Quantumness speeds up quantum thermodynamics processes

TL;DR

This paper introduces a new method to measure how quantum properties like coherence and entanglement speed up energy extraction in quantum thermodynamics.

Contribution

The paper introduces a thermodynamics speed metric to quantify the role of quantum coherence and entanglement in work extraction.

Findings

Coherence in quantum systems can speed up work extraction beyond incoherent states.

Entanglement can further increase work extraction speed beyond bi-separable states.

A new thermodynamic method is proposed to witness entangled systems using physical quantities.

Abstract

Quantum thermodynamic process involves manipulating and controlling quantum states to extract energy or perform computational tasks with high efficiency. There is still no efficient general method to theoretically quantify the effect of the quantumness of coherence and entanglement in work extraction. In this work, we propose a thermodynamics speed to quantify the extracting work. We show that the coherence of quantum systems can speed up work extracting with respect to some cyclic evolution beyond all incoherent states. We further show the genuine entanglement of quantum systems may speed up work extracting beyond any bi-separable states. This provides a new thermodynamic method to witness entangled systems with physical quantities.

•A quantum energy-exchange speed is introduced•The maximal energy-exchange speed of coherent batteries is proved•The maximal energy-exchange speed of entangled batteries is explored

A quantum energy-exchange speed is introduced

The maximal energy-exchange speed of coherent batteries is proved

The maximal energy-exchange speed of entangled batteries is explored

Physics; Quantum theory; Quantum physics

Figures

Schematic thermodynamic process of batteryA classical battery system undergoes a cyclic controlling with the Hamiltonian within a time period in the up protocol. The quantum battery system undergoes a cyclic controlling with the same probing Hamiltonian within the same time period in the down protocol. The final extracted energy is defined according to a given bare Hamiltonian. The goal is to characterize the difference of the maximal energy-exchange speed over time-independent Hamiltonians.

Operational work extractingFor an *n*-particle quantum state *ρ* the normalized extracting work $[eqn]$ and dual work $[eqn]$ consist of a unit cycle. When the state undergoes a unitary transformation $[eqn]$ , the present quantity $[eqn]$ denotes the quantum work distance (green line) while $[eqn]$ denotes the evolution distance (orange arc).

Witness of entangled battery with the maximum energy-exchange speedMaximum energy-exchange speed of fully separable states, $[eqn]$ (dots), probed by the Ising Hamiltonian. The present Hamiltonian $[eqn]$ provides a method to witness entanglement if the battery has larger energy-exchange speed than $[eqn]$ .

Full text

Introduction

Quantum thermodynamics provides a bridge to explore energy transfer and conversion of two systems at the microscopic level. By incorporating quantum effects into thermodynamic systems, it can gain insights into the fundamental limits of energy extraction and charging behavior of small-scale devices.1^,^2^,^3^,^4 So far, quantum thermodynamics has intrigued great improvements in energy storage and transfer, such as quantum heat engines5^,^6^,^7 and quantum refrigerators.7^,^8^,^9 Quantum thermodynamics sheds light on the fundamental principles governing the behavior of quantum systems, paving the way for quantum computing and quantum information processing.10^,^11

Exploiting the quantum features of the coherence or entanglement beyond the classical counterparts is one of the most important tasks in quantum thermodynamics.7^,^12^,^13 This can trace back to a basic problem from the birth of thermodynamics in 1824, i.e., what criteria is useful to compare different states with respect to their energies. In the thermodynamic process, a quantum system can provide energy for the other systems or be charged by others. This allows building battery-like quantum devices.14 Lots of potential examples are proposed from qubits,15 spins,16^,^17^,^18 and flywheels,19^,^20 to collision model.21^,^22^,^23 In general, it requires auxiliary systems to control the batteries.24^,^25^,^26^,^27^,^28 Moreover, with the coherence some quantum batteries show interesting features of faster and higher-power charging capability than the classical counterparts.14^,^25^,^26^,^27^,^28^,^29^,^30^,^31 This is recently extended for entangled quantum systems.32^,^33^,^34^,^35^,^36^,^37^,^38^,^39^,^40 Specially, most battery systems will generate entanglement under global evolution and then show an advantage over those with only local operations.32^,^37^,^41^,^42 These results show new quantum effects of the coherence and entanglement in the thermodynamic processes of quantum batteries.

