Clock/work tradeoff relation for coherence in quantum thermodynamics
Abstract
In thermodynamics, quantum coherences – superpositions between energy eigenstates – behave in distinctly nonclassical ways. Recently mathematical frameworks have emerged to account for these features and have provided a range of novel insights. Here we describe how thermodynamic coherence splits into two kinds – “internal” coherence that admits an energetic value in terms of thermodynamic work, and “external” coherence that does not have energetic value, but instead corresponds to the functioning of the system as a quantum clock. For the latter form of coherence we provide dynamical constraints that relate to quantum metrology and macroscopicity, while for the former, we show that quantum states exist that have finite internal coherence yet with zero deterministic work value. Finally, under minimal thermodynamic assumptions, we establish a clock/work tradeoff relation between these two types of coherences. This can be viewed as a form of timeenergy conjugate relation within quantum thermodynamics that bounds the total maximum of clock and work resources for a given system.
pacs:
Classical thermodynamics describes the physical behavior of macroscopic systems composed of large numbers of particles. Thanks to its intimate relations with statistics and information theory, the domain of thermodynamics has recently been extended to include small systems, and even quantum systems. One particularly pressing question is how the existence of quantum coherences, or superpositions of energy eigenstates, impacts the laws of thermodynamics Aberg14 (); Skrzypczyk13 (); Skrzypczyk14 (); Uzdin15 (); Korzekwa16 (), in addition to quantum correlations Groisman05 (); Jennings10 (); Park13 (); Reeb14 (); Bruschi15 (); Huber15 (); Llobet15 ().
We now have a range of results for quantum thermodynamics Janzing2000 (); Horodecki13 (); Brandao13 (); Brandao15 (); Lostaglio15 (); LostaglioX (); Cwiklinski15 (); Goold16 (); Bera16 (); Wilming16 (); LostaglioPRL (); Muller17 (); Gour17 () that have been developed within the resourcetheoretic approach. A key advantage of the resource theory framework is that avoids highly problematic concepts such as “heat” or “entropy” as its starting point. Its results have been shown to be consistent with traditional thermodynamics Weilmann16 () and has increasingly uncovered surprising connections with other theories such as entanglement theory Plenio07 (); Horodecki09 (), coherence Baumgratz14 (); Winter16 (), and reference frames (or asymmetry) Bartlett07 (); Gour08 (); MarvianPRA (); Marvian14 ().
Very recently, the first complete set of conditions for quantum thermodynamics with coherence was obtained in Gour17 (). The framework of thermal processes (TP) analysed in Gour17 () is based on the following 3 minimal physical assumptions: (i) That energy is conserved microscopically, (ii) that an equilibrium state exists, and (iii) that quantum coherence has a thermodynamic value. These are described in more detail in the Supplementary Material. Note that thermal processes contain thermal operations (TO) Janzing2000 (); Horodecki13 () as a subset, and coincide with TO on incoherent states, however in contrast to TO admit a straightforward description for the evolution of states with coherences between energy eigenspaces.
In this Letter we work under the same thermodynamic assumptions (i–iii) as above and show that quantum coherence in thermodynamics splits in two distinct types: internal coherences between quantum states of the same energy, and external coherences between states of different energies. This terminology is used because the external coherences in a system are only defined relative to an external phase reference frame , while internal coherences are defined within as relational coherences between its subcomponents.
We focus on the case of an partite system with noninteracting subsystems. The Hamiltonian is written as and we assume that each th local Hamiltonian has an energy spectrum with local energyeigenstates . Then a quantum state of this system may be represented as
where and . We also define the total energy of the string . Classical thermodynamic properties are determined by the probability distribution of the local energies, including their correlations. This information is contained in the diagonal terms of density matrix , where . The probability distribution of the total energy is . So every state has a corresponding classical state defined via the projection .
However, a quantum system is defined by more than its classical energy distribution – it may have coherence in the energy eigenbasis. This coherence is associated with nonzero offdiagonal elements in the density matrix, namely for . The internal coherence corresponds to offdiagonal terms of the same total energy (where ) and external coherence corresponds to terms with different energies (). For any state , we denote the corresponding state in which all external coherence is removed by , where is the projector onto the eigenspace of total energy .
