Relative Entropy Of Coherence Research Articles

Experimental Investigation of Quantum Uncertainty Relations With Classical Shadows

The quantum component in uncertainty relation can be naturally characterized by the quantum coherence of a quantum state, which is of paramount importance in quantum information science. Here, we experimentally investigate quantum uncertainty relations construed with relative entropy of coherence, l1 norm of coherence, and coherence of formation. Instead of quantum state tomographic technology, we employ the classical shadow algorithm for the detection of lower bounds in quantum uncertainty relations. With an all-optical setup, we prepare a family of quantum states whose purity can be fully controlled. We experimentally explore the tightness of various lower bounds in different reference bases on the prepared states. Our results indicate that the tightness of quantum coherence lower bounds depends on the reference bases and the purity of the quantum state.

Quantum coherence bounds the distributed discords

Establishing quantum correlations between two remote parties by sending an information carrier is an essential step of many protocols in quantum information processing. We obtain trade-off relations between discords and coherence within a bipartite system. Then we study the distribution of coherence in a bipartite quantum state by using the relations of relative entropy and mutual information. We show that the increase of the relative entropy of discord between two remote parties is bounded by the nonclassical correlations quantified by the relative entropy of coherence between the carrier and two remote parties, providing an optimal protocol for discord distribution and showing that quantum correlations are the essential resource for such tasks.

Dynamics of quantum Fisher information and quantum coherence of two interacting atoms under time–fractional analysis

In this paper, we give an exact general solution to the fractional Schrödinger equation of two interacting atoms. It consists of using the Riemann–Liouville fractional integral operator to examine the interaction between two identical atoms under temporal–fractional evolution. The dynamics of quantum Fisher information with respect to changes in the phase parameter and the quantum coherence by means of the l 1 -norm and relative entropy of coherence are discussed. Moreover, the analytical fractional solution allows us to evaluate the relationship between the estimation precision of the phase parameter and the coherence measures for various atom–atom states. Indeed, the obtained results show a good similarity between these measures for certain critical values of the phase parameter. In addition, it is observed that initially, for such coherent-like fractional states, the amount of quantum Fisher information may be sensitive or insensitive to changes in the phase parameter. Therefore, one can conclude that quantum Fisher information can be considered as an indicator of quantum coherence in a composite quantum state.

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

We investigate quantum phase transitions for q-state quantum Potts models (q = 2,3,4) on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme. We extend the universal order parameter to a two-dimensional lattice system, which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G. The universal order parameter is zero in the symmetric phase, and it ranges from zero to unity in the symmetry-broken phase. The ground-state fidelity per lattice site is computed, and a pinch point is identified on the fidelity surface near the critical point. The results offer another example highlighting the connection between (i) critical points for a quantum many-body system undergoing a quantum phase-transition and (ii) pinch points on a fidelity surface. In addition, we discuss three quantum coherence measures: the quantum Jensen–Shannon divergence, the relative entropy of coherence, and the l1 norm of coherence, which are singular at the critical point, thereby identifying quantum phase transitions.

Estimating coherence with respect to general quantum measurements

The conventional coherence is defined with respect to a fixed orthonormal basis, i.e., to a von Neumann measurement. Recently, generalized quantum coherence with respect to general positive operator-valued measurements has been presented. Several well-defined coherence measures, such as the relative entropy of coherence \(C_{r}\), the \(l_{1}\) norm of coherence \(C_{l_{1}}\) and the coherence \(C_{T,\alpha }\) based on Tsallis relative entropy with respect to general POVMs have been obtained. In this work, we investigate the properties of \(C_{r}\), \(C_{l_{1}}\) and \(C_{T,\alpha }\). We estimate the upper bounds of \(C_{l_{1}}\); we show that the minimal error probability of the least square measurement state discrimination is given by \(C_{T,1/2}\); we derive the uncertainty relations given by \(C_{r}\), and calculate the average values of \(C_{r}\), \(C_{T,\alpha }\) and \(C_{l_{1}}\) over random pure quantum states. All these results include the corresponding results of the conventional coherence as special cases.