Recently, most quantum work extracting protocols focus on exploiting the optimal final energy that can be exchanged for a given quantum battery under cyclic control.3 This intrigues us to define the maximum of the amount of extracted work as so-called ergotropy. As for the charging process, the maximum of the amount of charged work as so-called antiergotropy. Combining both quantities allows characterizing a given quantum battery using a fundamental quantity of quantum battery capacity.43 Dividing these quantities by the evolution time period provides bare Hamiltonian-based ways to characterize the energy transferring or charging processes.27

In this paper, we define a quantum energy-exchange speed as a figure of the thermodynamic energy transferring that will link to quantum features of the coherence44 and entanglement.45 We define an energy-exchange speed as the operational ratio of the Hellinger distance46 of two final extractable energies of one battery in two complement thermodynamic procedures. This quantity is independent of the governor Hamiltonian and shows an operational relationship to the statistical speed of quantum Fisher information.46^,^47 We further show the present quantity is the evolution speed of energies on a unit geometric sphere. We propose a novel approach to exploit the energy extraction of coherent or entangled battery systems based on the thermodynamics speed, as shown in Figure 1. We prove that quantum coherence can speed up the quantum thermodynamic process beyond all incoherent states. We then show the entangled battery systems can provide a speed up beyond separable states, even quadratic speeding up in an Ising model. This allows witnessing both coherence and entanglement with new thermodynamic features.Figure 1. Schematic thermodynamic process of batteryA classical battery system undergoes a cyclic controlling with the Hamiltonian within a time period in the up protocol. The quantum battery system undergoes a cyclic controlling with the same probing Hamiltonian within the same time period in the down protocol. The final extracted energy is defined according to a given bare Hamiltonian. The goal is to characterize the difference of the maximal energy-exchange speed over time-independent Hamiltonians.

Results

Quantum energy-exchange speed

Consider a single-particle quantum B which can exchange energy with external sources. Define the evolution of B is generated by a Hamiltonian $[eqn]$ which accounts for energy exchanging. Assume B is in a cyclic process, in which originally isolated is coupled at the time $[eqn]$ to external sources, and decouples from them at the time t^2^. The bare Hamiltonian is defined by $[eqn]$ . The battery is thermally isolated but may involve energy exchanges between its parts, i.e., it does not involve heat exchange with a thermal environment. The initial extractable energy is defined by

$$[eqn]$$

with an initial state $[eqn]$ . The final extractable energy is then given by $[eqn]$ , where the final state $[eqn]$ is defined according to the Liouville-Neumann equation

$$[eqn]$$

For continuous systems the optimal energy depends on von Neumann entropy of final Gibbsian state.3 But this is not always right for finite systems.

In what follows, we consider a finite-dimensional single-particle battery system. More precisely, define the evolution of B under a unitary operation U as $[eqn]$ . The bare Hamiltonian $[eqn]$ is decomposed into

$$[eqn]$$

where E denotes the unit energy and eigenenergies $[eqn]$ satisfy $[eqn]$ . Without loss of generality, we associate zero energy to the lowest energetic state $[eqn]$ . When the quantum battery goes a cyclic evolution $[eqn]$ with a time-independent Hamiltonian $[eqn]$ , the final extractable energy is given by48^,^49:

$$[eqn]$$

We extend the result to multiparticle quantum batteries. Suppose an isolated N-particle quantum battery is in the initial state $[eqn]$ on Hilbert space. Instead of the interaction-free Hamiltonian $[eqn]$ with Hamiltonian $[eqn]$ of the i-th particle, we consider the battery equipped with a total global Hamiltonian $[eqn]$ . Suppose the quantum battery is subjected to a cyclic evolution with a k-body correlated time-independent Hamiltonian $[eqn]$ given by

$$[eqn]$$

where $[eqn]$ account for the strength of local operations, and $[eqn]$ denote k-body correlating parameters satisfying the symmetry condition of $[eqn]$ for any different $[eqn]$ and permutation τ in the permutation group $[eqn]$ . γ is an arbitrary real number. Here, $[eqn]$ are $[eqn]$ generalized Pauli matrices of the j-th particle, where $[eqn]$ denotes the d-dimensional Gell-Mann matrix vector and $[eqn]$ is a unit vector on the Gell-Mann sphere.50 After the evolution period t, the final state is given by $[eqn]$ . The final extractable energy has the same form Equation 4. This intrigues us to identify the difference of extractable energies between a given evolution period. Specially, when $[eqn]$ , the difference of the extractable energy gives as $[eqn]$ , which characterizes the work extracted from the battery. On the other hand, it exhibits the amount of work being charged into the battery from other systems when $[eqn]$ . The corresponding maximal works under any unitary evolutions are defined as the ergotropy and the antiergotropy, respectively.3^,^43