As illustrated in Fig. 1, internal coherence may be used to extract work, however it has been shown that external coherences obey a superselection rule (called “worklocking”) that forbids work extraction, and is unavoidable if one wishes to explicitly account for all sources of coherence in thermodynamics Lostaglio15 (). We study this phenomenon by defining the process of extracting work purely from the coherence, without affecting the classical energy statistics. We find the conditions under which work can be deterministically extracted in this way from a pure state. Next, we show that external coherence is responsible for a system’s ability to act as a clock. The precision of the clock may be quantified by the quantum Fisher information (QFI) Braunstein94 (); we show that the QFI satisfies a secondlawlike condition, stating that it cannot increase under a thermal process. Finally, we derive a fundamental tradeoff inequality between the QFI and the extractable work from coherence demonstrating how a system’s potential for producing work is limited by its ability to act as a clock and viceversa.
Extractable work from coherence.— Here we demonstrate that, in a singleshot setting, work may be extracted from coherence without changing the classical energy distribution of the system. We consider the following type of work extraction process
in which the energy of a work qubit^{1}^{1}1Note that a “work qubit” is a convenient unit to quantify deterministically extracted, perfectly ordered energy. It does not place restrictions on assumed workbearing degrees of freedom and has been shown to be equivalent to other notions of work as ordered energy Faist12 (); Faistthesis (). with Hamiltonian () is raised from to .
For energyblockdiagonal states and , the work distance is defined as Brandao15 (), where is a generalized free energy based on the Rényi divergence
The work distance is the maximum extractable work that can be achieved by a thermal process by taking to Brandao15 ().
Even when the initial state is not blockdiagonal in the energy basis, the extractable work is still given by , so external coherence cannot be used to extract additional work Lostaglio15 (). In order to exploit external coherence for work, one needs multiple copies of Brandao13 (); Korzekwa16 () or ancilliary coherent resources Skrzypczyk13 (); Brandao13 (); Aberg14 (); Lostaglio15 (). Thus, the singleshot extractable work purely from coherence is given by
(1) 
For example, consider extracting work from coherence in the pure twoqubit state
where each qubit has local Hamiltonain . As shown in Fig. 2 using the concept of thermomajorization Horodecki13 (), we have only for sufficiently large . In this case, we have the necessary condition and the sufficient condition for , independent of and . See Supplemental Material for details.
We generalize this statement to arbitrary pure states:
Observation 1.
For a pure state, nonzero work can be extracted from coherence deterministically if and only if there exists internal coherence for the energy level such that .
Furthermore, in the manybody setting, internal coherence has some overlap with nonclassical correlations, namely quantum discord Modi12 (). To illustrate this, consider the following quantity which quantifies the sharing of free energy between subsystems:
where is the local state of the th subsystem. For nondegenerate local Hamiltonians, the extractable work from coherence can be written as
noting that the local free energies are the same for and . This is of the same form as discord as defined by Ollivier and Zurek Zurek01 (), expressed as a difference between total and classical correlations. Note that the classical correlations are defined here with respect to the energy basis, instead of the usual maximization over all local basis choices.
Unlike previous related studies Llobet15 (); Korzekwa16 (), our result requires that only coherence is consumed in the work extraction processes, leaving all energy statistics unchanged. We may also consider the “incoherent” contribution to the extractable work, , which is the achievable work from an incoherent state with the same energy statistics as , ending with a Gibbs state . The sum of the coherent and incoherent terms cannot exceed the total extractable work from to , i.e., . Equality holds when in Eq. (1) is given at . We also point out that this type of work extraction process operates without any measurement or information storage as in Maxwell’s demon Lloyd97 (); Zurek03 () or the Szilard engine Park13 () in the quantum regime.
Apart from the above example, a significant case is the socalled coherent Gibbs state Lostaglio15 (), defined for a single subsystem as . No work can be extracted from this state, as – an instance of worklocking. However, nonzero work can be unlocked Lostaglio15 () from multiple copies . In fact, from Observation 1, we see that is always sufficient to give . This is because is proportional to the degeneracy of the subspace – there always exists a degenerate subspace for , and this is guaranteed to have coherence.
