When the von Neumann entropy (VNE) of a system increases due to measurements, certain information is lost, some of which may be recoverable. We define information retrievability (IR) and information loss (IL) as functions of the density matrix through VNE to illustrate the relationship between gain and loss. We demonstrate that when a pure, unbiased m-qubit state collapses into a maximally mixed state, it experiences the maximal loss of information and the highest gain in entropy, equivalent to the m-bit classical Shannon entropy. We analyze the VNE, IR, and IL of single qubits, entangled photon pairs in Bell tests, three-qubit systems in quantum teleportation, multiple-qubit systems of GHZ and W states, and two-qubit Werner mixed states, emphasizing their IL dependence on parameters such as polarization bias and qubit count. Data exchange between two observers in Bell tests can recover some of the lost quantum information and eliminate the associated quantum entropy, even years later. The need to recover knowledge explains why no spooky action occurs at a distance. We show that measuring the Bell, GHZ, and marginally entangled Werner states yields the same minimum entropy gain (ln2) and equal minimal information loss (50 percent).
In the present paper we study the entropy gain $H(Φ[ρ])-H(ρ)$ for infinite-dimensional channels $Φ$. We show that unlike finite-dimensional case where the minimal entropy gain is always nonpositive \cite{al}, there is a plenty of channels with positive minimal entropy gain. We obtain the new lower bound and compute the minimal entropy gain for a broad class of Bosonic Gaussian channels by proving that the infimum is attained on the Gaussian states.
We are discussing a universal non-unitary map M subsequent to a generic unitary map U, whose von Neumann entropy gain coincides with the calculated thermodynamic entropy production. For many-body quantum reservoirs we prove that M can be the averaging over all translations of the spatial frame. Assuming the coincidence of microscopic and macroscopic entropy productions leads to a novel equation between entropy gain of frame averaging and relative entropy. Our map M turns out to coincide with the older one called twirl, used recently in the theory of quantum reference frames. Related results to…
There are numerous characterizations of Shannon entropy and Tsallis entropy as measures of information obeying certain properties. Using work by Faddeev and Furuichi, we derive a very simple characterization. Instead of focusing on the entropy of a probability measure on a finite set, this characterization focuses on the `information loss', or change in entropy, associated with a measure-preserving function. Information loss is a special case of conditional entropy: namely, it is the entropy of a random variable conditioned on some function of that variable. We show that Shannon entropy gives …
There are several inequalities in physics which limit how well we can process physical systems to achieve some intended goal, including the second law of thermodynamics, entropy bounds in quantum information theory, and the uncertainty principle of quantum mechanics. Recent results provide physically meaningful enhancements of these limiting statements, determining how well one can attempt to reverse an irreversible process. In this paper, we apply and extend these results to give strong enhancements to several entropy inequalities, having to do with entropy gain, information gain, entropic di…
We illustrate, through a series of prototypical examples, that linear parity-time (PT) symmetric lattices with extended gain/loss profiles are generically unstable, for any non-zero value of the gain/loss coefficient. Our examples include a parabolic real potential with a linear imaginary part and the cases of no real and constant or linear imaginary potentials. On the other hand, this instability can be avoided and the spectrum can be real for localized or compact PT-symmetric potentials. The linear lattices are analyzed through discrete Fourier transform techniques complemented by numerical …
We discuss the impact of gain and loss on the evolution of photonic quantum states and find that PT-symmetric quantum optics in gain/loss systems is not possible. Within the framework of macroscopic quantum electrodynamics we show that gain and loss are associated with non-compact and compact operator transformations, respectively. This implies a fundamentally different way in which quantum correlations between a quantum system and a reservoir are built up and destroyed.
