Convex Roof Research Articles

Two novel pure-state coherence measures in quantifying coherence

Quantification is one of the primary goals of quantum coherence resource theory. Here, we put forward two coherence measures, analytically prove their validity in the pure-state regime following the axiomatic definition of a legitimate pure-state coherence measure (PSCM), and provide their general normalized forms. We further discuss their extensions in the domain of mixed states with the assistance of the convex roof of coherence, and top coherence (quantification in terms of pure state coherence) and define their classes (coherence \emph{monotone} or \emph{measure}); in this regard, we thoroughly investigate the top coherence class. For the quantitative demonstration, we pick up different laser pulse-two-qubit interaction scenarios and compare the evolutions of these two coherence measures along with \(l_{1}\)-norm of coherence and relative entropy of coherence; this study also signifies the importance of coherence in probing an atomic or molecular system.

Quantum G-coherence factorization law under fully and strictly incoherent operations

Quantum coherence plays an important role in quantum resource theory, which is strongly related with entanglement. In order to quantify the full coherence of qudit states, we define G-coherence and convex roof of G-coherence, and prove that the G-coherence is a strong coherence monotone and the convex roof of G-coherence is a coherence measure under fully and strictly incoherent operation (FSIO), respectively. Similar to the entanglement factorization law, we prove a coherence factorization law for arbitrary d-dimensional quantum pure and mixed states under FSIO channels, which generalizes the entanglement factorization law for bipartite pure states. Our results will play an important role in the simplification of dynamical coherence measure.

Methods of constructing superposition measures

The resource theory of quantum superposition is an extension of the quantum coherent theory, in which linear independence relaxes the requirement of orthogonality. It can be used to quantify the nonclassical in superposition of finite number of optical coherent states. Based on convex roof extended, state transformation and weight, we give three methods of constructing superposition measures of quantum states, respectively. We also generalize the superposition resource theory from two perspectives.

Closed expressions for one-qubit states of convex roof coherence measures

We study the closed expressions of the convex roof coherence measures for one-qubit states in this paper. We present the analytical expressions for the convex roof coherence measures, , of one-qubit states with (where ) being convex with respect to the l 1 norm of coherence of ϕ (i.e. ), such coherence measures including the coherence of formation, the geometric measure of coherence, the coherence concurrence, and the coherence rank. We further present the operational interpretations of these measures. Finally, we present the usefulness of the convex roof coherence measures being non-convex with respect to by giving the necessary and sufficient conditions for the transformations via incoherent operations, where ϕ i , φ j are one-qubit pure states and .

Concurrence triangle induced genuine multipartite entanglement measure

We study the quantification of genuine multipartite entanglement (GME) for general multipartite states. A set of inequalities satisfied by the entanglement of N-partite pure states is derived by exploiting the restrictions on entanglement distributions, showing that the bipartite entanglement between each part and its remaining ones cannot exceed the sum of the other partners with their remaining ones. Then a series of triangles, named concurrence triangles, are established corresponding to these inequalities. Proper genuine multipartite entanglement measures are thus constructed by using the geometric mean area of these concurrence triangles, which are non-increasing under local operation and classical communication. The GME measures classify which parts are separable or entangled with the rest ones for non genuine entangled pure states. The GME measures for mixed states are given via the convex roof construction, and a witness to detect the GME of multipartite mixed states is presented by an approach based on state purifications. Detailed examples are given to illustrate the effectiveness of our GME measures.

