馬克斯威的惡魔和量子引擎

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(1)國立臺灣師範大學物理學系碩士論文 Department of Physics National Taiwan Normal University. 馬克斯威的惡魔和量子引擎 Maxwell demon and Quantum heat engine. 研究生：蔡孟儒. 指導教授：林豐利博士 Advisor: Feng-Li Lin, Ph.D..

(2) Acknowledgement 這篇文章要首先感謝林豐利老師的耐心指導。謝謝老師不斷的和我討論，而且持續地把我拉回研究的主軸上；若非老師花此極大的心力，我想這個工作是不可能完成的。還有感謝兩位口試委員的指導，感謝何俊麟老師啟發我對物理和量子資訊的興趣，以及謝謝高賢忠老師的提問和指正。接著我要感謝武壇的老師和師兄：感謝林仲曦老師帶我進入武壇和武術的世界，讓我在研究以外的時間，有一個既興趣又能紓壓的運動。謝謝廣福學長不斷鼓勵我向外發展，並且不斷告訴我他在台灣以外看到的各種現象和經驗。謝謝孤單和錯過的心兩位師兄，陪我上山下海練拳，讓我在六大開的進步之外，了解何謂真正的「大師」。謝謝大師兄偶爾的跳針提問，讓我不斷地以新角度去看待以前學習過的物理。另外我要感謝書昂，他對物理的熱忱會令我想起以前的自己，我才能在受挫折後持續堅持。謝謝楊教練對淡江社團的鼎力相助，減少了我一個責任，和他聊拳術也讓我獲益良多。謝謝甘道夫和我討論攻略問題，和他閒聊讓我拋開不少壓力和緊張。也同時要感謝蚯蚓，回台中和他聊完後，我才能重拾心態繼續努力。還有要特別感謝玉蘋，謝謝她不斷給我打氣和鼓勵，也從不厭煩的傾聽我的抱怨和訴苦。最後我要感謝我的家人，都是他們讓我能夠自私的追求夢想，而且沒有其他後顧之憂，也謝謝他們對我的包容。. I.

(3) Abstract In this thesis, we study the physics of Maxwell demon and the related issues in the context of quantum information sciences. We first review the basics of the Maxwell demon and Landauer’s principle, and then focus on the connection between thermodynamics and the information theory. This connection is manifested in a derivation of Holevo bound of quantum communication from the second law of thermodynamics. We review a classical gas model as a physical setup for this derivation. We then adopt the recent development in quantum thermodynamics and construct a simple quantum circuit model to check the Holevo bound in the context of quantum circuit model, in which the Maxwell demon is replaced by the quantum gates with or without physical memory. 關鍵字：Maxwell demon, quantum heat engine, Holevo bound.

(4) Contents Acknowledgement. I. 1 Introduction 1.1 Turing machine . . . . . . . . . . . . . . 1.2 Maxwell demon and Landauer’s principle 1.3 The circuit model and its reversibility . . 1.4 Information and entropy . . . . . . . . . 1.5 Density matrix and Holevo bound . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 1 2 5 6 9 12. 2 Review of Maxwell Demon and the related 2.1 Szilard engine . . . . . . . . . . . . . . . . . 2.2 Landauer‘s principle . . . . . . . . . . . . . 2.3 Resolution of the Maxwell demon paradox . 2.4 Information erasure in a gas model . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 14 14 17 18 20. . . . . .. 3 Information and thermodynamic 22 3.1 Holevo Bound in a thermodynamic cycle . . . . . . . . . . . . 22 3.2 Quantum thermodynamic and Maxwell demon . . . . . . . . . 28 4 The 4.1 4.2 4.3 4.4. check of Holevo Bound in two-level system The design of quantum circuit . . . . . . . . . . . Circuit with memory bit . . . . . . . . . . . . . . Circuit without the memory bit . . . . . . . . . . Holevo bound from quantum circuit model . . . .. 5 Conclusion. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 34 34 36 41 44 48. I.

(5) A Appendix A.1 Properties of the projection values Bibliography. 49 . . . . . . . . . . . . . . . 49 50. II.