Coherence as a clock resource.— Having discussed the thermodynamical relevance of internal coherence, we now turn to external coherence. Suppose we have an initial state . After free unitary evolution for time , this becomes , where each offdiagonal component rotates at frequency . Internal coherences do not evolve (), while external coherences with larger energy gaps, and hence higher frequencies, can be considered as providing more sensitive quantum clocks Chen10 (); Komar14 ().
By comparing with , one can estimate the elapsed time . More precisely, the resolution of a quantum clock can be quantified by , where is the time estimator derived from some measurement on . The resolution is limited by the quantum CramérRao bound Braunstein94 (), , where is the quantum Fisher information, and are the eigenvalues and eigenstates of , respectively. For the optimal time estimator saturating the bound, the larger the quantum Fisher information, the higher the clock resolution.
Another family of relevant measures of the clock resolution is the skew information for Wigner63 (); Braunstein94 (). For pure states, both the QFI and skew information reduce to the variance: . We also remark that a similar approach to “time references” in quantum thermodynamics has been recently suggested using an entropic clock performance quantifier Gour17 ().
We first note that, even though a quantum state might be very poor at providing work, it can still function as a good time reference. The coherent Gibbs state is a canonical example. As mentioned earlier, no work may be extracted from ; however, a precise time measurement is possible due to , proportional to the heat capacity Crooks12 ().
Furthermore, the QFI and skew information are based on monotone metrics Hansen08 (); Petz11 (), and monotonically decrease under timetranslationcovariant operations Marvian14 (); Yadin16 (). It follows that the resolution of a quantum clock gives an additional constraint of a secondlaw type:
Observation 2.
Under a thermal process, the quantum Fisher (skew) information of a quantum system cannot increase, i.e.
(2) 
We highlight that this condition is independent from those obtained previously, based on a family of entropy asymmetry measures Lostaglio15 () and modes of asymmetry LostaglioX (); MarvianPRA (). In Supplemental Material, we give an example of a state transformation that is forbidden by (2) but not by previous constraints.
Importantly, the measure is negligible in the manycopy, or independent and identically distributed (IID) limit: for all Lostaglio15 (). In contrast, the QFI and skew information are additive, thus for all , where is the additive copy Hamiltonian. Therefore these measures remain significant in the manycopy limit and provide nontrivial restrictions on asymptotic state transformations under thermal operations.
We can illustrate the physical implications of this condition in an particle twolevel systems with a local Hamiltonian for every th particle. As noted above, for a product state , the QFI and skew information scale linearly with . On the other hand, a GHZ state has quadratic scaling, . Thus the restriction given by Eq. (2) indicates that a thermal process cannot transform a product state into a GHZ state. More generally, it is known that for producible states in qubit systems Toth12 (); Hyllus12 (), so genuine multipartite entanglement is necessary to achieve a high clock precision of . Also note that the QFI has been used to quantify “macroscopicity”, the degree to which a state displays quantum behavior on a large scale Frowis2012 (); Yadin16 ().
Tradeoff between work and clock resources.— Having examined the two types of thermodynamic coherence independently, it is natural to ask if there is a relation between them. Here, we demonstrate that there is always a tradeoff between work and clock coherence resources. We first give the following bound in an particle twolevel system:
Theorem 1 (Clock/work tradeoff for twolevel subsystems).
For an particle twolevel system with energy level difference , the coherent work and clock resources satisfy
(3) 
where is the total Hamiltonian and is the binary entropy.
This shows that a quantum state cannot simultaneously contain maximal work and clock resources. Moreover, when the clock resource is maximal, , no work can be extracted from coherence . Conversely, if the extractable work form coherence is maximal, , the state cannot be utilized as a quantum clock as . For we derive a tighter inequality,
(4) 
We demonstrate that the GHZ state and Dicke states , summing over all permutations of subsystems, are limiting cases of this tradeoff relation. For a Dicke state, the extractable work from coherence is given by with and excitation rate . However, each Dicke state has since it has support on a single energy eigenspace with . In particular, when , , attaining the maximal value and saturating the bounds Eq. (3) and (4). The GHZ state behaves in the opposite way: has maximal QFI while having no internal coherence, thus . In this case, we can see the saturation of both bounds (3) and (4).
