Irreversibility and generalized second laws

Although often called a theory of heat and work, thermodynamics can be regarded also as a theory of irreversibility. Ultimately, one wants to understand how much heat can be converted into work. As we will see, such work yield is hindered exactly by the amount of irreversibility of a given process. In this section I discuss a generalized formulation of the entropy production [2, 81], the quantifier of irreversibility of a quantum mechanical process. Remarkably, it can be formulated in purely information-theoretical terms, independent of the notions of heat and work.

Suppose there are two quantum systems, prepared in a product state⇢ = ⇢_E ⌦ ⇢_S, whose non-degenerate spectral representations are⇢E = P

apa|aiha|and⇢S = P

⌫ p⌫|⌫ih⌫|. The system is initially measured, undergoes unitary evolution and then it is measured again. Then, we seek to reverse the process and compare the probability distributions obtained for the forward and backward trajectories.

For that sake, I introduce the two-point measurement(TPM) protocol [82]. In particular, I perform measurements in both system and environment, the first measurement is at the begin- ning according to the projector⇧_a⌫ ⌘ |a⌫iha⌫|and the second, after unitary evolutionU, at an arbitrary local basis⇧^⇤_a0⌫⁰ = | a⁰ ⌫⁰ih a⁰ ⌫⁰|, schematically

⇢= ⇢E ⌦⇢S

⇧_a⌫

!|a⌫iha⌫|!^U ⇢^{0 ⇧}

⇤a0⌫0

!| ^{a0 ⌫0}ih ^{a0 ⌫0}|, (4.6) where⇢⁰ =U⇢U^†. The probability associated to this process is given by the Born rule

P[ ]= Tr⇣

U^†⇧^⇤_a0⌫⁰U⇧a⌫⇢⌘

= pap⌫|h ^{a0 ⌫0}|U|a⌫i|², (4.7)

in which I introduced the trajectory notation = (a,⌫,a⁰,⌫⁰). It is noteworthy that the last factor of Eq. (4.7) satisfies all the requirements for a conditional probability|h ^{a0 ⌫0}|U|a⌫i|²= p(a⁰⌫⁰|ab) and we can see the Bayes’ ruleP[ ]= p_ap_⌫p(a⁰,⌫⁰|a,⌫).

We now have to reverse this procedure. Note that the unitary evolution is invertibleUU^† = UU ¹ = 1and time reversal is equivalent to reversing the sign of time in U. Yet, the measure- ment process is not.

Any quantum process is composed of two fundamental interventions: measurements and unitary evolution [6]. Since the latter is reversible, measurements are the key to understand irreversibility. In particular, depending on what information we can access, that is, what choice we make about the basis| ^{a0 ⌫0}ih ^{a0 ⌫0}|we know more or less about the end of the process and, hence, we can dispose of more information on how to reverse it.

I now make a choice of measurement scheme, to be justified once its implications are estab- lished. Let the end-point measurement of the environment be made at the initial basis and for the system I choose the end-point measurement at the basis in which⇢⁰_S = TrE{U(⇢E ⌦⇢S)U^†} is diagonal. That is, I assume⇧^⇤_a0⌫⁰ = |a⁰ _⌫⁰iha⁰ _⌫⁰| withha⁰ _⌫⁰|a⌫i = _aa⁰h ^⌫0|⌫i. In that man- ner, I can only reverse the process based on the information acquired through it and then the reverse process begins at ˜⇢ = ⇢_E ⌦ ⇢⁰_S³. Starting from this state, I perform the reverse TPM.

Schematically

˜

⇢= ⇢_E ⌦⇢⁰_S ^⇧

⇤a0⌫0

!|a⁰ _⌫⁰iha⁰ _⌫⁰| ^U!^† ⇢˜^{0 ⇧}!^a⌫ |a⌫iha⌫|, (4.8) whose associated probability is

P[ ]˜ = pa⁰p^⇤_⌫0|ha⌫|U^†|a⁰ ⌫⁰i|². (4.9)

To compare both processes, I introduce

[ ] ⌘lnP[ ] ln ˜P[ ]=ln P[ ]

P[ ]˜ . (4.10)

This quantity is so-calledstochastic entropy production; it quantifies the mismatch between a process and its reverse counterpart. Crucially, note that|ha⌫|U^†|a⁰ ⌫⁰i|² = |h ^{a0 ⌫0}|U|a⌫i|² so

3In other words, the final state is the state produced by the CPTP map ˜⇢=⇤(⇢) whose Kraus components are measurement operatorsM ⌘⇧^⇤_a0⌫⁰U⇧a⌫

that

[ ] =ln pap⌫

p_a⁰p^⇤_⌫0

*1 p(a⁰⌫⁰|a⌫)

p(a⌫|a⁰⌫⁰). (4.11)

We now perform the statistical average of this quantity⌃⌘ E[ ]=P P[ ] ln⇣

P[ ]/P[ ]˜ ⌘ .