(6) Chapter 1 Introduction In the development of computer sciences, one important issue is the limitation of the computational resource. One can not perform computation with infinite computational resource, it is then important to maximize the computational efficiency or minimize the essential required computational resource. Among the various requirements on the available resources, in this thesis we only focus on the issue about the energy resource, especially from the point of view of quantum information sciences. We will review how to obtain the minimal energy bound in a complete computational cycle. In the context of quantum information sciences, the energy consumption is caused by the erasure of information, and it is an inevitably operation. Why do we need to erase the information in the computation process? The most direct reason is that the computer memory is of finite amount. When the memory space is no enough, we need to erase the stored but unnecessary informations to accommodate the new ones and to proceed the further computation. On the other hand, we do not consume the energy if we could own the (or the nearly) infinitely large memory space. Since we have to consume energy to erase memory, the information and thermodynamic are linked apparently. Therefore, we will review the original connection between them initiated by the proposal of the Maxwell demon. Maxwell demon was once considered as a viable method to derive the infinite energy. If such a demon does exist, then we do not need to worry about the energy resource in computation. After a series of discussion, peoples found that it is impossible to build the mighty demon as the memory space owned by demon is also finite, then the demon eventually has to erase his informations about the system. 1.

(7) The other important concept discussed in this thesis is the Holevo bound. This is an upper bound on the accessible information of the quantum state in a binary communication scheme. This implies the impossibility of distinguishing a quantum state unambiguously. This is also related to impossibility of perfect cloning of a quantum state, thus we can only have a single chance to perform the measurement and guess the state by the outcome. The right guesses reach the maxima if the receivers use the same measurement basis as the one by the transmitter for encoding. As we will see that one can derive the Holevo bound from the consideration of quantum thermodynamics. More precisely, the 2nd law of thermodynamics for a quantum circuit model (quantum engine) with the Maxwell demon played by the quantum gates can be translated into the Holevo bound on the accessible information. The main goal of this thesis is to review this issues in the literatures and propose some preliminary models for quantum engine to derive the Holevo bound thermodynamically. This thesis is organized as follow: in Chapter one we introduce the basics of information theory necessary for our discussions such as the connection between information and thermodynamics, and sketch the main goal of this thesis. In Chapter two we will further review the issues related to the Maxwell demon, and at the same time introduce the setup such as Szilard engine and its variants for the further consideration in the rest of the thesis. In Chapter three we will further elaborate on the connection between information and thermodynamics with more specific model studies. In Chapter four we proposed two simple quantum circuit models to study the Holevo bound from the quantum thermodynamics. The Maxwell demon here is replaced by the unitary quantum gates. Finally we conclude this thesis with a short conclusion.. 1.1 Turing machine The first step in designing a computer is to construct the computation model. Computation model is an ideal machine which simulates the calculation procedure of the human beings. The first model was proposed by Alan Turing in 1950 who tried to perform the computational task on a machine, known as Turing machine now. Turing machine is an ideal device. It has ability to achieve any logical computation process with finite hypothetical steps. Thus Turing machine is an universal model for building a physical 2.

(8) computer, and establishes the foundation of modern day computer sciences. In the following we illustrate the composition of Turing machine, see also Fig. 1.1.. Figure 1.1: Composition of Turing Machine: Computation is the progress of mapping logical states to physical states, and vice versa. The basic composition of Turing machine contains four parts: (i)a program (ii)a set of finite logical states (iii)a blank tape (iv)a read-write head device. The program for a Turing machine plays the role of algorithm in computation. Algorithm provides precise recipes for particular computation tasks. The algorithm usually takes a form of step-by-step list, which serves as the instructions to solve our problems. The instructions usually start from initial state with input values and produce output values at the end of computation. The logical states are a set of finite abstract variables. These variables are controlled by algorithm. These variables transform its state by the order of algorithm. Usually, logical states are set to the initial state and end up in the halting state at the end of a computation cycle. A series of unceasing operations force the states of variables to transform from initial states to the halting states in the computation process. These operations mimic the 3.