Furthermore, our twolevel tradeoff relation can be generalized for an arbitrary noninteracting particle system.
Theorem 2 (Clock/work tradeoff for arbitrary subsystems).
Let be a noninteracting Hamiltonian of subsystems, where the th subsystem has an arbitrary (possibly degenerate) level spectrum . Also define with . Then
(5) 
This is more generally applicable than (4), but is weaker for twolevel subsystems – maximal does not imply via (5).
For systems with identical local level Hamiltonians, (5) reduces to
(6) 
where is extractable work per particle and where is the maximum energy difference between the local energy eigenvalues. Our bounds do not limit the extractable work in the IID limit, since for .
Remarks.— We have found that thermodynamic coherence in a manybody system can be decomposed into time and energyrelated components. Manybody coherence contributing to the thermodynamic free energy has been shown to be convertible into work by a thermal process, without changing the classical energy statistics. We have illustrated that this workyielding resource comes from correlations due to coherence in a multipartite system. We have also shown that coherence may take the form of a clock resource, and we have quantified this with the quantum Fisher (skew) information. Our main result is a tradeoff relation between these two different thermodynamic coherence resources.
Acknowledgements.— This work was supported by the UK EPSRC (EP/KO34480/1) the Leverhulme Foundation (RPG2014055), the NRF of Korea grant funded by the Korea government (MSIP) (No. 20100018295), and the Korea Institute of Science and Technology Institutional Program (Project No. 2E2668016P025). DJ and MSK were supported by the Royal Society.
References
 (1) J. Åberg, Phys. Rev. Lett. 113, 150402 (2014).
 (2) P. Skrzypczyk, A. J. Short, and S. Popescu, arXiv:1302.2811.
 (3) P. Skrzypczyk, A. J. Short, and S. Popescu, Nat. Commun. 5, 4185 (2014).
 (4) R. Uzdin, A. Levy, and R. Kosloff, Phys. Rev. X 5, 031044 (2015).
 (5) K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings, New J. Phys. 18, 023045 (2016).
 (6) B. Groisman, S. Popescu, and A. Winter, Phys. Rev. A 72, 032317 (2005).
 (7) D. Jennings and T. Rudolph, Phys. Rev. E 81, 061130 (2010).
 (8) J. J. Park, K.H. Kim, T. Sagawa, and S. W. Kim, Phys. Rev. Lett. 111, 230402 (2013).
 (9) D. Reeb and M. M. Wolf, New J. Phys. 16, 103011 (2014).
 (10) D. E. Bruschi, M. PerarnauLlobet, N Friis, K V. Hovhannisyan, and M Huber, Phys. Rev. E 91, 032118 (2015).
 (11) M Huber, M. PerarnauLlobet, K. V. Hovhannisyan, P. Skrzypczyk, Claude Klöckl, N. Brunner, and A. Acín, New J. Phys. 17 065008 (2015).
 (12) M. PerarnauLlobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Acín, Phys. Rev. X 5, 041011 (2015).
 (13) D. Janzing, P. Wocjan, R. Zeier, R. Geiss, T. Beth, Int. Journ. Th. Phys. 39 (12):22172753, (2000).
 (14) M. Horodecki and J. Oppenheim, Nat. Commun. 4, 2059 (2013).
 (15) F. Brandão, M. Horodecki, J. Oppenheim, J. M. Renses, and R. W. Spekkens, Phys. Rev. Lett. 111 250404 (2013).
 (16) F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, Proc. Natl. Acad. Sci. U.S.A. 112, 3275 (2015).
 (17) M. Lostaglio. D. Jennings, and T. Rudolph, Nat. Commun. 6, 6383 (2015).
 (18) M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Phys. Rev. X 5, 021001 (2015).
 (19) P. Ćwikliński, M. Studziński, M. Horodecki, and J. Oppenheim, Phys. Rev. Lett. 115 210403 (2015).
 (20) M Lostaglio, M. P. Müller, and M. Pastena, Phys. Rev. Lett. 115, 150402 (2015).