Our goal in what follows is to characterize the extractable energy (4) in terms of the quantum features of battery systems. The main idea is using the thermodynamic speed to characterize the extractable energy that can be charged or extracted from the battery systems within any cyclic control protocols, as shown in Figure 1. Suppose that a given quantum battery in the state ρ will be probed by applying a global transformation $[eqn]$ with an evolution time t. Define an operational measure to quantify the difference of extractable energies $[eqn]$ and $[eqn]$ during the evolution period $[eqn]$ as

$$[eqn]$$

where $[eqn]$ defines the complement Hamiltonian of $[eqn]$ , with the corresponding final extractable energy $[eqn]$ . Both quantities satisfy the operational relationship of

$$[eqn]$$

for any $[eqn]$ and $[eqn]$ . The present distance (6) is zero, if and only if two final states are same. The maximal difference is $[eqn]$ if one final extractable energy is zero and the other is $[eqn]$ . The metric (6) can be regarded as a Hellinger distance of extractable energies.46^,^47 This allows us to define a thermodynamic speed to figure out how fast can the extractable energy be exchanged under the cyclic control of the given battery system as:

$$[eqn]$$

i.e., the ratio at which $[eqn]$ changes with $[eqn]$ around the reference time t. From the Taylor expansion, it follows that

$$[eqn]$$

with $[eqn]$ . The present speed will be used to explore the capability of extracting energies.

Define a maximal quantum energy-exchange speed as

$$[eqn]$$

where the maximum is over all equipped Hamiltonians in the unitary evolution protocols. This means the present quantity $[eqn]$ is independent of the Hamiltonian and provides a general feature of the controlling protocol while the known definition of quantum speed depends on the controlling Hamiltonian.8^,^27^,^35

Inspired by the classical Fisher information,47 the present quantity (10) can be evaluated according to the symmetric logarithmic derivative as

$$[eqn]$$

where the symmetric logarithmic derivative $[eqn]$ is uniquely defined on the support of ρ via the relation $[eqn]$ . This implies the quantity $[eqn]$ is convex from the convexity of the Fisher information.51

From Equation 9, it follows that

$$[eqn]$$

if $[eqn]$ . By integral over time period $[eqn]$ this implies a new formula of the operational charging work as (Method):

$$[eqn]$$

under a cyclic charging evolution $[eqn]$ . Instead, if $[eqn]$ we get a new form of the operational extracting work as

$$[eqn]$$

under a cyclic discharging evolution $[eqn]$ . Both quantity $[eqn]$ and $[eqn]$ provide different metrics of quantum work procedure from previous definitions,43 as shown in Figure 2. This further intrigues us to define the maximal works of ergotropy and antiergotropy with the present metric, which is valuable for further exploration.Figure 2. Operational work extractingFor an n-particle quantum state ρ the normalized extracting work $[eqn]$ and dual work $[eqn]$ consist of a unit cycle. When the state undergoes a unitary transformation $[eqn]$ , the present quantity $[eqn]$ denotes the quantum work distance (green line) while $[eqn]$ denotes the evolution distance (orange arc).

Now, we present the main result to show the energy-exchange speed of quantum battery systems in the superposition states44 and multipartite entangled states.45 Equation 10 implies this quantity is independent of all bare Hamiltonians. Given two batteries in the states ρ and ϱ on the same Hilbert space, suppose both are under the same control with Hamiltonian $[eqn]$ . From Equation 11, if $[eqn]$ , it means the battery in the state ρ has a larger maximum energy-exchange speed beyond the battery in the state ϱ. Informally, as for any isolated quantum battery systems, its coherence can speed up the fastest energy exchange over all the incoherent systems. This provides a new state-independent supremacy in quantum thermodynamics over all classical counterparts. As for multiparticle quantum batteries, the optimal work extraction may not generate multipartite entanglement.33 However, we will show that genuine entanglement52 may show a state-dependent supremacy to speed up the fastest energy exchange over biseparable systems. Here, the genuine entanglement means it cannot be decomposed into a mixture of any biseparable state $[eqn]$ , where $[eqn]$ and $[eqn]$ denote a bipartition of all particles.

