We study the relationship between the entanglement, mixedness and energy of two-qubit and two-mode Gaussian quantum states. We parametrize the set of allowed states of these two fundamentally different physical systems using measures of entanglement, mixedness and energy that allow us to compare and contrast the two systems using a phase diagram. This phase diagram enables one to clearly identify not only the physically allowed states, but the set of states connected under an arbitrary quantum operation. We pay particular attention to the maximally entangled mixed states (MEMS) of each system. Following this we investigate how efficiently one may transfer entanglement from two-mode to two-qubit states.
To explore the properties of a two-qubit mixed state, we consider quantum teleportation. The fidelity of a teleported state depends on the resource state purity and entanglement, as characterized by concurrence. Concurrence and purity are functions of state parameters. However, it turns out that a state with larger purity and concurrence, may have comparatively smaller fidelity. By computing teleportation fidelity, concurrence and purity for two-qubit X-states, we show it explicitly. We further show that fidelity changes monotonically with respect to functions of parameters - other than concur…
We investigate the influence of AC driving fields on the coherence properties of one- and two-qubit gate operations. In both cases, we find that for suitable driving parameters, the gate purity improves significantly. A mapping of the time-dependent system-bath model to an effective static model provides analytical results. The resulting purity loss compares favorably with numerical results.
The dynamical map of entanglement and mixedness in four-qubit maximally entangled GHZ state paired with classical channels driven by fractional Gaussian noise is investigated. The qubit-channel coupling is assumed in four distinct ways: common, bipartite, tripartite, and independent local channel-qubit configurations comprising single, double, triple, or independent noisy sources. Using entanglement witness, negativity, purity and von Neumann entropy, except for the independent configuration, we show that indefinite entanglement and purity preservation may be simulated in multipartite GHZ-like…
Whereas single- and two-photon wave packets are usually treated as pure states, in practice they will be mixed. We study how entanglement created with mixed photon wave packets is degraded. We find in particular that the entanglement of a delocalized single-photon state of the electro-magnetic field is determined simply by its purity. We also discuss entanglement for two-photon mixed states, as well as the influence of a vacuum component.
Knowledge of the relationships among different features of quantumness, like entanglement and state purity, is important from both fundamental and practical viewpoints. Yet, this issue remains little explored in dynamical contexts for open quantum systems. We address this problem by studying the dynamics of entanglement and purity for two-qubit systems using paradigmatic models of radiation-matter interaction, with a qubit being isolated from the environment (spectator configuration). We show the effects of the corresponding local quantum channels on an initial two-qubit pure entangled state i…
In this paper we study the decoherence and purity of a driven solid-state qubit in the Ohmic bath by using the method based on the master equation. At first, instead of solving the master equation we investigate the coefficients of the equation which describe the shift in frequency, diffusive, decoherence, and so on. It is shown that one of the coefficients (we called it decoherence coefficient) is crucial to the decoherence of the qubit in the model. Then we investigate the evolution of the purity of the state in the model. From the analysis of the purity we see that the decoherence time of t…
We show the existence of dynamical purity swapping phenomena in the Jaynes-Cummings model. Moreover we show that purity swapping between a qubit and a generic quantum system is possible, provided they are coupled via non-unitary matrix elements interaction. We particularize to the case of a quantum optical Ramsey interferometer. We show that using purity swapping, a perfect interference pattern can be obtained at the output port of the interferometer even if we start from totally unpolarized sources. This feature is shown to be associated with the phenomena of recreation of the state vector at…
The entanglement, purity and energy of two isolated two-level atoms which are initially prepared in Bell state and each interacts with a thermal cavity field are investigated by considering the atomic motion and the field-mode structure. We achieve the analytical solutions of the atomic qubits by using the algebraic dynamical approach and the influences of the field-mode structure parameter, the strength of the thermal field and the detuning on the entanglement, purity and energy are discussed. We also investigate the state evolution of the atomic qubits based on the entanglement-purity-energy…
We provide a graphical description of the entanglement of pure-state multiparty systems based on an analogy between a bipartite purity analysis and the centroid of a collection of point masses. This description applies to quantum systems with $N$ parties, each with an arbitrary number of (discrete) states. The case of $N$ qubits is highlighted for simplicity. This geometric description illustrates some of the restrictions in the form of inequalities that apply to entanglement in multiparty systems.