 (21) J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, J. Phys. A 49, 143001 (2016).
 (22) M. N. Bera, A. Riera, M. Lewenstein, A. Winter, arXiv:1612.04779.
 (23) H. Wilming, R. Gallego, and J. Eisert, Phys. Rev. E 93, 042126 (2016).
 (24) M. P. Müller, arXiv:1707.03451.
 (25) G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Marvian, arXiv:1708.04302 (2017).
 (26) M. Weilenmann, L. KrÃ¤mer, P. Faist, R. Renner Phys. Rev. Lett. 117, 260601 (2016).
 (27) M. B. Plenio and S. Virmani, Quantum Inf. Comput. 7, 1 (2007).
 (28) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
 (29) T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
 (30) A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016).
 (31) S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev. Mod. Phys. 79, 555 (2007).
 (32) G. Gour and R. W. Spekkens, New J. Phys. 10, 033023 (2008).
 (33) I. Marvian and R. W. Spekkens, Phys. Rev. A 90, 062110 (2014).
 (34) I. Marvian and R. W. Spekkens, Nat. Commun. 5, 3821 (2014).
 (35) S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439, (1994).
 (36) P. Faist, F. Dupuis, J. Oppenheim, R. Renner Nat. Commun. 6 7669 (2015).
 (37) P. Faist, PhD thesis, arXiv:1607.03104 (2016).
 (38) K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, Rev. Mod. Phys. 84, 1655 (2012)
 (39) H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001).
 (40) S. Lloyd, Phys. Rev. A 56, 3374 (1997).
 (41) W. H. Zurek, Phys. Rev. A 67, 012320 (2003).
 (42) P. Chen and S. Luo, Theor. Math. Phys. 165, 1552 (2010).
 (43) P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, Nat. Phys. 10, 582 (2014).
 (44) E. P. Wigner and M. M. Yanase, Proc. Natl. Acad. Sci. U.S.A. 49, 910 (1963).
 (45) G. E. Crooks, Tech. Note 008v4 (2012), http://threeplusone.com/CrooksFisherInfo.pdf.
 (46) F. Hansen, Proc. Natl. Acad. Sci. U.S.A. 105, 9909 (2008).
 (47) D. Petz and C. Ghinea, Introduction to Quantum Fisher Information, QPPQ: Quantum Probability and White Noise Analysis. Vol. 27, edited by R. Rebolledo and M. Orszag (World Scientific, Singapore, 2011), pp. 261â281.
 (48) B. Yadin and V. Vedral, Phys. Rev. A 92 022356 (2015).
 (49) G. Tóth, Phys. Rev. A 85, 022322 (2012).
 (50) P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezze, and A. Smerzi, Phys. Rev. A 85, 022321 (2012).
 (51) F. Fröwis and W. Dür, New J. Phys. 14, 093039 (2012).
Appendix A Supplemental Material
a.1 Physical assumptions for the analysis.
For the class of free operations that define the thermodynamic framework, we make the following three physical assumptions.

Energy is conserved microscopically. We assume that any quantum operation that is thermodynamically free admits a Stinespring dilation of the form
(7) where the isometry conserves energy microscopically, namely we have
(8) Here denotes the Hamiltonian for the system .

An equilibrium state exists. We assume that for any systems and there exist states and such that for all thermodynamically free operations between these two quantum systems. In the case where one admits an unbounded number of free states within the theory, this together with energy conservation essentially forces one to take
(9) at some temperature . However one may also consider scenarios in which the thermodynamic equilibrium state deviates from being a Gibbs state. Here we restrict our analysis to thermal Gibbs states.

Quantum coherences are not thermodynamically free. Since one is interested in quantifying the effects of coherence in thermodynamics, one must not view coherence as a free resource that can be injected into a system without being included in the accounting. We therefore assume that the thermodynamically free operations do not smuggle in coherences in the following sense: if has a microscopic description of the form 7 then the same operation is possible with and being some other energy conserving isometry. In other words no coherences in are exploited for free. It can be shown this assumption has the mathematical consequence that
(10) for any translation through a time interval , namely covariance under timetranslations.