Theorem 1. Given an n-particle quantum battery in the pure state $[eqn]$ , the following results hold:

• (1)there exists a time-independent Hamiltonian $[eqn]$ such that $[eqn]$ for any incoherent states $[eqn]$ if $[eqn]$ is a coherent state;
• (2)there exists a time-independent Hamiltonian $[eqn]$ such that $[eqn]$ for any bi-separable state $[eqn]$ if $[eqn]$ is genuinely entangled state.

The proof is inspired by recent methods.47^,^53 For any pure states and a control protocol with the time-independent Hamiltonian $[eqn]$ , the quantum energy-exchange speed can be evaluated as

$$[eqn]$$

To show the effect of quantum coherence, from the convexity of quantum energy-exchange speed, it is sufficient to prove the result for all the isolated systems, i.e., the pure states. A simple fact is that any two different incoherent states are orthogonal. This allows finding a simple Hamiltonian for each coherent state such that the average energy-exchange speed is larger than its of any incoherent states. The detailed proof is shown in Method. For multiparticle entangled batteries, the quantum energy-exchange speed depends on specific Hamiltonian H and its state but is no less than that of any bi-separable batteries.

Coherent quantum batteries

Quantum coherence refers to the state of a quantum system where its constituent particles are in a superposition.44^,^54 Quantum coherent states are characterized by their ability to exhibit interference effects, making them valuable for applications such as quantum computing, quantum communication, and quantum metrology. We quantify the maximal energy-exchange speed of a given coherent battery. We estimate the largest energy-exchange speed over all classically correlated states. From the convexity, we obtain the following inequality

$$[eqn]$$

where the maximum is over all pure product incoherent states $[eqn]$ . Specially, for a given quantum battery the maximum energy-exchange speed can be saturated by optimal Hamiltonians.47

As its derived in Method, we prove that the maximum energy-exchange of incoherent state $[eqn]$ under the cyclic evolution with the Hamiltonian (5) is given by

$$[eqn]$$

where $[eqn]$ , $[eqn]$ and $[eqn]$ with local Hamiltonians $[eqn]$ and nonlocal Hamiltonians $[eqn]$ , i.e., $[eqn]$ . All the details of these correlations are shown in Method. As $[eqn]$ bounds the energy-exchange speed over all Hamiltonians and all incoherent states, the coherent battery system has a larger maximum energy-exchange speed than incoherent systems if the battery states violate the inequality Equation 16.

Example 1. Consider a multiple-qubit battery with a constant Hamiltonian, i.e., $[eqn]$ . From Equation 17 we obtain the maximum energy-exchange speed as

$$[eqn]$$

This implies that for a product coherent pure state $[eqn]$ , with the homogeneous Hamiltonian of $[eqn]$ , from Equation 15 the local Hamiltonian $[eqn]$ gives the maximum energy-exchange speed satisfies $[eqn]$ , which provides a quadratic speed up beyond the maximum energy-exchange speed $[eqn]$ for all incoherent states. For general coherent batteries, the maximum energy-exchange speed depends on nonlocal Hamiltonians.

Entangled batteries

Entanglement is a fundamental concept in quantum mechanics.45^,^55^,^56 For an entangled two particles measuring one particle can instantaneously affect the state of the other.55 This phenomenon cannot be described by a classical understanding of cause and effect.57^,^58 Here, we quantify the maximum energy-exchange speed for an entangled battery. From the convexity of the maximum energy-exchange speed we obtain the following inequality59

$$[eqn]$$

where the maximum is over all product pure states $[eqn]$ or all bi-separable product states $[eqn]$ for witnessing entangled or genuinely entangled batteries, respectively.52 Similar to the proof for coherent systems in Method, the maximum energy-exchange speed of $[eqn]$ with the Hamiltonian $[eqn]$ has the form Equation 17 with respect to separable states. The entangled battery may violate the inequality Equation 19. The general case depends on the Hamiltonians.