We establish contact between the delocalization properties of pure quantum states, as quantified by their number of principal components, and the average generalized entanglement properties, as quantified by purity measures relative to different observable sets. We find that correlations between products of state vector components with respect to Hamming distance play an important role in the structure of subsystem-based purity measures. In particular, we derive general conditions under which the amount of global multipartite entanglement relates to the inverse participation ratio averaged ove…
A generic model of measurement device which is able to directly measure commonly used quantum-state characteristics such as fidelity, overlap, purity and Hilbert-Schmidt distance for two general uncorrelated mixed states is proposed. In addition, for two correlated mixed states, the measurement realizes an entanglement witness for Werner's separability criterion. To determine these observables, the estimation only one parameter - the visibility of interference, is needed. The implementations in cavity QED, trapped ion and electromagnetically induced transparency experiments are discussed.
In this article, we investigate the purity dynamics of entangled 2 two-level atoms interacting with a single quantized electromagnetic field. We show that the purity of the qubit pairs depends on the initial state of the atomic system. It is found that the superposition case is the best choice to generate entangled states with high purity and hence high entanglement. It is clear that the purity of one qubit can be purified at the expense of the other pair through the phenomena of purity swapping. The mean photon number plays an important role in increasing the purity. The robustness of the qua…
We show that entanglement can always arise in the interaction of an arbitrarily large system in any mixed state with a single qubit in a pure state. This small initial purity is enough to enforce entanglement even when the total entropy is close to maximum. We demonstrate this feature using the Jaynes-Cummings interaction of a two level atom in a pure state with a field in a thermal state at an arbitrarily high temperature. We find the time and temperature variation of a lower bound on the amount of entanglement produced and study the classical correlations quantified by the mutual information…
We analyze and show experimental results of the conditional purity, the quantum discord and other related measures of quantum correlation in mixed two-qubit states constructed from a pair of photons in identical polarization states. The considered states are relevant for the description of spin pair states in interacting spin chains in a transverse magnetic field. We derive clean analytical expressions for the conditional local purity and other correlation measures obtained as a result of a remote local projective measurement, which are fully verified by the experimental results. A simple exac…
Given a finite number $N$ of copies of a qubit state we compute the maximum fidelity that can be attained using joint-measurement protocols for estimating its purity. We prove that in the asymptotic $N\to\infty$ limit, separable-measurement protocols can be as efficient as the optimal joint-measurement one if classical communication is used. This in turn shows that the optimal estimation of the entanglement of a two-qubit state can also be achieved asymptotically with fully separable measurements. The relationship between our global Bayesian approach and the quantum Cramer-Rao bound is also di…
The relation between the symplectic and Lorentz groups is explored to investigate entanglement features in a two-mode bipartite Gaussian state. We verify that the correlation matrix of arbitrary Gaussian states can be associated to a hyperbolic space with a Minkowski metric, which is divided in two regions - separablelike and entangledlike, in equivalence to timelike and spacelike in special relativity. This correspondence naturally allows the definition of two insightful invariant squared distances measures - one related to the purity and another related to amount of entanglement. The second …
The success of quantum information processing applications relies on accurate and efficient characterization of quantum states, especially nearly-pure states. In this work, we investigate a procedure for adaptive qubit state tomography which achieves $O(1/N)$ scaling in accuracy for large $N$. We analyze the performance of the adaptive protocol on the characterization of pure, mixed and nearly-pure states, and clarify the interplay between the purity of the state and the statistical fluctuations of the measurement results. Our results highlight the subtle difference between the characterizatio…
We analyze how a maximally entangled state of two-qubits (e.g., the singlet $ψ_s$) is affected by action of local channels described by completely positive maps $\cE$ . We analyze the concurrence and the purity of states $\varrho_\cE=\cE\otimes\cI[ψ_s]$.Using the concurrence-{\it vs}-purity phase diagram we characterize local channels $\cE$ by their action on the singlet state $ψ_s$. We specify a region of the concurrence-{\it vs.}-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even most general (including non-unital) local channels act…