A variational quantum algorithm for approximating convex roofs

Many entanglement measures are first defined for pure states of a bipartite Hilbert space, and then extended to mixed states via the convex roof extension. In this article we alter the convex roof extension of an entanglement measure, to produce a sequence of extensions that we call $f$-$d$ extensions, for $d \in \mathbb{N}$, where $f:[0,1]\to [0, \infty)$ is a fixed continuous function which vanishes only at zero. We prove that for any such function $f$, and any continuous, faithful, non-negative function, (such as an entanglement measure), $\mu$ on the set of pure states of a finite dimensional bipartite Hilbert space, the collection of $f$-$d$ extensions of $\mu$ detects entanglement, i.e. a mixed state $\rho$ on a finite dimensional bipartite Hilbert space is separable, if and only if there exists $d \in \mathbb{N}$ such that the $f$-$d$ extension of $\mu$ applied to $\rho$ is equal to zero. We introduce a quantum variational algorithm which aims to approximate the $f$-$d$ extensions of entanglement measures defined on pure states. However, the algorithm does have its drawbacks. We show that this algorithm exhibits barren plateaus when used to approximate the family of $f$-$d$ extensions of the Tsallis entanglement entropy for a certain function $f$ and unitary ansatz $U(\theta)$ of sufficient depth. In practice, if additional information about the state is known, then one needs to avoid using the suggested ansatz for long depth of circuits.

Exponential Clustering of Bipartite Quantum Entanglement at Arbitrary Temperatures

Macroscopic quantum effects play central roles in the appearance of inexplicable phenomena in low-temperature quantum many-body physics. Such macroscopic quantumness is often evaluated using long-range entanglement, i.e., entanglement in the macroscopic length scale. The long-range entanglement not only characterizes the novel quantum phases but also serves as a critical resource for quantum computation. Thus, the problem that arises is under which conditions can the long-range entanglement be stable even at room temperatures. Here, we show that bi-partite long-range entanglement is unstable at arbitrary temperatures and exponentially decays with distance. Our theorem provides a no-go theorem on the existence of the long-range entanglement. The obtained results are consistent with the existing observations that long-range entanglement at non-zero temperatures can exist in topologically ordered phases, where tripartite correlations are dominant. In the derivation of our result, we introduce a quantum correlation defined by the convex roof of the standard correlation function. We establish an exponential clustering theorem for generic quantum many-body systems for such a quantum correlation at arbitrary temperatures, which yields our main result by relating quantum correlation to quantum entanglement. As a simple application of our analytical techniques, we derived a general limit on the Wigner-Yanase-Dyson skew information and the quantum Fisher information, which will attract significant attention in the field of quantum metrology. Our work reveals novel general aspects of low-temperature quantum physics and sheds light on the characterization of long-range entanglement.

L 1 norm of coherence is not equal to its convex roof quantifier

Since a rigorous framework for quantifying quantum coherence was established by Baumgratz et al (2014 Phys. Rev. Lett. 113 140401), many coherence measures have been found. For a given coherence measure C, extending the values of C on pure states to mixed states by the convex roof construction, we will get a valid coherence measure , we call the corresponding convex roof quantifier of C. Whether for a given coherence measure is an important question. In this work, we show that for the widely used coherence measure, l 1 norm of coherence , it holds that .

Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices

We present several inequalities related to the Robertson-Schr\"odinger uncertainty relation. In all these inequalities, we consider a decomposition of the density matrix into a mixture of states, and use the fact that the Robertson-Schr\"odinger uncertainty relation is valid for all these components. By considering a convex roof of the bound, we obtain an alternative derivation of the relation in Fr\"owis et al. [Phys. Rev. A 92, 012102 (2015)], and we can also list a number of conditions that are needed to saturate the relation. We present a formulation of the Cram\'er-Rao bound involving the convex roof of the variance. By considering a concave roof of the bound in the Robertson-Schr\"odinger uncertainty relation over decompositions to mixed states, we obtain an improvement of the Robertson-Schr\"odinger uncertainty relation. We consider similar techniques for uncertainty relations with three variances. Finally, we present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states based on the variances of the canonical position and momentum operators for two-mode continuous variable systems. We show that the violation of well-known entanglement conditions in these systems discussed in Duan et al., [Phys. Rev. Lett. 84, 2722 (2000)] and Simon [Phys. Rev. Lett. 84, 2726 (2000)] implies that the state is more useful metrologically than certain relevant subsets of separable states. We present similar results concerning entanglement conditions with angular momentum operators for spin systems.

The Tightness of Multipartite Coherence from Spectrum Estimation.