(9) procedures of computation in mind or brain of a human being. We can not accomplish the computational assignments only with these hypothetic steps, they have to co-operate with physical devices to print out the outcomes. The read-write head and the tape are the physical parts of Turing machine. The tape is an one-dimension object with infinite length which functions as computer memory. Since the tape is infinitely long, the Turing machine can operate with unlimited memory. It stores all intermediate outcomes produced during the whole computation process. The read-write head is a device to manipulate the contents on the tape. Its actions are controlled by the algorithm. The read-write head is similar as our hands, and the tape is similar as the scratch papers. The states of the read-write head and the tape are defined as the physical states. In the computation process, logical states and physical states are mapped into each other by following the algorithm. When the logical states turn into the halting state, the read-write shut down at the same time. This mimic the cooperative operations of our hands and minds when performing calculations. If the mapping between the logical and physical states is a one-to-one mapping, then the computation process is a reversible one. Reversible process allows the machines to retrace the any intermediate state from the final one. Therefore, the input values can be uniquely determined by performing the inverse operations in a reversible computation process. On the contrary, the many-to-one mapping yields a irreversible process. We can not decide the unique prior states in such a process. The energy expense of a computation process is deeply related to the reversibility of the process. Since the reversible process allows one to determine the unique input from a given output, any energy loss in an operation can be recovered by performing its inverse. That is, the energy is conserved if the whole computation cycle is reversible. On the contrary, we can not recover the energy loss in an irreversible process as the earlier states are not uniquely determined by the final one in such a process. That is, the energy is not conserved in the irreversible computation process. Therefore, we can attribute the energy consumption to the irreversible process. Fortunately, most operations are reversible. Only a few irreversible operations are inevitably involved in the computation process, such as erasure or dephasing.. 4.

(10) 1.2 Maxwell demon and Landauer’s principle In this subsection we review the issues related to Maxwell demon 11 , which is also related to the role of energy in a computation machine. Maxwell in 1867 proposed a thought experiment as follows: there is a mighty microscopic creature, i.e., a demon, can monitor the velocities and positions of each molecule in a many-particle system. 10 Consider a chamber full of ideal gas molecules is surrounded by a thermal bath. The chamber reaches thermal equilibrium with the heat bath in temperature T . Therefore, we can denote the average velocity of the molecules inside the chamber by ⟨v⟩T . Then the chamber is suddenly divided into two equal half parts by a partition inserted in the middle of the chamber. Besides, there is a hole on the partition with a movable door attached to it. The hole is so small that only one molecule is allowed to pass through at a time. The Maxwell demon stationed near the door and can observe all molecules’ motion in a precise way. Moreover, the demon can control the door to open or close without energy consumption. This setup is shown in Fig. 1.2 Now consider such a situation: if a molecule in the left chamber moves toward the door on the partition with the velocity less than ⟨v⟩T , then the demon opens the door and let it pass through to right. Otherwise, the demon keeps the door close. The Maxwell demon also observes the right chamber in a similar way: it allows the molecule moving faster than ⟨v⟩T to pass through the door to enter into the left chamber, and keep the slow molecule in the right section. By doing this many times, the Maxwell demon can separate the rapidly moving molecules from the slow ones, i.e., the fast molecules are collected in the left chamber and the slow ones in the right. Consequently, the Maxwell demon causes a temperature difference between left and right chamber without any energy consumption. If the Maxwell demon uses the resultant two chambers as a set of heat baths, it can design a thermodynamic cycle to produce the positive work. Then the demon can extract infinite amount of energy if it repeats the above procedure again and again. The above conclusion is obviously not physical and leads to a paradox. The resolution of this paradox turns out to be an important step in the development of quantum information sciences. This is the so-called Landauer’s principle 8 . The resolution of the paradox seems lying on the following fact: Maxwell demon has to perform the measurements on molecules’ motion status before 5.