The Uhlmann-Jozsa fidelity (or, equivalently, the Bures distance) is a basic concept of quantum communication and quantum information, which however is very difficult to measure efficiently without recourse to quantum tomography. Here we propose a direct experimental method to estimate the fidelity between two unknown two-qubit mixed states via the measurement of the upper and lower bounds of the fidelity, which are referred to as the superfidelity and subfidelity, respectively. Our method enables a direct measurement of the first- and second-order overlaps between two arbitrary two-qubit stat…
In this work, we show that while all measures of mixedness may be used to witness entanglement, all such entangled states must have a negative partial transpose (NPT). Though computing the negativity of the partial transpose scales well at high dimension, it relies on knowing the complete quantum state, which does not. To address this, we compare different measures of mixedness over uniform ensembles of joint quantum states at varying dimension to gauge their relative success in witnessing entanglement. In doing so, we find that comparing joint and marginal purities is overwhelmingly more succ…
Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pul…
We study the evolution of purity, entanglement and total correlations of general two--mode Gaussian states of continuous variable systems in arbitrary uncorrelated Gaussian environments. The time evolution of purity, Von Neumann entropy, logarithmic negativity and mutual information is analyzed for a wide range of initial conditions. In general, we find that a local squeezing of the bath leads to a faster degradation of purity and entanglement, while it can help to preserve the mutual information between the modes.
The SU(2) and SU(3) Lie algebras lend themselves naturally to studies of two- and three-well Bose-Einstein condensates, with the group operators being expressed in terms of bosonic annihilation and creation operators at each site. The success of these representations has led to the purities associated with these algebras to be promoted as a measure of entanglement between the atomic modes in each well. In this report, we show that these purities do not provide an unambiguous measure, but instead give results which depend on the quantum statistical states of each atomic mode. Using the example …
We present methods for evaluating the rate of change in quantities during quantum evolution due to coupling to the environment (dissipation hereafter). The protocol is based on repeating a given quantum circuit (or quantum operation) twice, thrice, and so on, and measuring an expectation value after each number of repetitions. We start by applying this method for measuring the rate of purity changes in quantum circuits. This provides direct information on the quality of the circuit. Furthermore, the presented scheme enables to distill the dissipative contribution in the changes of quantities s…
We study the quantum entanglement produced by a head-on collision between two gaussian wave packets in three-dimensional space. By deriving the two-particle wave function modified by s-wave scattering amplitudes, we obtain an approximate analytic expression of the purity of an individual particle. The loss of purity provides an indicator of the degree of entanglement. In the case the wave packets are narrow in momentum space, we show that the loss of purity is solely controlled by the ratio of the scattering cross section to the transverse area of the wave packets.
We consider an arbitrary qubit channel depending on a single parameter, which is to be estimated by a physical process. Using the quantum Fisher information per channel invocation to quantify the estimation accuracy, we consider various estimation protocols when the available initial states are mixed with very low purity. We compare a protocol using a single channel invocation on one out of $n$ qubits prepared in a particular correlated input state to the optimal protocol using uncorrelated input states, with the same initial state purity. We show that, to lowest order in initial-state purity,…
We consider the problems of maximizing the entanglement negativity of X-form qubit-qutrit density matrices with (i) a fixed spectrum and (ii) a fixed purity. In the first case, the problem is solved in full generality whereas, in the latter, partial solutions are obtained by imposing extra spectral constraints such as rank-deficiency and degeneracy, which enable a semidefinite programming treatment for the optimization problem at hand. Despite the technically-motivated assumptions, we provide strong numerical evidence that three-fold degenerate X states of purity $P$ reach the highest entangle…
We study production of entanglement with highly-mixed states. We find that entanglement between highly mixed states can be generated via a direct unitary interaction even when both states have purities arbitrarily close to zero. This indicates that purity of a subsystem is not required for entanglement generation. Our result is in contrast to previous studies where the importance of the subsystem purity was emphasized.
