We will review the erasure process in a gas model with energy consump-tion in this subsecconsump-tion12. By the definition of Landauer, erasure is a “restore-to-one” process. It shall go through four steps to accomplish the erasure.
The first step is the preparation of a gas chamber. Assume that a cham-ber is brought to contact with a temperature T heat bath. The volume of chamber is V , and the volume of heat bath is huge enough to keep the temperature constant during the cycle. Inside the chamber a small molecule bounces randomly. The volume of chamber is divided into two equal halves by a fixed partition. Then we set the occupied volume at left as the state ONE, and the occupied volume at right as the state ZERO. This setting forms the binary information in a chamber.
At the second step, the Maxwell demon drops out the partition in the middle from chamber. Then the occupied volume is the whole chamber V .
At the third step, the demon inserts the partition at the right end. Note that this operation does not consume the energy.
At the final step, the demon attaches a piston with the partition and push toward left. The piston moves very slowly so that the demon makes a quasistatic compression in an isothermal way. The demon stops pushing when the partition reaches the center of chamber. Now the occupied volume becomes V2 and the whole system is fixed to the ONE state. Note that we do not need to know the prior state as either the ONE and the ZERO state will be transferred to the ONE state after the erasure process. The whole procedure is summarized in Fig. 2.2
The Maxwell demon needs to do the extra work to compress the occupied volume. The occupied volume are compressed from V to V2, thus the energy expense in the erasure process is
Werasure =
From the above erasure process, we need to spend the amount of kT ln 2 energy at least to erase a bit of information. This value also verifies Lan-dauer’s principle that k ln 2 is the minima entropy dissipation.
Figure 2.2: The thermodynamic operations of the erasure process. It is a general procedure that can fixes the initial state to the standard state.
This procedure also agrees with the standpoint of Bennett: Demon have to perform external work to erase the information.
Chapter 3
Information and thermodynamic
There are verifications of Landauer‘s principle see (18 and13), and even in the experiments2. These verifications indicate the deep link between in-formation and energy.
In the following two subsections we will review some examples to further reinforce this link. In the first part, we review how to derive the Holevo bound from a thermodynamic cycle. The basic idea is to translate the consequence of the 2nd law of thermodynamics into the form of the information theory so that we can re-interpret it as the Holevo bound7,10.
The second part introduces the basics of the quantum thermodynamics.
We use quantum matter as the working substance so that we can design a thermodynamic cycle between two heat baths and extract the positive work14,15.
3.1 Holevo Bound in a thermodynamic cycle
In the following we will review the works7,10. The Fig 3.1 shows the ther-modynamic cycle which implies the Holevo bound. This cycle is composed by three thermodynamic paths. The path A is the returning path from the final state to the initial state. The path B and C are the forward operations from the initial state to the final state. The three paths can form a cycle by the order B → C → A.
The working substance of this cycle is a gas of molecules inside a chamber
Figure 3.1: The thermodynamic cycle for deriving the Holevo bound.
with two movable semi-permeable membranes in the middle. There are two thermodynamic paths from the initial state to the final state. The difference between the two paths is the choice of the semi-permeable membranes. These semi-permeable membranes function as Maxwell demon. It can be opaque to the certain state of molecules and transparent to the others. The ability of selecting molecules is the same as Maxwell demon so that we can design a mechanism to extract the work by functioning the membranes in Fig 3.2.
We start from the path A. We can see that the expansion of the occupied volume can extract the work from the gas chamber as shown in Fig. 3.2. On the contrary, the compression of occupied volume consumes energy. There-fore, reversing the path A can extract the work, and we will do that here.
Note that all the three paths are reversible, thus the amount of energy lost and energy gain is the same during the thermodynamic path.
Initially, a chamber of volume V is divided into two sections, p1V and p2V by a partition. p1 and p2 are the fractions of occupied volume and particle number, and p1 + p2 = 1. This chamber is brought to contact with a heat bath of temperature T , and the volume of the heat bath is large enough so that it can keep constant temperature during the process. As uniform density of the molecules, there are p1N of particles in the left chamber and
Figure 3.2: Extract work in a gas chamber with semi-permeable movable membrane partitions: A set of semi-permeable membranes are initially in-serted at the center of chamber. The chamber is brought to contact with a infinitely large heat bath of temperature T . The membranes are tied with the weights by massless ropes, respectively. During the expansion of the gas molecules, the membranes moves toward the bottoms of the vessel, re-spectively. Then we can raise the weights up by the expansion of the gas molecules.
p2N of particles in the right chamber. The state of particles in the left is denoted by |ψ1⟩, and state of particles in the right by |ψ2⟩. We can describe the whole chamber by a density matrix ρA:
ρA= p1|ψ1⟩ ⟨ψ1| + p2|ψ2⟩ ⟨ψ2| (3.1) where p1 and p2 are the probability distributions of |ψ1⟩ and |ψ2⟩ states, respectively.
If a powerful demon can perfectly distinguish |ψ1⟩ from |ψ2⟩ in a direct way, we can then insert a set of membranes to extract the complete work.