(11) Figure 1.2: The whole chamber are surrounded with a heat bath of temperature T . The Maxwell demon classifies the molecules in the chamber through the following 4 steps:(i) measurement—to observe velocity of molecule. (ii) memorize—to take down the measurement outcome. (iii) operation—to open or close the small door on partition. (iv) erasure—to erase the memory to return to the initial state. he can decide to open the small door or not. It was widely believed that the measurements would cause energy consumption, thus the Maxwell demon needs energy to classify the molecules. However, this turns out to be more subtle than the above, and Rolf Landauer 8 and Charles Bennett 4,3 clarified the issue in such a way: the Maxwell demon must take record of its measurement outcome into its memory, it has to erase the memory eventually with a finite amount of the memory space. The erasure of information itself must accompany with entropy generation and cause the process to become irreversible. Note that the entropy generation is not caused by the measurement but by the erasure of the information. Landauer’s principle implies a deep connection between the information loss and the energy loss, and shed a new light on the relation between information characterized by the Shannon entropy 17 and the thermodynamic entropy. From this point of view, we can see the abstract manipulations of information as a concrete operations of energy and thermodynamic.. 1.3. The circuit model and its reversibility. Turing machine is considered as a computer with unbounded size, thus it is a ideal computer after all. An alternative computation model is more. 6.

(12) realistic: the circuit model. Circuit model is constructed by the wires and logic gates. Logic gate is the core of circuit model. The following discussion reviews the relation between the logic gates and Turing machine. Logic gates are responsible for all the computational assignments. Each logic gate contains specific control state and physical parts of Turing machine. The input bits and output bits can be seen as the tape of Turing machine as shown in Fig. 1.3. We can think a single logic gate as a tiny Turing machine which operates a specific function. The algorithm depends on the arrangement of the logic gates, different arrangements lead to different algorithm. One can see a specific example of algorithm in 5 realized by the quantum circuit given in 6 .. Figure 1.3: Each logic gate can be regraded as a small computer of specific function. The feature of this design is that it can deal with many inputs at a time. To imitate the calculation behavior of human, the first task is to realize the most fundamental operations of arithmetic on machines. Such operations are addition and multiplications which can be realized on the circuit model. The mighty power of fundamental operations is that we can produce all the other more complicated functions from them. Similarly, we can compute all the complicated functions on machine only by the appropriate arrangement of few fundamental logic gates. Therefore, we can just focus on the realization of these fundamental operations: the OR gate, the AND gate and the NOT 7.

(13) gate. These gate are also called the universal logic gates. The addition is realized by the OR gate. Suppose that there are two inputs A and B, the operation of OR gate is denoted as A + B. On the other hand, the multiplication is realized by the AND gate, denoted as A · B. The truth table of the AND and the OR gates are given in Fig. 1.4. The third elementary logic gate is the NOT gate. It flips the input value and is the only gate which operates with one input. It is denoted by A → A and B → B.. Figure 1.4: The truth table of the AND gate and the OR gate. Note that these gates have two inputs but only one output. This implies the erasure of the information during the computation, and thus they are irreversible gates. Now we discuss the universal logic gates in quantum computing. The universal logic gate in quantum computing is C-NOT gate. The C-NOT is a controlled operation. The typical controlled operation is as follows: if A is true, then do B. Such an operation is acting on the two-qubit state in the form of |control, target⟩ in quantum computing. The C-NOT functions in the following way:|a, b⟩ → |a, a ⊕ b⟩. The target qubit |b⟩ flips if the control qubit |a⟩ is in the state |1⟩, otherwise the target qubit remains intact. We can reach the same truth table as for the classical logic gate by the multiply-controlled operations. For examples, we can simulate the classical AND gate and OR gate only by using the combination of the C-NOT gates, see Fig. 1.5. The difference between the classical and quantum logic gates is their numbers of outputs. Most classical logic gates have two input but only one output. One bit of information is erased away obviously in this case. In 8.