Evolution of quantum coherence of qutrit-qutrit system under correlated depolarizing channels

In order to study the decoherence process of quantum system and find the ways to delay it, in this paper we investigate the decoherence process under correlated depolarizing channels by calculating the evolution of norm coherence, relative entropy coherence and basis-independent coherence of the three initial states: special initial state, maximum coherent state and isotropic state. From the analytic results, numerical results and evolution images, it can be found that 1) completely correlated channels can suppress decoherence to a greatest extent,. 2) the time point at which maximum decoherence occurs is irrelevant to the degree of correlation, but system’s decoherence evolution behavior depends on it. 3)the correlation of depolarizing channels can reinforce the collective quantum coherence between subsystems, and an inequality relation of basis-independent coherence is proved in this system.

Quantumness of ensemble via coherence of Gram matrix

The Gram matrix of a set of vectors, which encapsulates the relations between the constituent vectors, plays an important role in the exploration of both geometric and information-theoretic aspects of quantum state space. In view of its usefulness and importance, we study the Gram matrix of an ensemble of pure states (a set of pure states with a prior probability distribution) and reveal its fundamental properties. We highlight and exploit the fact that the Gram matrix of an ensemble can be formally regarded as a bona fide quantum state. The key idea is to employ coherence (with respect to the computational basis) of the Gram matrix (regarded as a quantum state) to quantify quantumness of the corresponding ensemble. In particular, we propose to use the l 1-norm of coherence and the relative entropy of coherence, of the Gram matrix, as two significant quantifiers of quantumness. We illustrate the effectiveness and power of these quantifiers of quantumness by evaluating them for several important ensembles arising from quantum measurement and quantum cryptography. We further compare the quantumness based on the Gram matrix with various other quantifiers of quantumness in the literature, and show that although they are closely related in general, they have subtle differences and capture different aspects of ensembles, which shed light on the complexity of quantum ensembles.

Uncertainty and certainty relations for quantum coherence with respect to design-structured POVMs

The concept of quantum coherence and its possible use as a resource are currently the subject of active research. Uncertainty and complementarity relations for quantum coherence allow one to study its changes with respect to other characteristics of the process of interest. Protocols of quantum information processing often use measurements that have a special inner structure. Quantum designs are considered as tools with a perspective of fruitful applications in quantum information processing. We obtain uncertainty and certainty relations for coherence averaged with respect to a set of design-structured POVMs of rank one. To characterize the degree of coherence, the relative entropy of coherence is utilized. The derived relations are exemplified with quantum designs in the two-dimensional Hilbert space.

Verification of complementarity relations between quantum steering criteria using an optical system

The ability that one system immediately affects another one by using local measurements is regarded as quantum steering, which can be detected by various steering criteria. Recently, Mondal et al. [Phys. Rev. A 98, 052330 (2018)] derived the complementarity relations of coherence steering criteria, and revealed that the quantum steering of a system can be observed through the average coherence of a subsystem. Here, we experimentally verify the complementarity relations between quantum steering criteria by employing two-photon Bell-like states and three Pauli operators. The results demonstrate that if prepared quantum states can violate two setting coherence steering criteria and turn out to be steerable states, then they cannot violate the complementary settings criteria. Three measurement settings inequalities, which establish a set of complementarity relations between these two coherence steering criteria, are always obeyed by all prepared quantum states in experiment. In addition, we experimentally certify that the strengths of coherence steering criteria depend on the choice of coherence measure. In comparison with two setting coherence steering criteria based on ${l}_{1}$ norm of coherence and relative entropy of coherence, our experimental results show that the steering criterion based on skew information of coherence is the strongest in detecting the steerability of two-photon Bell-like states. Thus, our experimental demonstration can deepen the understanding of the relation between the quantum steering and quantum coherence.