A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$-norm, and is denoted by $ν_q$. The multiplicativity problem is the question whether two channels used in parallel have a combined $ν_q$ that is the product of the $ν_q$ of the two channels. A positive answer would …
We investigate the entanglement dynamics and purity of a two-level atom, which is additionally driven by a classical field, interacting with a coherent field in a dissipative environment. It is shown that the amount of entanglement and the purity of the system can be improved by controlling the classical field.
Quantum decoherence due to imperfect manipulation of quantum devices is a key issue in the noisy intermediate-scale quantum (NISQ) era. Standard analyses in quantum information and quantum computation use error rates to parameterize quantum noise channels. However, there is no explicit relation between the decoherence effect induced by a noise channel and its error rate. In this work, we propose to characterize the decoherence effect of a noise channel by the physical implementability of its inverse, which is a universal parameter quantifying the difficulty to simulate the noise inverse with a…
We provide a general method to calculate and maximize the purity of a qubit interacting with an anisotropic non-Markovian environment. Counter to intuition, we find that the purity is often maximized by preparing and storing the qubit in a superposition of non-interacting eigenstates. For a model relevant to decoherence of a heavy-hole spin qubit in a quantum dot or for a singlet-triplet qubit for two electrons in a double quantum dot, we show that preparation of the qubit in its non-interacting ground state can actually be the worst choice to maximize purity. We further give analytical result…
The quantum correlations, including entanglement and discord with its geometric measure, and classical correlation are studied for a bipartite partition of a open or closed quantum system. It is found that the purity of the initial state plays an important role in the dynamics of quantum and classical correlations. In the dephasing model, the quantum correlations loss and the classical correlation gain are instantaneously happen. While, the purity of the initial state destroys the quantum correlations which is resulted by the unitary interaction. Therefore, with the purity parameter, a particu…
We derive a monogamy inequality for entanglement and local contextuality, for any finite bipartite system. It essentially results from the relations between the purity of a local state and the entanglement of the global state, and between the purity of a state and its ability to violate a given noncontextuality inequality. We build an explicit entanglement monotone that satisfies the found monogamy inequality. An important consequence of this inequality, is that there are global states too entangled to violate the local noncontextuality inequality.
Quantum channels depending on a number of classical control parameters are considered. Assuming the stochastic fluctuations of the control parameters in the small errors limit it is shown that the channel fidelity is equal to the average value of the channel purities calculated in the cases of the control errors presence and absence respectively. The result validity is demonstrated on two particular examples, namely, one-qubit quantum gates implemented in the framework of the ion trap setup and the single qubit anisotropic depolarizing quantum channel.
Given a large number N of copies of a qubit state of which we wish to estimate its purity, we prove that separable-measurement protocols can be as efficient as the optimal joint-measurement one if classical communication is used. This shows that the optimal estimation of the entanglement of a two-qubit state can also be achieved asymptotically with fully separable measurements. Thus, quantum memories provide no advantage in this situation. The relationship between our global Bayesian approach and the quantum Cramer-Rao bound is discussed.
Some features of the global entanglement of a composed quantum system can be quantified in terms of the purity of a balanced bipartition, made up of half of its subsystems. For the given bipartition, purity can always be minimized by taking a suitable (pure) state. When many bipartitions are considered, the requirement that purity be minimal for all bipartitions can engender conflicts and frustration arises. This unearths an interesting link between frustration and multipartite entanglement, defined as the average purity over all (balanced) bipartitions.