(14) Figure 1.5: Simulate the elementary classical logic gates by the quantum gates—the quantum circuits at the left part function exactly the corresponding ones at the right part. However, unlike the classical ones, there is no erasure of information for the quantum version. the view of Landauer principle, the erasure must accompany with the energy consumption. On the contrary, the number of inputs and outputs are the same for the quantum circuit. Thus, there is no energy consumption in the computation process. The logic gates without erasure are also called the unitary gates, and the advantage of quantum computing is that all the quantum gates are unitary. That is, in principle there is no energy expense in performing the quantum computing by wiring quantum gates.. 1.4 Information and entropy After reviewing the connection between information and energy, we now briefly review the basic information theory. The first task is to quantify the amount of information in a string of messages 17 . The basic ideas is to re-interpret the amount of information as the measure of uncertainty of the state of system. For example, in a system A there are two possible events a1 and a2 with 9.

(15) the corresponding prior probabilities p1 and p2 . Assume a1 and a2 are statistically independent, thus p1 + p2 = 1. The information of A shows an uncertainty beyond its expectation value for the event to happen. This leads to the corresponding entropy for this even to happen 19 , the so-called Shannon entropy for the binary prior probabilities: Hbin (p) ≡ −p1 log p1 − p2 log p2 .. (1.1). By this definition, the system reaches maximal Shannon entropy when p1 = p2 = 12 . On the contrary, the system has no entropy (no information) if one of the two events does not occur at all, see Fig. 1.6.. Figure 1.6: The Shannon entropy as a function of one of the binary prior probabilities. The next step is to quantify the information relation between two systems. Denote these two sub-systems by X and Y , then the measure of uncertainty of the joint system (X, Y ) is called joint entropy: ∑ H(X, Y ) ≡ − p(x, y) log p(x, y). (1.2) x,y. It is easy to see that H(X, Y ) is larger than Hbin (X) or Hbin (Y ). Thus the entropy of two systems are larger than the one of the single sub-system as expected. If we only have information about one part of the joint system, we have to quantify the lack of information. Suppose that we already have the the 10.

(16) events of Y , we still lack the information of the other set X. The lack of information is defined as conditional entropy of by: H(X|Y ) ≡ H(X, Y ) − H(Y ) .. (1.3). Conditional entropy measures the information about X when Y is given. The second quantity is the measure of the common information shared between X and Y , i.e., the so-called mutual information. Follow the concept of set theory: (A ∩ B) = A + B − (A ∪ B) , (1.4) the mutual information is defined as: I(X : Y ) ≡ H(X) + H(Y ) − H(X, Y ) .. (1.5). Since H(X|Y ) = H(X, Y )−H(Y ), thus I(X : Y ) can also be re-expressed as I(X : Y ) = H(X) − H(X|Y ) = H(Y ) − H(Y |X) .. (1.6). By the definition of H(X) and H(X, Y ), we can express H(X|Y ) in terms of the prior and joint probabilities, viz, H(X|Y ) = H(X, Y ) − H(Y ) ∑ ∑ =− p(x, y) log p(x, y) + p(y) log p(y) x∈X,y∈Y. =−. ∑. y∈Y. p(x, y) log(p(x)p(y|x)) +. x∈X,y∈Y. ∑. p(y) log p(y) .. (1.7) (1.8) (1.9). y∈Y. We can further rewrite the above in terms of the conditional probability p(y|x) which satisfies the two conditions: p(x, y) = p(x)p(y|x) and ∑ x p(x|y) = 1. This then yields ∑ ∑ H(X|Y ) = − p(x, y) log p(x, y) + p(y) log p(y) (1.10) x∈X,y∈Y. =−. ∑. y∈Y. p(x, y) log p(x, y) +. x∈X,y∈Y. =−. ∑. =−. p(y). y∈Y. p(x, y) log p(x, y) +. x∈X,y∈Y. ∑. ∑. ∑. ∑. p(x|y) log p(y) (1.11). x. p(x, y) log p(y). (1.12). x∈X,y∈Y. p(x, y) log p(x|y) .. x∈X,y∈Y. 11. (1.13).