Example 2. Consider an Ising model with the k-paired nearest-neighbor interaction,60 i.e., $[eqn]$ with Dirac delta function δ and $[eqn]$ . γ denotes the ferromagnetic coupling factor. For the homogeneous case of $[eqn]$ and $[eqn]$ , the optimal fully separable state for maximizing Equation 17 is given by $[eqn]$ , where $[eqn]$ is a superposition state in terms of two eigenstates of $[eqn]$ . We obtain the maximum energy-exchange speed from Equation 17 satisfy (Method):

$$[eqn]$$

where $[eqn]$ . The critical value $[eqn]$ is defined with equal of two speeds in Equation 20. Numerical evaluations are shown in Figure 3.Figure 3. Witness of entangled battery with the maximum energy-exchange speedMaximum energy-exchange speed of fully separable states, $[eqn]$ (dots), probed by the Ising Hamiltonian. The present Hamiltonian $[eqn]$ provides a method to witness entanglement if the battery has larger energy-exchange speed than $[eqn]$ .

Under local Hamiltonians, the maximum energy-exchange speed of special entangled states may be larger than that of fully separable batteries. For exploring the quantum supremacy of general entangled battery, it requires both local and nonlocal Hamiltonians.

Example 3. Consider the Dicke state $[eqn]$ with k number of excited spins.61 If the local Hamiltonian $[eqn]$ satisfies $[eqn]$ and $[eqn]$ , from Equation 15 its maximum energy-exchange speed is given by $[eqn]$ . This is larger than the maximum energy-exchange speed $[eqn]$ for all fully separable states when $[eqn]$ .

Consider the maximally entangled Greenberger-Horne-Zeilinger (GHZ) state,62 it is easy to show the maximum energy-exchange speed $[eqn]$ from Equation 15, which shows a quadratic speed up over all fully separable states with nonlocal nearest-neighbor Hamiltonians.

Discussion

In applications, if there are some averaged Hamiltonians $[eqn]$ , it is useful to extend the Hellinger distance and energy-exchange speed to the work distribution $[eqn]$ , where $[eqn]$ and $[eqn]$ are extracted works from different bare Hamiltonians $[eqn]$ . This allows to characterize the statistical mixture of quantum work speed. But we can not obtain the operational standard forms (13) and (14) of extracting work. This intrigues new problems to explore the relationships between the Hamiltonians and average energy-exchange speed.

The present energy-exchange speed reveals the quantum features of quantum batteries in a unified manner. This requires to probe a battery system with a generic multi-particle Hamiltonian. The present method shares several important properties. One is that the quantum features of the coherence and entanglement allow quantum supremacy of the maximum energy-exchange speed even quadratic speed up. This shows the importance of quantum features in quantum-size batteries for extracting or charging energies. The second is that the present quantity provides a new way to witness the quantum properties of coherence and entanglement hold in given quantum batteries. Beside the possibility to witness a larger class of entangled states, the extracting work speed allows to take in account the residual coupling among neighboring particles.63

To conclude, the method discussed in this manuscript allows the experimental characterization of a larger class of quantum states. The present results show that entanglement can be detected even when the Hamiltonian is nonlinear and therefore generates entanglement. This opens the way to study entanglement near quantum phase transition points by quenching the parameters of the governing many-body Hamiltonian.

Limitations of the study

This paper aims to show the quantum properties of coherence and entanglement can speed up quantum thermodynamics processes. The main limitation of the proposed method is from the Fish information.

STAR★Methods

Resource availability

Lead contact

Further information and requests for resources should be directed to the lead contact Ming-Xing Luo (mxluo@swjtu.edu.cn).

Materials availability

This study did not generate new materials.

Data and code availability

This study has no data and code available.

Method details

The operational charging work

Since $[eqn]$ for any state ρ and Hamiltonian $[eqn]$ , define $[eqn]$ and $[eqn]$ . This implies that

$$[eqn]$$

Proof of Theorem 1

The proof is inspired by recent methods.47^,^53 Given a quantum battery in the pure state $[eqn]$ , and a unitary transformation $[eqn]$ under the time-independent probing Hamiltonian $[eqn]$ , the energy-exchange speed is given by

$$[eqn]$$

We firstly prove the result for quantum batteries in the coherent states. For a given quantum battery in the coherent state $[eqn]$ , let the probing Hamiltonian be $[eqn]$ , where $[eqn]$ satisfies $[eqn]$ . This implies from Equation 22 that

$$[eqn]$$

But for any incoherent state $[eqn]$ , we obtain that

$$[eqn]$$

where we have used the fact $[eqn]$ for incoherent states $[eqn]$ . This implies that $[eqn]$ .

