with multiphoton emissions. In particular, we present a compact expression for the fidelity of the Bell state produced by a repeater chain of arbitrary length including noise from double-pair emissions.

Driven by the need to develop efficient atom-photon interfaces, recent efforts have proposed replacing cavities by large arrays of cold atoms that can support subradiant or superradiant collective states. In practice, subradiant states are decoupled from radiation, which constitutes a hurdle to most applications. In this work, we study theoretically a protocol that bypasses this limit using a one dimensional chain composed of $N$ three-level atoms in a V-shaped configuration. Throughout the protocol, the chain behaves as a time-varying metamaterial: enabling absorption, storage, and on-demand emission in a spectrally and spatially controlled mode. Taking into account the quantum nature of atoms, we establish the boundary between the linear regime and the nonlinear regime. In the nonlinear regime, we demonstrate that doubly excited states can be coherently transferred from superradiant to subradiant states, opening the way to the optical characterization of their entanglement.

We present a combined experimental and theoretical study of the spectrum of magnetically tunable Feshbach resonances in NaLi $\left(a^3{\mathrm{\Sigma }}^{+}\right)+$ Na collisions. In the accompanying paper, J. J. Park, H. Son, Y.-K. Lu, T. Karman, M. Gronowski, M. Tomza, A. O. Jamison, and W. Ketterle, Phys. Rev. X 13, 031018 (2023), we observe experimentally 8 and 17 resonances occurring between $B=0$ and 1400 G in upper and lower spin-stretched states, respectively. Here, we perform ab initio calculations of the NaLi $+$ Na interaction potential and describe in detail the coupled-channel scattering calculations of the Feshbach resonance spectrum. The positions of the resonances cannot be predicted with realistic uncertainty in the state-of-the-art ab initio potential, but our calculations yield a typical number of resonances that is in near-quantitative agreement with experiment. We show that the main coupling mechanism results from spin-rotation and spin-spin couplings in combination with the anisotropic atom-molecule interaction. The calculations furthermore explain the qualitative difference between the numbers of resonances in either spin state.

We propose theoretically a quantum interferometer in which the NOON state interferometer is combined with the Hong-Ou-Mandel interferometer. With this interferometer, the temporal interference patterns associated with biphoton frequency sum and difference can be shown in different parts of a single interferogram. It is thus possible to simultaneously obtain the spectral correlation information of biphotons in both frequency sum and difference by taking the Fourier transform from such an interferogram. This provides a method for complete spectral characterization of an arbitrary two-photon state with exchange symmetry and might be useful in quantum Fourier-transform spectroscopy where direct spectral measurement is difficult. Furthermore, as it can realize the measurement of time intervals on three scales at the same time, we expect that it can provide a method in quantum metrology. Finally, we discuss another potential application of such an interferometer in the generation and characterization of high-dimensional and phase-controlled frequency entanglement.

We analyze the formation of transient patterns and spin-spin correlations in quasi-two-dimensional spin-1 homogeneous Bose-Einstein condensates subjected to parametric driving of $s$-wave scattering lengths. The dynamics for an initial ferromagnetic phase is identical to that of a scalar condensate. In contrast, intriguing dynamics emerges for an initial polar state. For instance, we show that competition exists between density patterns and spin-mixing dynamics. Dominant spin-mixing dynamics lead to a gas of polar core vortices and antivortices of different spin textures. The density modes of the Bogoliubov spectrum govern the wave-number selection of Faraday patterns. The spin modes determine the vortex density and the spatial dependence of spin-spin correlation functions. When the density patterns outgrow the spin-mixing dynamics, the spin-spin correlations decay exponentially with a correlation length of the order of a spin healing length; otherwise, they exhibit a Bessel function dependence. Strikingly, competing instabilities within density and spin modes emerge when both scattering lengths are modulated at different frequencies and appropriate modulation amplitudes. The competing instability leads to a superposition of density patterns or correlation functions of two distinct wavelengths. Our studies reveal that fine control over the driven dynamics can be attained by tuning interaction strengths, quadratic Zeeman field, driving frequencies, and amplitudes.

We generalize the theory of resolving a mixture of two closely spaced spontaneous emission lifetimes to include pure dephasing contributions to decoherence, leading to the resurgence of Rayleigh's curse at small lifetime separations. Considerable resolution enhancement remains possible when lifetime broadening is more significant than that due to pure dephasing. In the limit that lifetime broadening dominates, one can achieve super-resolution either by a tailored one-photon measurement or by Hong-Ou-Mandel interferometry. We describe conditions for which either choice is superior.