Detecting multipartite quantum coherence usually requires quantum state reconstruction, which is quite inefficient for large-scale quantum systems. Along this line of research, several efficient procedures have been proposed to detect multipartite quantum coherence without quantum state reconstruction, among which the spectrum-estimation-based method is suitable for various coherence measures. Here, we first generalize the spectrum-estimation-based method for the geometric measure of coherence. Then, we investigate the tightness of the estimated lower bound of various coherence measures, including the geometric measure of coherence, the -norm of coherence, the robustness of coherence, and some convex roof quantifiers of coherence multiqubit GHZ states and linear cluster states. Finally, we demonstrate the spectrum-estimation-based method as well as the other two efficient methods. We observe that the spectrum-estimation-based method outperforms other methods in various coherence measures, which significantly enhances the accuracy of estimation.

An Extension of Entanglement Measures for Pure States

AbstractTo quantify the entanglement is one of the most important topics in quantum entanglement theory. An entanglement measure is built from measures for pure states. Conditions when the entanglement measure is entanglement monotone and convex are presented, as well as the interpretation of smoothed one‐shot entanglement cost. Next, a difference between the measure under the local operation and classical communication and the separability‐preserving operations is presented. Then, the relation between the convex roof extended method and the way here for the entanglement measures built from the geometric entanglement measure for pure states, as well as the concurrence for pure states in two‐qubit systems are considered. It is also shown that the measure is monogamous for system.

Coherence of assistance and assisted maximally coherent states

Coherence and entanglement are fundamental concepts in resource theory. The coherence (entanglement) of assistance is the coherence (entanglement) that can be extracted assisted by another party with local measurement and classical communication. We introduce and study the general coherence of assistance. First, in terms of real symmetric concave functions on the probability simplex, the coherence of assistance and the entanglement of assistance are shown to be in one-to-one correspondence. We then introduce two classes of quantum states: the assisted maximally coherent states and the assisted maximally entangled states. They can be transformed into maximally coherent or entangled pure states with the help of another party using local measurement and classical communication. We give necessary conditions for states to be assisted maximally coherent or assisted maximally entangled. Based on these, a unified framework between coherence and entanglement including coherence (entanglement) measures, coherence (entanglement) of assistance, coherence (entanglement) resources is proposed. Then we show that the coherence of assistance as well as entanglement of assistance are strictly larger than the coherence of convex roof and entanglement of convex roof for all full rank density matrices. So all full rank quantum states are distillable in the assisted coherence distillation.

Resource distillation in convex Gaussian resource theories

It is known that distillation in continuous variable resource theories is impossible when restricted to Gaussian states and operations. To overcome this limitation, we enlarge the theories to include convex mixtures of Gaussian states and operations. This extension is operationally well-motivated since classical randomness is easily accessible. We find that resource distillation becomes possible for convex Gaussian resource theories-albeit in a limited fashion. We derive this limitation by studying the convex roof extension of a Gaussian resource measure and then go on to show that our bound is tight by means of example protocols for the distillation of squeezing and entanglement.

Generalized coherence vector applied to coherence transformations and quantifiers

One of the main problems in any quantum resource theory is the characterization of the conversions between resources by means of the free operations of the theory. In this work, we advance on this characterization within the quantum coherence resource theory by introducing the generalized coherence vector of an arbitrary quantum state. The generalized coherence vector is a probability vector that can be interpreted as a concave roof extension of the pure states coherence vector. We show that it completely characterizes the notions of being incoherent, as well as being maximally coherent. Moreover, using this notion and the majorization relation, we obtain a necessary condition for the conversion of general quantum states by means of incoherent operations. These results generalize the necessary conditions of conversions for pure states given in the literature, and show that the tools of the majorization lattice are useful also in the general case. Finally, we introduce a family of coherence quantifiers by considering concave and symmetric functions applied to the generalized coherence vector. We compare this proposal with the convex roof measure of coherence and others quantifiers given in the literature.

Some Remarks on the Entanglement Number

Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Quanta 2020; 9: 22–36.

Quantifying entanglement in terms of an operational way* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11775040, 12011530014 and 11375036) and the Fundamental Research Funds for the Central Universities.China (Grant No. DUT20LAB203).