Now, we show the result for genuinely entangled batteries. Consider a given isolated battery system in the entangled pure state $[eqn]$ . If one makes use of the special entanglement witness operator45 of $[eqn]$ as the probing Hamiltonian $[eqn]$ , it follows that $[eqn]$ and $[eqn]$ , where $[eqn]$ . This means the well-known entanglement witness is useless to design the probing Hamiltonian. Instead, we use the following fact, i.e., there is a biseparable pure state $[eqn]$ such that $[eqn]$ . This can be proved by using Schmidt decomposition of $[eqn]$ in terms of any bipartition of all particles. With this state, we define the probing Hamiltonian as $[eqn]$ . It follows that

$$[eqn]$$

For any biseperable pure state $[eqn]$ , we obtain that

$$[eqn]$$

This has proved the result for genuinely entangled batteries.

Similar proof holds for entangled batteries by ruling out all fully separable batteries if there is a fully separable state $[eqn]$ such that $[eqn]$ .

The maximum energy-exchange speed of incoherent states

In this section, we evaluate the energy-exchange speed of incoherent systems as

$$[eqn]$$

where $[eqn]$ are given by

$$[eqn]$$

with $[eqn]$ , $[eqn]$ , $[eqn]$ and $[eqn]$

Inspired by recent methods,47^,^53 for pure states and unitary transformation $[eqn]$ , the energy-exchange speed is given by $[eqn]$ . Taking $[eqn]$ , this equality implies that

$$[eqn]$$ $$[eqn]$$ $$[eqn]$$

From the convexity of the energy-exchange speed, it is sufficient to show $[eqn]$ for all product incoherent pure states.

We first evaluate $[eqn]$ with respect to product incoherent pure states. For the single-particle Hamiltonian $[eqn]$ given by $[eqn]$ , we obtain that

$$[eqn]$$

where $[eqn]$ . If all vectors $[eqn]$ are the same, it follows a special case of $[eqn]$ for $[eqn]$ ; and 0 for other cases. Moreover, we obtain that

$$[eqn]$$

where $[eqn]$ . This implies from the equality (28) that

$$[eqn]$$

as $[eqn]$ for product incherent pure states.

Now, we evaluate $[eqn]$ with respect to product incoherent pure states. Let $[eqn]$ be given by $[eqn]$ . We obtain that

$$[eqn]$$

where $[eqn]$ . Moreover, we obtain that

$$[eqn]$$

This implies from the equality (28) that

$$[eqn]$$

Now, we evaluate $[eqn]$ with respect to product incherent pure states. We obtain that

$$[eqn]$$

where $[eqn]$ . Moreover, we obtain that

$$[eqn]$$

This implies from the equality (30) that

$$[eqn]$$

Given a product incoherent state, for $[eqn]$ with any $[eqn]$ , we obtain that

$$[eqn]$$

If $[eqn]$ , we obtain that

$$[eqn]$$

This implies that

$$[eqn]$$

Ising model

Let us consider an Ising model with the k-pair nearest-neighbor interaction, i.e., $[eqn]$ with Dirac delta function δ and $[eqn]$ . For γ smaller than a critical value $[eqn]$ , the maximum speed $[eqn]$ is achieved when $[eqn]$ are all equal to s. From Equation 17, with forward tedious evaluations the energy-exchange speed for fully separable states is given by

$$[eqn]$$

where $[eqn]$ are given by

$$[eqn]$$

For a large N, $[eqn]$ and $[eqn]$ , we find $[eqn]$ for optimizing $[eqn]$ as

$$[eqn]$$

where $[eqn]$ .

For $[eqn]$ , we obtain the optimized energy-exchange speed when $[eqn]$ for all i, and $[eqn]$ and $[eqn]$ for all j. This implies that $[eqn]$ and $[eqn]$ , i.e.,

$$[eqn]$$

Indeed, in the limit $[eqn]$ , Equation 45 goes as $[eqn]$ and thus is larger than Equation 43 for any k, which goes as $[eqn]$ . The critical value of γ is obtained setting Equation 43 is equal to the maximum over s of Equation 45, and given by

$$[eqn]$$

where $[eqn]$ are defined by

$$[eqn]$$

The upped bound of $[eqn]$ for the inhomogeneous case ( $[eqn]$ ) can be obtained by maximizing each term in Equation 17 separately. This gives

$$[eqn]$$

with $[eqn]$ .