Coherences in mutually unbiased bases of states of an isolated quantum system follow a complementarity relation. The nonlocal advantage of quantum coherence (NAQC), defined in a bipartite scenario, is a situation in which the average quantum coherences of the ensembles of one subsystem, effected by a measurement performed on the other subsystem, violate the complementarity relation. We analyze two criteria to detect NAQC for bipartite quantum states. We construct a more generalized version of the criterion to detect NAQC that is better than the standard criterion as it can capture more states exhibiting NAQC. We prove the local unitary invariance of these NAQC criteria. Further on, we focus on investigating the monogamy properties of NAQC in the tripartite scenario. We check for monogamy of NAQC from two perspectives, differentiated by whether or not the nodal observer in the monogamy relation performs the measurement for the nonlocal advantage. We find in particular that in the case where the nodal observer does not perform the measurement, a strong monogamy relation—an exclusion principle—is exhibited by NAQC.

In the design of stochastic models, there is a constant trade-off between model complexity and accuracy. Here we prove that quantum models enable a more favorable trade-off. We present a technique for identifying fundamental upper bounds on the predictive accuracy of dimensionality-constrained classical models. We identify quantum models that surpass this bound by creating an algorithm that learns quantum models given time-series data. We demonstrate that this quantum accuracy advantage is attainable in a present-day noisy quantum device. These results illustrate the immediate relevance of quantum technologies to time-series analysis and offer an instance where their resulting accuracy advantage can be provably established.

Quantum computers now show the promise of surpassing any possible classical machine. However, errors limit this ability and current machines do not have the ability to implement error correcting codes due to the limited number of qubits and limited control. Therefore, dynamical decoupling (DD) and encodings that limit noise with fewer qubits are more promising. For these reasons, we put forth a model of universal quantum computation that has many advantages over strategies that require a large overhead such as the standard quantum-error correcting codes. First, we separate collective noise from individual noises on physical qubits and use a decoherence-free subspace that uses just two qubits for its encoding to eliminate collective noise. Second, our bath model is very general as it uses a spin-boson-type bath but without any Markovian assumption. Third, we are able to either use a steady global magnetic field or to devise a set of DD pulses that remove much of the remaining noise and commute with the logical operations on the encoded qubit. This allows removal of noise while implementing gate operations. Numerical support is given for this hybrid protection strategy which provides an efficient approach to deal with the decoherence problems in quantum computation and is experimentally viable for several current quantum computing systems. This is emphasized by a recent experiment on superconducting qubits which shows promise for increasing the number of gates that can be implemented reliably with some realistic parameter assumptions.

We present a method for implementing quantum temperature sensing for extremely low temperatures in a quasi-1D dipolar Bose-Einstein condensate reservoir with a magnetic field-driven impurity atom acting as a quantum sensor. By analyzing the quantum signal-to-noise ratio (QSNR) as a metric for temperature sensing performance, we demonstrate that the presence of an attractive dipolar interaction in the reservoir, which includes the effects of non-Markovian dynamics on the sensor, significantly enhances estimation efficiency. We also investigate the steady-state estimation efficiency for long-encoding times through an analytical expression, which shows that the optimal QSNR depends on the driving magnetic field of the impurity atom. Our method can achieve high-efficiency temperature sensing for any low temperature by tuning the magnetic field. These findings suggest that our approach has potential applications in high-resolution quantum thermometry.

We demonstrate that the transition between the fine-structure splitting of the ground state of triply ionized zirconium (Zr iv) is suitable for a terahertz (THz) atomic clock. Its transition frequency is about 37.52 THz and is mainly guided by the magnetic dipole (M1) transition, and it is accessible by a readily available laser. We suggest considering stable even isotopes of Zr and $M_J=\pm 1/2$ sublevels (i.e., $|4D_{3/2},M_J=\pm 1/2\rangle \to |4D_{5/2},M_J=\pm 1/2\rangle$ clock transition) for experimental advantage. By performing necessary calculations, we estimate possible systematics due to blackbody radiation and ac Stark, electric quadrupole, and second-order Zeeman shifts along with shifts due to the second-order Doppler effects. The proposed THz atomic clock could be very useful in quantum thermometry and frequency metrology.

Dynamical encirclement around higher-order exceptional point (EP) singularities can provide control over higher-order modal transport, which holds promise for on-chip device applications. For future photonic technologies, controlled light steering is in demand for communication, data processing, and computing. Nonlinearity-driven dynamical encirclement of second-order EPs (EP2s) and corresponding chirality or nonchirality driven nonadiabatic modal dynamics have attracted attention in the topological study of various non-Hermitian photonic systems. Here, we propose the concept of nonlinearity-induced higher-order conjugate EPs, which are analytically associated with multiple pairs of conjugate EP2s. Moreover, in the presence of saturable nonlinearity, dynamical encirclement around higher-order conjugate EPs in two $\mathcal{T}$-symmetric systems are yet to be explored. Here, we report the hosting of higher-order conjugate EPs in two complementary photonic systems in the presence of nonlinearity connected with $\mathcal{T}$-symmetry by using the framework of a multilayer gain-loss assisted planar optical waveguide. We establish that, in the absence of chirality, the same amount of focusing and defocusing types of saturable nonlinearity leads to different dominating higher-order output modes, irrespective of the choice of the inputs, in two $\mathcal{T}$-symmetric devices. An analytical model to describe this anomalous higher-order mode collapsing phenomenon in two $\mathcal{T}$-symmetry systems owing to the interplay between nonlinearity and the topology of the higher-order conjugate EPs has been developed. Our findings provide a promising opportunity to switch higher-order modes and open up a credible platform to study the physics of nonlinearity-induced higher-order conjugate EPs in $\mathcal{T}$-symmetry systems.

