The Maxwell demon operates as the semi-permeable membranes in the last subsection. It might be argued that the demon is not physical in the thermodynamic path. Replacing the Maxwell demon by a physical one is the subject of quantum thermodynamics. We will now review the works14,15 on how a physical demon assists a quantum heat engine.
Before discussing the operations of quantum engine, we have to find the quantum version of those fundamental thermodynamic variable, such as work W or heat Q.16Consider an adiabatic thermodynamic path between two heat baths. Let a potential well couples to the heat bath of temperature Th at the beginning, and then couples to the heat bath of temperature Tl at the end. We set ρ as the density matrix of our potential well, and ˆH is the Hamiltonian. Then the expectation value of Hamiltonian yields the internal
energy
U = T r(ρ ˆH) , (3.17)
⇒ dU = T r(ρd ˆH) + T r( ˆHdρ) . (3.18) During the adiabatic process, the variation of eigenenergies is denoted by R:= (R1, R2, ..., Rn) where R1 is the change of the ground state energy Eg during the process, i.e., R1 = ∆Eg,·, Rn = ∆En.
R can be thought as the thermodynamical variable, thus the infinitesimal variation of Hamiltonian with respect to R can be written as
dRH = dˆ RQ + dˆ RW .ˆ (3.19) The first term is the off-diagonal part of dRH, and the second term is theˆ diagonal one. We then express the second term in the expansion of the basis, i.e.,
dRW =ˆ ∑
m
dEm|m⟩ ⟨m| (3.20)
where dEm =⟨m|dRHˆ|m⟩ with Em the eigen-energy of |m⟩.
Then the variation of work done during the process is given by dW = T r(ρdRW ) =ˆ ∑
m
PmdEm . (3.21)
In deriving the above we have assumed R changes adiabatically. However, the the potential well is brought to contact with different thermal baths at two different instants to reach thermal equilibrium, thus the heat transfer could be discontinuous. By definition, the variation of the hear is given by
dQ := dU − dW (3.22)
The above expressions are consistent with physical intuition. In classical thermodynamic, the heat dissipation is equivalent to the entropy dissipation.
The variation of heat dQ and entropy dS are also related to the probability distribution in quantum thermodynamic. Thus we know that the work done
on the system leads the variation of eigen-energies. On the other hand, the heat flow only changes the occupation probability of the eigen-states.
We have reviewed the basics of quantum thermodynamics, and based on that we will consider a specific model of quantum engine with a potential well of two-level system as the working substance. Let us denote the ground state by |0⟩ and the excited state |1⟩, and the corresponding Hamiltonian is
H =ˆ ∑
n=0
(En− E0)|n⟩ ⟨n| = ∆ |1⟩ ⟨1| (3.25)
where ∆ = E1 − E0. Later on we will have many two-level systems so that we will add the specific subscript, for example, for system A we have HˆA= ∆A|1⟩A⟨1|.
We prepare two potential wells of two-level systems: the system ρS and the memory of demon ρD. We can extract work via the interaction of ρS and ρD. Initially, ρS and ρD are decoupled and reach thermal equilibrium with two different heat baths of temperature TS and TD, respectively. See Fig.
3.5. Let us assume the initial state of of ρS and ρD are
ρS = PS1|1⟩ ⟨1| + PS0|0⟩ ⟨0| (3.26) ρD = PD1 |1⟩ ⟨1| + PD0 |0⟩ ⟨0| . (3.27)
Figure 3.5: The quantum heat engine of two-level systems.
Therefore, the initial joint state is
ρSD(1) = PS,D1,1 |1, 1⟩ ⟨1, 1| + PS,D1,0 |1, 0⟩ ⟨1, 0|
+PS,D0,1 |0, 1⟩ ⟨0, 1| + PS,D0,0 |0, 0⟩ ⟨0, 0| . (3.28) There are two operations in our heat engine cycle. The first operation is to measure the state of ρS, and take down the outcome to ρD. This can be achieved by the C-NOT operation. The second operation performs a feedback control on ρS to extract work. Thus we can realized the two operations by the interaction Hamiltonian ˆHA→B:
HˆA→B = g
Figure 3.6: The circuit of quantum heat engine. The first part is to measure the information of ρS and take down the outcome to ρD. The second part is the feedback control of demon. The joint system extract work by the second step.
For the first operation, A = S and B = D, but A = D and B = S for the second operation. After coupling ρD to ρS by ˆHA→B, the demons perform a measurement via the C-NOT operation, and take down the outcome to its memory ρD, this then results a new state as
ρSD(2) = PS,D1,1 |1, 0⟩ ⟨1, 0| + PS,D1,0 |1, 1⟩ ⟨1, 1|
+PS,D0,1 |0, 1⟩ ⟨0, 1| + PS,D0,0 |0, 0⟩ ⟨0, 0| . (3.30) Note that the joint entropy is conserved in this process, which agrees with Bennett’s assertion that the measurement is a reversible process. During this process, the internal energy of demon changes so that it causes an expense of work WD :
WD = TrD[TrS(ρSD)(2) ˆHD]− TrD[TrS(ρSD)(1) ˆHD] , (3.31)
= ∆D(PD1 − PS,D1,0 − PS,D0,1) (3.32)
where ∆D is the energy difference between excited and ground states for the demon, and T rSD is the trace for the full system.