Quantum coherence and spin nematic to nematic quantum phase transitions in biquadratic spin-1 and spin-2 XY chains with rhombic single-ion anisotropy

We investigate quantum phase transitions and quantum coherence in infinite biquadratic spin-1 and -2 XY chains with rhombic single-ion anisotropy. All considered coherence measures such as the ${l}_{1}$ norm of coherence, the relative entropy of coherence, and the quantum Jensen-Shannon divergence, and the quantum mutual information show consistently that singular behaviors occur for the spin-1 system, which enables to identify quantum phase transitions. For the spin-2 system, the relative entropy of coherence and the quantum mutual information properly detect no singular behavior in the whole system parameter range, while the ${l}_{1}$ norm of coherence and the quantum Jensen-Shannon divergence show a conflicting singular behavior of their first-order derivatives. Examining local magnetic moments and spin quadrupole moments lead to the explicit identification of novel orderings of spin quadrupole moments with zero magnetic moments in the whole parameter space. We find the three uniaxial spin nematic quadrupole phases for the spin-1 system and the two biaxial spin nematic phases for the spin-2 system. For the spin-2 system, the two orthogonal biaxial spin nematic states are connected adiabatically without an explicit phase transition, which can be called quantum crossover. The quantum crossover region is estimated by using the quantum fidelity. Whereas for the spin-1 system, the two discontinuous quantum phase transitions occur between three distinct uniaxial spin nematic phases. We discuss the quantum coherence measures and the quantum mutual information in connection with the quantum phase transitions including the quantum crossover.

Geometry of skew information-based quantum coherence

We study the skew information-based coherence of quantum states and derive explicit formulas for Werner states and isotropic states in a set of autotensors of mutually unbiased bases (MUBs). We also give surfaces of skew information-based coherence for Bell-diagonal states and a special class of X states in both computational basis and in MUBs. Moreover, we depict the surfaces of the skew information-based coherence for Bell-diagonal states under various types of local nondissipative quantum channels. The results show similar as well as different features compared with relative entropy of coherence and l1 norm of coherence.

Renormalization of quantum coherence and quantum phase transition in the Ising model

Quantifying quantum coherence of a given system not only plays an important role in quantum information science but also promotes our understanding of some basic problems, such as quantum phase transition. Conventional quantum coherence measures, such as l1-norm of coherence and relative entropy of coherence, are widely used to study quantum phase transition. Here we adopt a basis-independent coherence measure that is a quantum version of the Jensen–Shannon divergence to investigate the property of total quantum coherence, as well as its two contributions in quantum critical systems. Based on the quantum renormalization group method, we propose an analysis of the distribution of quantum coherence in the Ising system near the quantum critical point. We directly obtain the tradeoff relation, singular property, and scaling behavior in the Ising system. Furthermore, the monogamy relation is also studied in detail. These results further expand our understanding of quantum coherence as well as to enlarge the applications in using quantum coherence to reflect quantum critical phenomena.

Wave-particle-duality relations based on entropic bounds for which-way information

We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an entropic bound for the which-way information, which we derive in this letter. We show that this latter crucially depends on the choice of the measurement strategy to obtain the path information. In particular, we present results for two strategies: zero-error identification of the path-detector states, which never produces an error but sometimes fails to return a conclusive answer, and a mixed strategy where both errors and failure are allowed.

The resource theory of coherence for quantum channels

We define the quantum-incoherent relative entropy of coherence ($\mathcal{QI}$ REC) of quantum channels in the framework of the resource theory by using the Choi-Jamiolkowsky isomorphism. Coherence-breaking channels are introduced as free operations and their corresponding Choi states as free states. We also show the relationship between the coherence of channel and the quantum discord and find that basis-dependent quantum asymmetric discord can never be more than the $\mathcal{QI}$ REC for any quantum channels. {Also}, we prove the $\mathcal{QI}$ REC is decreasing for any divisible quantum incoherent channel and we also claim it can be considered as the quantumness of quantum channels. Moreover, we demonstrate that for qubit channels, the relative entropy of coherence (REC) can be equivalent to the REC of their corresponding Choi states and the basis-dependent quantum symmetric discord can never exceed the coherence.