We consider a line with noise in the simplest case. Loss does not add noise. Amplification via phase insensitive amplifiers do add noise. A lower bound of this capacity is the quantum analog to the Shannon capacity of a linear channel with additive white Gaussian noise, namely the difference of the Von Neumann entropy of the signal plus noise at the output of the line and the entropy of the noise alone. We show that this expression is indeed the capacity for the case of an amplifier with infinitesimal gain $G = 1+ε$, and for a cascade of an amplifier with arbitrary gain and a large loss, such …
We consider the problem of quantum noise extraction from polarization swapping in a gain-switched VCSEL. The principle of operation of a quantum random number generator is based on the generation of laser pulses with one of two orthogonal polarization states, followed by digitization of polarization-resolved pulses with a comparator. At intensity values of laser pulses close to the threshold value of the comparator, the contribution of the classical noise of the photodetector will have a crucial role in making a decision on the choice of a logical zero or one. We show how to evaluate the contr…
The kinetic energy term of Hamiltonian systems with balanced loss and gain is not semi-positive-definite, leading to instabilities at the classical as well quantum level. It is shown that an additional Lorentz interaction in the Hamiltonian allows the kinetic energy term to be semi-positive-definite and thereby, improving the stability properties of the system. Further, a consistent quantum theory admitting bound states may be obtained on the real line instead of Stoke wedges on the complex plane. The Landau Hamiltonian in presence of balanced loss and gain is considered for elucidating the ge…
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument, positive operator valued measure, etc., by using quantum stochastic differential equations and by using classical stochastic differential equations (SDE's) for vectors in Hilbert spaces or for trace-class operators. In the same times Ozawa made developments in the theory of instruments and introduced the related notions of a posteriori states and of information gain [1]. In this paper we introduce a simple class of SDE's relevant to the the…
We study a one-dimensional topological superconductor, the Kitaev chain, under the influence of a non-Hermitian but $\mathcal{PT}$-symmetric potential. This potential introduces gain and loss in the system in equal parts. We show that the stability of the topological phase is influenced by the gain/loss strength and explicitly derive the bulk topological invariant in a bipartite lattice as well as compute the corresponding phase diagram using analytical and numerical methods. Furthermore we find that the edge state is exponentially localized near the ends of the wire despite the presence of ga…
The experimental realization of balanced gain and loss in a quantum system has been a long standing goal in quantum mechanics since the introduction of the concept of $\mathcal{PT}$ symmetry and has only recently been achieved. In this paper we analyze balanced gain and loss in Gaussian multi-well potentials with either only gain or loss in each well. By means of symmetrization via matrix models we can construct asymmetric extended potentials with partially real or complex conjugate spectra. This will be demonstrated explicitly for double-well and triple-well systems. Such systems can be reali…
Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential averag…
Balanced gain and loss renders the mean-field description of Bose-Einstein condensates PT symmetric. However, any experimental realization has to deal with unbalancing in the gain and loss contributions breaking the PT symmetry. We will show that such an asymmetry does not necessarily lead to a system without a stable mean-field ground state. Indeed, by exploiting the nonlinear properties of the condensate, a small asymmetry can stabilize the system even further due to a self-regulation of the particle number.
We consider the propagation of Gaussian beams in a waveguide with gain and loss in the paraxial approximation governed by the Schrödinger equation. We derive equations of motion for the beam in the semiclassical limit that are valid when the waveguide profile is locally well approximated by quadratic functions. For Hermitian systems, without any loss or gain, these dynamics are given by Hamilton's equations for the center of the beam and its conjugate momentum. Adding gain and/or loss to the waveguide introduces a non-Hermitian component, causing the width of the Gaussian beam to play an impor…
In systems with ``balanced loss and gain'', the PT-symmetry is broken by increasing the non-hermiticity or the loss-gain strength. We show that finite lattices with oscillatory, PT-symmetric potentials exhibit a new class of PT-symmetry breaking and restoration. We obtain the PT phase diagram as a function of potential periodicity, which also controls the location complex eigenvalues in the lattice spectrum. We show that the sum of PT-potentials with nearby periodicities leads to PT-symmetry restoration, where the system goes from a PT-broken state to a PT-symmetric state as the average loss-g…
We show that a dynamic gain-loss modulation in an optical structure can lead to a direction-dependent parity-time (PT) phase transition. The phase transition can be made thresholdless in the forward direction, and yet remains with a non-zero threshold in the backward direction. As a result, non-reciprocal directional amplification can be realized.