We establish entanglement monotones in terms of an operational approach, which is closely connected with the state conversion from pure states to the objective state by the local operations and classical communications. It is shown that any good entanglement quantifier defined on pure states can induce an entanglement monotone for all density matrices. Particularly, we show that our entanglement monotone is the maximal one among all those having the same form for pure states. In some special cases, our proposed entanglement monotones turn to be equivalent to the convex roof construction, which hence gain an operational meaning. Some examples are given to demonstrate different cases.

Evaluating single-mode nonclassicality

Quantifying nonclassicality of a bosonic mode is an important but challenge task in quantum optics. Recently, the first nonclassicality measure based on the concept of operational resource theory has been proposed [Phys. Rev. Research 2, 023400 (2020)], which shows several crucial properties as a resource for quantum metrology. Here we apply the measure to evaluate and categorize different classes of nonclassical states. We discover a class of states that can achieve the maximum nonclassicality in the asymptotic limit of large mean number of excitations. These states can provide the same sensitivity in place of a squeezed vacuum in a sensing scheme, e.g., the LIGO experiments. We also discover that the nonclassicality of certain states can be greatly improved with a single-photon addition. Besides, we explore some examples on how to evaluate analytically the nonclassicality of mixed states, which are in general difficult to calculate due to the convex roof nature of the measure. Our results will be useful for preparing and utilizing nonclassical states for applications in precision sensing tasks, such as quadrature sensing and phase sensing in a Mach-Zehnder interferometer.

Quantifying coherence in terms of the pure-state coherence

Quantifying quantum coherence is a key task in the resource theory of coherence. Here we establish a good coherence monotone in terms of a state conversion process, which automatically endows the coherence monotone with an operational meaning. We show that any state can be produced from some input pure states via the corresponding incoherent channels. It is found that the coherence of a given state can be well characterized by the least coherence of the input pure states, so a coherence monotone is established by only effectively quantifying the input pure states. In particular, we show that our proposed coherence monotone is the supremum of all the coherence monotones that give the same coherence for any given pure state. We also prove that our coherence monotone is continuous. Considering the convexity, we prove that our proposed coherence measure is a subset of the coherence measure based on the convex roof construction. The similarities and differences between our coherence monotone and coherence cost are studied in detail. As applications, we give a concrete expression of our coherence measure by employing the geometric coherence of a pure state. We also give a thorough analysis of the states of the qubit and finally obtain a series of analytic coherence measures. The numerical examples are also given to show the difference between our coherence monotone and that based on the convex roof construction.

Experimentally Accessible Lower Bounds for Genuine Multipartite Entanglement and Coherence Measures

Experimentally quantifying entanglement and coherence are extremely important for quantum resource theory. However, because the quantum state tomography requires exponentially growing measurements with the number of qubits, it is hard to quantify entanglement and coherence based on the full information of the experimentally realized multipartite states. Fortunately, other methods have been found to directly measure the fidelity of experimental states without quantum state tomography. Here we present a fidelity-based method to derive experimentally accessible lower bounds for measures of genuine multipartite entanglement and coherence. On the one hand, the method works for genuine multipartite entanglement measures including the convex-roof extended negativity, the concurrence, the G-concurrence, and the geometric measure for genuine multipartite entanglement. On the other hand, the method also delivers observable lower bounds for the convex roof of the $l_{1}$-norm of coherence, the geometric measure of coherence, and the coherence of formation. Furthermore, all the lower bounds are based on the fidelity between the chosen pure state and the target state, and we obtain the lower bounds of several real experimental states as examples of our results.

Tighter polygamy relations for a superposition of the generalized W class state and vacuum beyond qubits

In this article, we present generalized polygamy relations for a superposition of the generalized W class state and vacuum. This class of states were shown to satisfy the strong monogamy relations in terms of the squared convex roof extended negativity [14]. Here we first present a polygamy relation in terms of the β-th power of the linear entanglement of assistance when β∈(0,1] for the superposition of the generalized W class state and vacuum. Then we present a tighter polygamy relation for the class of states in terms of the linear entanglement of assistance under some constraints. At last, we present a class of mixed states in higher dimensional systems satisfying the tighter polygamy relations we present.