Dynamics of quantum coherence and quantum Fisher information after a sudden quench

The dynamics of relative entropy and $l_{1}$-norm of coherence, as well as, the Wigner-Yanase-skew and quantum Fisher information are studied for a time-dependent coupled XY spin chain in presence of a time-dependent transverse magnetic field. Independent of the initial state of the system and while the relative entropy of coherence, $l_{1}$-norm of coherence, and quantum Fisher information are incapable, surprisingly, the dynamic Wigner-Yanase-skew information can truly spotlight the equilibrium critical point. We also observe that when the system is quenched to the critical point, these quantities show suppressions and revivals. Moreover, the first suppression (revival) time scales linearly with the system size and its scaling ratio is unique for all quenches independent to the initial phase. This is the promised universality of the first suppression (revival) time.

Transformations of superpositions by means of incoherent operations

In this paper we study how the coherence of a superposition of pure states is related with the coherence of its components. We consider two pure initial states and two pure final coherent states, such that the former ones cannot be transformed into the latter ones by means of incoherent transformations. In this situation, we analyze conditions for the existence of superpositions of the initial states that can be transformed into superpositions of the final states. In particular, we consider superpositions formed by quantum states belonging to orthogonal subspaces. By appealing to the majorization theory, we obtain necessary and sufficient conditions for such transformations to be possible. Finally, we provide some examples that illustrate the difference between the obtained conditions and the necessary criterion based on the relative entropy of coherence.

Quantum renormalization of $${l_1}$$-norm and relative entropy of coherence in quantum spin chains

We investigate quantum-phase transitions (QPT) in the Ising transverse field model, the XY-Heisenberg model with staggered Dzyaloshinskii–Moriya (DM) interaction, and the bond alternating Ising model with DM interaction, on a one-dimensional periodic chain using the quantum renormalization group (QRG) method. Based on $${l_1}$$-norm and relative entropy, we employ two quantum coherence measures to identify QPT. Our results show that the QPT can be characterized by the coherence measure calculated for each QRG step. It is found that after large number of QRG iterations, the coherence in the ground state develops two saturated values corresponding to two different phases and changes abruptly at a quantum critical point (QCP). The scaling behavior of the system is investigated by computing the derivative of the coherence measure around the QCP that shows a logarithmic dependence on the length of the chain. The critical exponent of the correlation length is also calculated.

Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence.

Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. To quantify coherence, two proper measures were introduced in the literature, the one is the relative entropy of coherence and the other is the -norm of coherence . In this paper, we obtain a symmetry-like relation of relative entropy measure of coherence for an n-partite quantum states , which gives lower and upper bounds for . As application of our inequalities, we conclude that when each reduced states is pure, is incoherent if and only if the reduced states and are all incoherent. Meanwhile, we discuss the conjecture that for any state , which was proved to be valid for any mixed qubit state and any pure state, and open for a general state. We observe that every mixture of a state satisfying the conjecture with any incoherent state also satisfies the conjecture. We also observe that when the von Neumann entropy is defined by the natural logarithm ln instead of , the reduced relative entropy measure of coherence satisfies the inequality for any state .

Restoration of Coherence by Local PT-Symmetric Operator

In this work, we mainly investigate effect of PT-symmetric operation on the dynamics of the relative entropy of coherence for a two-level system within non-Markovian environments, and put forward a feasible physical scheme to recover coherence by means of optimal PT-symmetric operation. The results show that the damaged quantum coherence can be restored to a large extent. Furthermore, the freezing phenomenon of the coherence can be detected by using the optimal PT-symmetric operation strength within the non-Markovian environments.

Influence of the quantum-jump-based feedback on the Rydberg atoms system

We investigated the influence of the quantum-jump-based feedback on the behaviors of the three-level Rydberg system by making use of relative entropy coherence and the square root of quantum Jensen–Shannon divergence. It is shown that quantum relative entropy coherence and quantum Jensen–Shannon divergence are enhanced by particular quantum-jump-based feedbacks and increase with the enhancement of the detuning δ. We also explored the behavior of the entropic uncertainty relationship, and lower uncertainty bounds are obtained with quantum-jump-based feedbacks. The effect of different detunings δ on the entropic uncertainty relationship is also discussed, and it is found that the uncertainty bound decreases when the detuning δ increases.