We theoretically propose a scheme to realize rotation sensing based on two coupled whispering-gallery-mode resonators with loss and gain. We consider that the active resonator with gain is rotated while the passive one with loss is stationary. The rotation will induce Sagnac effect and we show that the eigenfrequencies of the supermodes are sensitive to the Sagnac-Fizeau shift. Therefore, we can measure the average photon number in the steady state or the fluctuation spectrum of the output fields to detect the angular velocity of the rotation. We hope that our investigation will be useful in t…
Motivated by recent experiments, we analyse the phonon-assisted steady-state gain of a microwave field driving a double quantum-dot in a resonator. We apply the results of our companion paper, which derives the complete set of fourth-order Lindblad dissipators using Keldysh methods, to show that resonator gain and loss are substantially affected by dephasing-assisted dissipative processes in the quantum-dot system. These additional processes, which go beyond recently proposed polaronic theories, are in good quantitative agreement with experimental observations
As has been shown elsewhere, a reasonable model of the loss of entanglement or correlation that occurs in quantum computations is one which assumes that they can effectively be predicted by a framework that presupposes the presence of irreversibilities internal to the system. It is based on the steepest-entropy-ascent principle and is used here to reproduce the behavior of a controlled-PHASE gate in good agreement with experimental data. The results show that the loss of entanglement predicted is related to the irreversibilities in a nontrivial way, providing a possible alternative approach th…
Loss-induced transmission in waveguides, and reversed pump dependence in lasers, are two prominent examples of counter-intuitive effects in non-Hermitian systems with patterned gain and loss. By analyzing the eigenvalue dynamics of complex symmetric matrices when a system parameter is varied, we introduce a general set of theoretical conditions for these two effects. We show that these effects arise in any irreducible system where the gain or loss is added to a subset of the elements of the system, without the need for parity-time symmetry or for the system to be near an exceptional point. The…
Balanced gain and loss leads to stationary dynamics in open systems. This occurs naturally in PT-symmetric systems, where the imaginary part of the potential describing gain and loss is perfectly antisymmetric. While this case seems intuitive, stationary dynamics are also possible in asymmetric open systems. Open multi-well quantum systems can possess completely or partly real spectra if their Hamiltonian is symmetrised or semi-symmetrised, respectively. In contrast to similar concepts, symmetrisation allows for the description of physical multi-well potentials with gain and loss. A simple mat…
We investigate PT -symmetry breaking transitions in a dimer comprising two LC oscillators, one with loss and the second with gain. The electric energy of this four-mode model oscillates between the two LC circuits, and between capacitive and inductive energy within each LC circuit. Its dynamics are described by a non-Hermitian, PT -symmetric Hamiltonian with three different phases separated by two exceptional points. We systematically measure the eigenfrequencies of energy dynamics across the three regions as a function of gain-loss strength. In addition to observe the well-studied PT transiti…
We consider a non-Hermitian generalization of the Kitaev model and study the existence of stable Majorana zero energy modes. We show that they exist in the limit of zero chemical potential even if balanced gain and loss are randomly distributed along the lattice. We show that Majorana zero modes also appear if the chemical potential is different from zero provided that not the full Hamiltonian but the non-Hermitian part of the Hamiltonian is PT symmetric.
Parity-time ($\mathcal{PT}$) symmetric systems are classical, gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues of a $\mathcal{PT}$-symmetric Hamiltonian change from real to complex conjugates at a critical value of gain-loss strength that is called the $\mathcal{PT}$ breaking threshold. Here, we obtain the $\mathcal{PT}$-threshold for a one-dimensional, finite Kitaev chain -- a prototype for a p-wave superconductor -- in the presence of a single pair of gain and loss potentials as a function of the superconduc…
We use continuous weak measurements of a driven superconducting qubit to experimentally study the information dynamics of a quantum Maxwell's demon. We show how information gained by a demon who can track single quantum trajectories of the qubit can be converted into work using quantum coherent feedback. We verify the validity of a quantum fluctuation theorem with feedback by utilizing information obtained along single trajectories. We demonstrate, in particular, that quantum backaction can lead to a loss of information in imperfect measurements. We furthermore probe the transition between inf…
We investigate the Su-Schrieffer-Heeger model in presence of an injection and removal of particles, introduced via a master equation in Lindblad form. It is shown that the dynamics of the density matrix follows the predictions of calculations in which the gain and loss are modeled by complex $\mathcal{PT}$-symmetric potentials. In particular it is found that there is a clear distinction in the dynamics between the topologically different cases known from the stationary eigenstates.
We consider particle-hole symmetric photonic graphene with balanced gain and loss. We show that edge states with purely imaginary eigenvalues appear along the zigzag edge. We propose an idea that these edge states are protected by spontaneously broken particle-hole symmetry. We discuss that the edge states are topological in the sense that the exceptional rings are robust against symmetry protecting disorder. If the disorder is too strong to restore the spontaneously broken particle-hole symmetry, then such protection of the edge states is lost.
The existence and stability of the nonlinear spatial localized modes are investigated in parity-time symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with competing gain and loss profile. The exact analytical expressions of the localized modes are found for all values of the competing parameter and in the presence of both the self-focusing and self-defocusing Kerr nonlinearity. The effect of competing gain/loss profile on the stability structure of these localized modes are discussed with the help of linear stability analysis followed by the d…