The next operation of the heat engine is the conditional evolution (CEV) which is defined as follows:
|1⟩ → |e1⟩ = cos θ |1⟩ + sin θ |0⟩ (3.33)
|0⟩ → |e0⟩ = − sin θ |1⟩ + cos θ |0⟩ . (3.34) Thus the feedback action of the demon to the system ρS yields the final state as follows:
ρSD(3) = PS,D1,1 |1, 0⟩ ⟨1, 0| + PS,D1,0 |e1, 1⟩ ⟨e1, 1|
+PS,D0,1 |e0, 1⟩ ⟨e0, 1| + PS,D0,0 |0, 0⟩ ⟨0, 0| . (3.35) Similarly, the work done by the system ρS is
WS = TrS[TrD(ρSD)(2) ˆHS]− TrS[TrD(ρSD)(3) ˆHS] , (3.36)
= ∆S(PS1− PS,D1,1 − PS,D1,0| ⟨e1|1⟩ |2− PS,D0,1| ⟨e0|1⟩ |2) . (3.37) When the two operations are accomplished, then ρS and ρD are decou-pled. Then, ρS and ρD couple to their own heat baths to reach the thermal equilibrium, respectively. We can get the heat flow of ρS and ρD as follows:
Qin = TrS[TrD(ρSD)(1) ˆHS]− TrS[TrD(ρSD)(3) ˆHS] (3.38)
= ∆S(PS,D1,0 − PS,D1,0| ⟨e1|1⟩ |2− PS,D0,1| ⟨e0|1⟩ |2) (3.39) for the heat absorbed by ρS, and
Qout = TrD[TrS(ρSD)(3) ˆHD]− TrD[TrS(ρSD)(1) ˆHD] (3.40)
= ∆D(PS,D1,0 + PS,D0,1 − PD1) (3.41) for the heat released by the demon.
Finally, the net work done in one cycle of this heat engine is
Wcycle=Qin− Qout , (3.42)
=WS − WD , (3.43)
=∆S(PS1− PS,D1,1 − PS,D1,0| ⟨e1|1⟩ |2− PS,D0,1| ⟨e0|1⟩ |2)
− ∆D(PD1 − PS,D1,0 − PS,D0,1) . (3.44)
As shown above, the quantum heat engine operates in a similar manner as its classical version: the heat flows from thermal bath of high temperature to the one of lower temperature. Moreover, the end results also obey the same relations as its classical version.
Chapter 4
The check of Holevo Bound in two-level system
In this section we propose simple models of quantum demon modeled by quantum circuit and then derive the Holevo bound from the thermodynamic consideration. Schematically, the relation between thermodynamics and the information theory for our specific setup is depicted in Fig. 4.1. We replace the working substance by two-level systems in this process. This process tries to imitate the thermodynamic cycle in section 3.1. We will design a quantum circuit which functions as semi-permeable membranes, viz, the quantum demon.
More specifically, first, we consider the more detailed behaviors of the semi-permeable membranes. Then the desired quantum circuit follows. This will be done in the next subsection. After this, we go through the cycle considered in section 3.1 under two different conditions. In the first case we include the physical memory bit of Maxwell demon. As the memory space is finite, we have to incorporate the irreversible process to complete the cycle of the quantum circuit. In the second case, we consider a demon without the need of using memory so that the whole thermodynamic cycle is a reversible process.
4.1 The design of quantum circuit
This subsection introduces the behavior of semi-permeable membranes in more details. The membranes go through three steps to classify the
.
Figure 4.1: Alice and Bob are the two-level systems with different eigen-spectrum. The operation basis of Alice and Bob are not the same when thermalizing with the heat bath. For simplicity, we will use density matrix to depict Alice and Bob in the detailed circuit operations in the main text.
molecules: perform the measurement, take down the information and de-cide the motion of the molecules, see Fig. 4.2.
The first step is the state transformation. When a molecule moves toward the membranes, the demon performs a measurement on the molecule by the basis of membranes. The measurement helps the demon to observe the state of the coming particles. This observation forces the states of molecules to be transformed. The outcomes of transformation depend on the conditional probability, which was discussed in section 3.1. It is similar to an operation that the demon acts a set of projectors on the quantum states.
The operation of projector can be realized by the C-NOT operation in the quantum circuit. Note that the operation basis of the target bit and the controlled bit are different. If the C-NOT gate can not distinguish the state of the controlled bit unambiguously, then the outcomes of the C-NOT operation will depend on the probability of the projection.
At the second step, the demon takes down the outcome to its memory.
Since there is no physical memory bit in gas model, the erasure is not an essential process.
At the third step, the demon has to decide to let the molecule to pass through or not according its memory. In the setting considered in7, when the membrane allows a molecule to move into the chamber σ, it also means to transfer the information of the molecule to σ. Thus, this operation does increase the entropy of σ in an obvious way. On the contrary, if the molecule bounces back by the membranes, then the entropy of either chamber does
not change.
Therefore, we can prepare two different potential wells. Each hosts a two-level system and can be thought as representing a chamber in the gas model:
the chamber A is represented by it density matrix ρA, and the chamber B by ρB, see Fig. 4.1. The goal of our circuit is that tries to decrease the entropy of ρA and increase the entropy of ρB.
Apparently, the design of quantum circuit must include the above three steps above. There are three parts in this circuit as shown in Fig. 4.2
Figure 4.2: The circuit for semi-permeable membrane (or the demon).
Semi-permeable membranes function as a state selector in section 3.1.
The behavior of state selection is similar to the operation of C-NOT gate.
Thus we can use the C-NOT operation to manipulate the entropy of each two-level system in the quantum circuit. Fig. 4.3 illustrates the equivalence between the C-NOT gate and the semi-permeable membranes.