The set of quantum operations defined by these physical assumptions are called thermal processes. It can be shown that for states blockdiagonal in energy, the state transformations possible under thermal processes coincides with those of TOs. It is an open question as to whether there is a natural physical assumption that separates the two classes of operations in terms of their state interconversion power over all quantum states.
Note that the third assumption only accounts for external coherences within the framework. The way in which we account for internal coherences in our analysis is to demand that the diagonal components (in the basis ) of the state are left invariant by the evolutions considered. Also note that if assumptions (1) and (3) hold then the isometry can be taken to be equal to .
a.2 Proof of Observation 1
We find the conditions under which work can be deterministically extracted from coherence in a pure state. After energy block diagonalizing, the state can be written as , where are pure eigenstates of energy . Suppose gives the maximum value of . Then we have
(11)  
where is a fully dephased state in the energy eigenspace . Then we notice that for any , unless is incoherent (i.e. no internal coherence for ). This leads to for any finite value of . In the limit , , unless is incoherent. Here, are eigenvalues of . Thus if a pure state does not contain internal coherence for , . Conversely, if the state contains internal coherence for , for all , thus positive work can be extracted.
Appendix B Work extraction from a pure state coherence
We consider work extraction from coherence in a twoqubit system with local energy difference in each qubit, starting from a pure state of the form
Observation 1 says that should be large enough to extract work under a singleshot thermal operation. In this case, the condition from Observation 1 can be written as
This leads to a necessary condition for extracting a positive amount of work from coherence:
Thus if , we cannot extract work from coherenc,e even though the state definitely contains internal coherence of the form . On the other hand, a sufficient condition for can be obtained:
b.1 An example of the quantum Fisher (skew) information imposing independent constraints from or
We present an example showing that our asymmetry quantifiers give constraints on quantum thermodynamics independent from those due to the free energies or the coherence measures . Let us consider the transformation by a thermal process of the initial state
to the final state
with the Hamiltonian . It can be checked that the free energies and coherence measures of the initial state are larger than those of the final state . Furthermore, each mode of coherence is decreased from to . However, the skew information values for are given by and ; the quantum Fisher information values are and (all in units of ). Thus a thermal process cannot transform into , but such a transformation is not disallowed by the restrictions given by or .
Appendix C Tradeoff relation between work and clock resources
We first prove the following proposition.
Proposition 1 (Work bound).
For a given energy distribution , the extractable work from coherence is upper bounded as follows:
(12) 
where is the dimension of the eigenspace of energy level .
Proof.
Note that
(13)  
Since both and are energyblock diagonal, we can express and for with . Then we have
(14)  
since for dimensional states and . ∎
Now we prove Theorem 1:
(15) 
Proof.
In an particle twolevel system with energy difference , the degeneracy of the energy level is given by , where . By using the fact , we obtain
(16) 
Furthermore, we can express the binary entropy as
For a given probability distribution and we have
where with and . Then we have
By substituting this result into Eq. (16), we obtain
(17) 
where and . Note that is symmetric about and monotonically increasing for . We also note that for every quantum state . These observations lead to , which completes the proof. ∎
We can numerically verify that for every and up to when we define for both even and odd numbers of using the Gamma function . The proof for a general case has not been found. Note that when , this bound approaches to the bound of Theorem. 1.
We also show the tighter bound for case:
(18) 
Proof.
Suppose the state has probability , , and for each energy level , and . By using Eq. (16) for , we have , since the state has a degeneracy in the energyeigenspace only for . In this case, energy variance is given by
which leads to the maximum value of ,
for a given value of . Again, we can use to get
which is the desired inequality. ∎
Appendix D Proof of Theorem 2
We prove the statement of Theorem 2:
(19) 
where with is the maximum energy difference of the th subsystem.
Proof.
In this case, the degeneracy of the energy is given by
where is a probability (or frequency) to have the energy in the particle system, since is total possible numbers of . Then can be considered as a probability distribution of a variable from the distribution of independent random variables of for th party. In our case, is strictly bounded by and it has the same probability for every and zero for all other cases. Hoeffding’s inequality Heffding63 (), then shows that
where and . Using this, the upper bound of is given by