⌃ =X

(lnpap⌫ lnpa⁰p^⇤_⌫0)pap⌫|h ^{a0 ⌫0}|U|a⌫i|² (4.12)

=Tr(⇢E⇢Sln⇢E⇢S) Tr ⇢⁰ln⇢E⇢⁰_S , (4.13) where⇢⁰ = U(⇢_E ⌦⇢_S)U^†, I used that⇢⁰_S =P

⌫⁰ p^⇤_⌫0| ^⌫0ih ^⌫0|and⇢_E =P

a⁰ pa⁰|a⁰iha⁰|. Employing the definition of entropy (2.14) we can write

⌃ = S(⇢⁰)+S(⇢⁰_S) Tr⇢⁰_Eln⇢_E ±(S(⇢⁰_E)+S(⇢⁰_S)), (4.14) Finally, this expression can be cast as

⌃= I^⇢0(S :E)+S(⇢⁰_E||⇢E), (4.15) whereI^⇢0(S : E)= S(⇢⁰_S)+S(⇢⁰_E) S(⇢⁰) is the mutual information andS(⇢⁰_E||⇢E) the relative entropy, both introduced in pg.11. The first term quantifies the informational loss in performing local measurements,i.e., the destruction of correlations ofS E state. The second term quantifies the change underwent by the environment, which we could not access due to our choice of basis inE. This formula was first presented in [80] and the above derivation comes originally from [81].

Let me discuss the meaning of our choice of basis. You can think, for example, that the environment is a collisional model and at the end of the process we either discarded the ancilla or it bounced back to the reservoir. In either case, information about the bath state would be hard to track and that is why we perform a measurement in a basis which only requires knowledge about the initial state. Performing the second measurement in the particular basis in which the end-point state is diagonal implies prior knowledge of such basis and, in general, this is an information we do not have about a bath. In practice, measuring in the end w.r.t. the initial basis formalizes formalizes our ignorance about the bath or, equivalently, implements the bath as a highly entropic entity.

More generally, one can also allow the environment to be multipartite, say⇢_E = ⇢_E1⌦...⌦⇢_EN; in this case we have

⌃ =I^⇢0s(S : E₁ :...: E_N)+X

i

S(⇢⁰_Ei||⇢_Ei), (4.16)

whereI^⇢0s(S :E1 :... :EN)=S(⇢⁰_S)+P

iS(⇢⁰_Ei) S(⇢⁰) [2].

The expression ⌃ in Eq. (4.15) is so-called a generalized second law of thermodynamics, once one realizes that both terms can be written as relative entropies (see pg.11) and therefore are always positive

⌃ 0, (4.17)

in which no mention was made to Gibbs states so far.

One may then find it peculiar that another choice of backward process could lead to a di↵er- ent second law. This is indeed the case, and these possibilities are explored in [81]. However, the specific choice I made is the one which treats the environments closer to a thermal bath and is thus suitable to make contact with the usual formulations of the second law, as follows.

First, Eq. (4.15) can be written in another form [2], which emphasizes the splitting between system and environment quantities. For this sake, I explicitly write Eq. (4.15) as

⌃ =S(⇢⁰_S)+S(⇢⁰_E) S(⇢⁰)+Tr ⇢⁰_E[ln⇢⁰_E ln⇢E] (4.18)

=S(⇢⁰_S) S(⇢S) S(⇢E) Tr⇢⁰_Eln⇢E , (4.19) where I have considered thatS(⇢⁰)= S(⇢ =⇢_E ⌦⇢_S). Then, we can recognize that

⌃ = SS Tr (⇢⁰_E ⇢E) ln⇢E

| {z }, (4.20) where SS = S(⇢⁰_S) S(⇢S) and is so-called entropy flux. At last, if we introduce the Gibbs state for the environment, we find that the last term writes

⌃= SS + Tr HE(⇢⁰_E ⇢E), (4.21)

whereHE is the local Hamiltonian of the environment. Assuming that the environment is at a

maximum entropy state thus renders a special way of entropy flux, proportional to the energy change in the environment. This motivates our definition ofheat⁴

QE ⌘Tr HE(⇢⁰_E ⇢_E) , (4.22)

the energy change in the environment. Therefore we can write the second law in the form

⌃ = SS + Q^E 0. (4.23)

Notably, Gibbs states are a rather special in the sense that they define an entropy flux directly proportional to the energy flux = Trn

HE(⇢⁰_E ⇢E)o

. However, the derivation of Eq. (4.15) suggests that there are other ways to implement our ignorance about the environment and if one adopts our definition of heat with more information about the environment she could have an entropic flux which is not proportional to the energy flux in to the environment.

Equation 4.23 also generalizes to multiple environments [2]. For example, in the case of two environments

⌃= SS + _AQ^A+ _BQ^B. (4.24)