The relativity of simultaneity together with the definition of a quantum state collapse results in experimental situations where multiple measurements can be taken on an uncollapsed quantum state. The quantum state's collapse is defined to be instantaneous in a rest inertial frame of a detector performing measurements on the quantum system. The definition is consistent with the Copenhagen interpretation and in agreement with all measurements performed with detectors at rest in an arbitrary Lorentz (laboratory) frame. From the introduced collapse model follows that under certain conditions multiple measurements are allowed on the same uncollapsed quantum state. An application of the developed approach is shown on measurement of photon-pair state entangled in polarization and energy. Conditions under which two measurements can be taken on the uncollapsed photon-pair state are derived. Serious consequences follow from the allowance of multiple measurements on the same uncollapsed state. For example, the measurements taken by both detectors in this situation are uncorrelated. Moreover, all the conservation laws could be violated in individual measurements, but not in mean values. This statement is proved on the two-photon state entangled in energy. This is in contradiction with experimental results observed by the detectors in rest relative to each other. It is shown that the property of measuring uncorrelated results with detectors in relative movement is related solely to the proposed collapse model. The remaining collapse models—the preferred Lorentz frame, Aharonov-Albert, and Hellwig-Kraus—are examined and discussed with respect to the designed experiment, which involves spacelike separated measurements.

This paper proposes and analyzes a theoretical concept of a parity-time ($\mathcal{P}\mathcal{T}$) symmetry structure of two coupled microresonators. We propose a particle sensing scheme based on exceptional points (EPs), degeneracy points of non-Hermitian systems, where both eigenvalues and eigenvectors coalesce and where a transition from the exact $\mathcal{P}\mathcal{T}$ phase to the broken $\mathcal{P}\mathcal{T}$ phase occurs. The abrupt nature of this phase transition is used here for enhanced frequency splitting, based on the spectral response characteristics of the resonator electromagnetic field coupled to a single emitter. The method proposes the usage of a strong coupling of an external emitter to a $\mathcal{P}\mathcal{T}$-symmetric resonator structure at and near an EP.

Dephasing processes significantly impact the performance of deterministic single-photon sources. Dephasing broadens the spectral line and suppresses the indistinguishability of the emitted photons, which is undesirable for many applications, primarily for quantum computing. We consider a light emitted by a two-level system with a pulsed incoherent pump in the presence of the spectral filter. The spectral filter allows control of the second-order autocorrelation function, indistinguishability, and quantum yield. We show that narrow spectral filters can increase the indistinguishability of the emitted light while undermining the quantum yield. The influence of the spectral filter on the second-order autocorrelation function depends on the duration of the pump. When the pump pulse is long compared to the lifetime of the two-level system, the narrow spectral filters lead to a rapid increase in the second-order autocorrelation function. In this limit, the statistics of the light from the two-level system inherit the statistics of the incoherent pump. In the case of the short duration of the pump pulse, it is possible to preserve single-photon properties to some degree for the sublifetime width of the spectral filter. Moreover, the single-photon properties of the light manifest themselves differently, depending on the response time of the quantum system affected by this light. In particular, in the case of long response time, the spectral filter with a sublifetime width can provide the near-zero second-order autocorrelation function.

Motivated by the recent growing interest in the field of $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian systems we theoretically study the emergence of singularities called exceptional points (EPs) in the eigenspectrum of a pseudo-Hermitian Hamiltonian as the strength of Hermiticity-breaking terms increases. Using general symmetry arguments, we characterize the separate energy levels by a topological index which corresponds to the signs $\pm 1$ of the eigenvalues of pseudometric operator $\widehat{\zeta }$ in the absence of Hermiticity-breaking terms. After that, we show explicitly that the formation of second-order EPs is governed by this index: only the pairs of levels with opposite index can provide second-order EPs. Our general analysis is accompanied by a detailed study of EPs' appearance in an exemplary $\mathcal{P}\mathcal{T}$-symmetric pseudo-Hermitian system with the parity operator in the role of $\widehat{\zeta }$: a transverse-field Ising spin chain with a staggered imaginary longitudinal field. Using analytically computed parity indices of all the levels, we analyze the eigenspectrum of the model in general, and the formation of third-order EPs in particular.