We set the membrane M1 to be opaque to the state|ψ1⟩ and transparent to the state |ψ2⟩. No molecule can be rebound in this expansion. Suppose that the demon deals with a molecule at a time, then the expansion procedure
goes very slowly and can be assumed to be quasistatic. Thus we can derive the extractable work Wext by the state equation of ideal gas. For molecules in the state |ψ1⟩, the extractable work is
Wψ1 =
∫ V
p1V
p1N KT
V dV =−p1N KT ln(p1) . (3.2) Similarly, the extractable work for the molecules in the state|ψ1⟩ is Wψ2 =
−p2N KT ln(p2). Thus the total extractable work in the complete expansion is
Wtotal= Wψ1 + Wψ2 (3.3)
=−NKT (p1N KT ln(p1) + p2N KT ln(p2)) (3.4)
= H(A) . (3.5)
It is interesting to see that the extractable work of complete expansion can be exactly expressed as the Shannon entropy of the whole system ρA. These operations correspond to the path A of Fig. 3.1.
All thermodynamic paths in this cycle are reversible. The reversibility allows us to consume the same amount of energy from the final state to the initial state, and we do not need to perform the extra work. Thus the backward operation of path A also spends the H(A) bits of work.
Next, we consider a less powerful demon, which can not perfectly distin-guish ψ1 from ψ2. At this time, we set the membrane M1 to be opaque for the
|e1⟩ molecules, but transparent for the |e2⟩ molecules. And |e2⟩ is orthogonal to |e1⟩, i.e., ⟨e2|e1⟩ = 0. Therefore, supposes that a molecule goes toward to membrane M1. The M1 membrane has the chance of the probability P (e1|ψi) to reflect the molecules, and has the chance of the probability P (e2|ψ1) to let the molecule to pass through. Note that P (e2|ψ1) + P (e1|ψ1) = 1.
We have to go through two stages of extraction to obtain the extractable work. In the first stage, the demon inserts a set of movable membranes which operate with basis {|e1⟩ , |e2⟩}. The imperfectness of the demon’s ability in distinguishing the particles yields incomplete expansion so that there is amount of margin to reach H(A).
In the second stage, the demon replaces the membranes by another set.
This hypothetical set can distinguish the states of |Left⟩ and |Right⟩. Thus we can extract the amount of marginal work to reach H(A). According to the occupied volume of each state in Fig. 3.3, we can get the extractable
Figure 3.3: Two stages of extracting work via the imperfect semi-permeable membranes (demon).
work for the state ψ2 measured in the|e1⟩ basis as follows:
Wp(ψ2|e1) =
∫
P dV (3.6)
=
∫ V
p(ψ2|e1)
p(e1)p(ψ2|e1)N KT
V dV (3.7)
=−p(e1)p(ψ2|e1)N KT ln p(ψ2|e1) . (3.8) Similarly, the extractable works for the other three cases are
Wp(ψ2|e2) =−p(e2)p(ψ2|e2)N KT ln p(ψ2|e2) , (3.9) Wp(ψ1|e1) =−p(e1)p(ψ1|e1)N KT ln p(ψ1|e1) , (3.10) Wp(ψ1|e2) =−p(e2)p(ψ1|e2)N KT ln p(ψ1|e2) . (3.11) Therefore, the total extractable work for such an imperfect demon is
Wmargin (3.12)
= Wp(ψ1|e2)+ Wp(ψ2|e1)+ Wp(ψ1|e1)+ Wp(ψ2|e2) (3.13)
= H(A|B) . (3.14)
In the last expression, we have translated the energy extraction to the form of conditional Shannon entropy. The amount of margin part is H(A|B) bits of
work. Since the second stage is a hypothetical process, then the extractable work at the first stage is
H(A)− H(A|B) = I(A : B) . (3.15)
This is nothing but the mutual information shared between system A and B. From all the above discussions, we explicit observe that there is a close connection between energy and information.
Figure 3.4: The detailed argument to reach the final state, i.e., the Path C in Fig. 3.1. This figure is taken from10.
After the path B, we need to consume ∆S bits of work to reach the final state, the Path C in Fig. 3.1 which is also a reversible process. This path tries to let all the molecules pass through the semi-permeable membranes, see Fig. 3.4.
In the following, we review the arguments in10: In the initial state (a), the demon singe out the state of molecules |e1⟩ and |e1⟩ to reach (b). First it attaches a vacuum chamber with the same volume at the bottom of the
vessel. Then the demon push the membrane M2 but with M1fixed. Since the occupied volume of each molecule is invariant, the operations form (a) to (b) do not consume the energy. The process (c) to (d) is to compress the volume of the chamber. The compression of volume is associate with the numbers of states. This compression can be seen as the data compression of information, then the (a) to (d) process is the encoding process and consumes S(σ) bits of work. After σ reaches (d), the demon inserts the semi-permeable the set of membranes M2′ and M1′ with the chosen eigen-basis. This special basis allows the membranes to reach the complete expansion of the gas molecules.
The process (d) to (e) can extract S(ρ) bits of work, thus the whole path C process consumes S(σ)− S(ρ) bits of work.
Finally, we can form a thermodynamic cycle from the above operation.
Due to the restriction of 2nd law of thermodynamics, the energy extraction must be less than consumption. We can translate this constraint into the language of information theory, this turns out to yield the Holevo bound, viz,
I(A : B)≤ S(A) + ∆S . (3.16)