Entanglement is the backbone of quantum information science and its applications. Entangled states of light are necessary for distributed quantum protocols, quantum sensing and quantum internet. A distributed quantum network requires entanglement between light modes of different colours optimized for interaction with the nodes as well as for communication between them. Here we demonstrate a high-purity Einstein-Podolsky-Rosen (EPR) entangled state between light modes with the wavelengths separated by more than 200 nm. The modes display $-7.7\pm0.5$ dB of two-mode entanglement and an overall st…
Modeling and simulations of entanglement-based quantum-networking protocols commonly assume perfect entangled states. Some investigations have been performed which show how imperfections cause the efficiency of the protocols to rapidly deteriorate. For polarization-entangled states created by SPDC, the fundamentals of phase matching lead to a trade-off problem for the optimal properties of the state. We present a new SPDC method which circumvents the trade-off problem and allows one to obtain degeneracy, indistinguishability, and heralded-single-photon spectral purity greater than 99% for any …
We classify the entanglement of two--mode Gaussian states according to their degree of total and partial mixedness. We derive exact bounds that determine maximally and minimally entangled states for fixed global and marginal purities. This characterization allows for an experimentally reliable estimate of continuous variable entanglement based on measurements of purity.
Bipartite entanglement between two parties of a composite quantum system can be quantified in terms of the purity of one party and there always exists a pure state of the total system that maximizes it (and minimizes purity). When many different bipartitions are considered, the requirement that purity be minimal for all bipartitions gives rise to the phenomenon of entanglement frustration. This feature, observed in quantum systems with both discrete and continuous variables, can be studied by means of a suitable cost function whose minimizers are the maximally multipartite-entangled states (MM…
Trapped atomic ions are an ideal candidate for quantum network nodes, with long-lived identical qubit memories that can be locally entangled through their Coulomb interaction and remotely entangled through photonic channels. The integrity of this photonic interface is generally reliant on purity of single photons produced by the quantum memory. Here we demonstrate a single-photon source for quantum networking based on a trapped $^{138}\mbox{Ba}^+$ ion with a single photon purity of $ g^{2}(0)=(8.1\pm2.3)\times 10^{-5} $ without background subtraction. We further optimize the tradeoff between t…
We analyze the quantum-state purity of heralded single photons produced from frequency-anti-correlated biphotons. We find that the quantum-state purity in time-frequency domain depends strongly on the response time uncertainty of the trigger-photon detector that heralds the generation of its paired photon. If the trigger response time is much shorter than the two-photon coherence time, the time-frequency quantum-state purity of heralded single photons approaches unity and the heralded single photon is in a nearly pure state. If the trigger response time is much longer than the two-photon coher…
Photonic time-frequency entanglement is a promising resource for quantum information processing technologies. We investigate swapping of continuous-variable entanglement in the time-frequency degree of freedom using three-wave mixing in the low-gain regime with the aim of producing heralded biphoton states with high purity and low multi-pair probability. Heralding is achieved by combining one photon from each of two biphoton sources via sum-frequency generation to create a herald photon. We present a realistic model with pulsed pumps, investigate the effects of resolving the frequency of the h…
High-fidelity two-qubits gates are essential for the realization of large-scale quantum computation and simulation. Tunable coupler design is used to reduce the problem of parasitic coupling and frequency crowding in many-qubit systems and thus thought to be advantageous. Here we design a extensible 5-qubit system in which center transmon qubit can couple to every four near-neighbor qubit via a capacitive tunable coupler and experimentally demonstrate high-fidelity controlled-phase (CZ) gate by manipulating center qubit and one near-neighbor qubit. Speckle purity benchmarking (SPB) and cross e…
Quantum programming languages enable developers to implement algorithms for quantum computers that promise computational breakthroughs in classically intractable tasks. Programming quantum computers requires awareness of entanglement, the phenomenon in which measurement outcomes of qubits are correlated. Entanglement can determine the correctness of algorithms and suitability of programming patterns. In this work, we formalize purity as a central tool for automating reasoning about entanglement in quantum programs. A pure expression is one whose evaluation is unaffected by the measurement outc…
We study dynamical generation of entanglement in bipartite quantum systems, characterized by purity (or linear entropy), and caused by the coupling between the two subsystems. Explicit semiclassical theory of purity decay is derived for integrable classical dynamics of the uncoupled system, and for localized (general Gaussian wave-packet) initial states. Purity decays as an algebraic function of time times strength of perturbation, independently of the Planck's constant.