(17) Similarly, we can express the mutual information I(A : B) in the following form: ∑ p(x, y) I(X : Y ) = p(x, y) log . (1.14) p(x)p(y) x∈X,y∈Y If all the events of X and Y are independent, i.e., p(x, y) = p(x) × p(y), then there is clearly no mutual information, i.e., (I(X : Y ) = 0) of X and Y from the above expression.. 1.5. Density matrix and Holevo bound. In quantum computing, we replace the probability by density matrix to measure the information of quantum states. Density matrix can be thought as the generalization of wave function for the mixed states, i.e., an ensemble of quantum states. Thus, the time evolution of the quantum state of a system is completely encoded in the time evolution of its density matrix. Moreover, the details about the each sub-system is also encoded in the reduced density matrix, i.e., partial trace of the density matrix of the whole system. In 1995, von Neumann established a measure of quantum information: von Neumann entropy S(ρ) ≡ −tr(ρ log ρ) (1.15) where ρ denotes the density matrix of the system. If λi ’s are the eigenvalues of the density matrix ρ, then the associated von Neumann entropy can be obtained as ∑ S(ρ) = − λi log λi . (1.16) i. This is the quantum version of Shannon entropy given in (1.1). The basic properties of von Neumann entropy is the same as the ones of Shannon entropy. Suppose that we have two sub-systems A and B, the conditional entropy and mutual entropy satisfy S(A|B) ≡ S(A, B) − S(B) S(A : B) ≡ S(A) + S(B) − S(A, B). (1.17) (1.18). A key difference between classical and quantum information is the ability to distinguish the states by the accessible information. Let us illustrate this as follows. 12.

(18) Assume that Alice and Bob, the famous couple in information theory, deliver a string of message to each other. Alice prepares a set of quantum states ρ0 , ..., ρn with the corresponding prior probabilities p0 , ..., pn . Next, Alice randomly picks up a state ρX with the probability pX and delivers it to Bob. When Bob gets the state ρX , he performs a measurement on ρX with his own basis. Then Bob makes a guess according to his measurement outcome. The question is how precise can Bob guess Alice’s chosen state. The information gain of Bob is nothing but the mutual information I(A : B). Bob’s tough mission is to choose the appropriate basis that can maximize the information I(A : B). The maxima information gain by Bob’s operation, i.e., the maximum I(A : B) that Bob can reach, is called the accessible information. The Holevo bound is the limitation on the accessible information. As discussed above, Alice’s ∑ prepared quantum state can be described by the density matrix ρA = i pi ρi . Then Holevoe bound is expressed as follows: ∑ H(X : Y ) ≤ S(ρA ) − pi S(ρA (1.19) i ) . i. In the above description Alice can be seen as a random information resource, and Bob is the receiver. Surprisingly, later one we will see that one can obtain the same upper bound from the consideration of thermodynamics. This result reinforce the relation between information and energy initiated by the dissuasions of the Maxwell demon.. 13.

(19) Chapter 2 Review of Maxwell Demon and the related The 2nd law of the thermodynamics implies that energy will convert into heat eventually. This one-way process restricts the usage of our energy, because we can not convert the heat into work as we wish. Since Maxwell demon in principle can extract infinite amount of energy by repeating its ingenious operations, thus it is in contradiction with the 2nd law apparently. The conflict with has been considered for almost 150 years. This demon can be seen as a possible method to violate the thermodynamic 2nd law. In the debates of resolving the paradox, unexpectedly the Maxwell demon bridges the physic of thermodynamics and the information theory gradually. The following subsections introduce the evolution of this connection. It started from the paradox of Szilard engine and ended up with the solution of Bennett with the setup of a single molecule in a chamber. This explicitly demonstrate the power of Landauer’s principle.. 2.1. Szilard engine. The connection between thermodynamics and information firstly appeared in the discussions of the Sizlard engine proposed in 1929 10 . This engine attempts to built up a cycle that can do positive work with a single heat bath. At the initial stage of the Szilard engine cycle, a chamber contacts with a heat bath. The volume of the chamber is V and the temperature of heat bath is T . A molecule bounces in the chamber randomly. The kinetic energy of 14.

(20) Figure 2.1: The four stages of the Szilard engine:(a) the molecule occupies the whole chamber initially. (b) the Maxwell demon inserts a partition at the center of chamber and divides the vessel into two sections. Demon has to perform a measurement and record the outcome into its memory. (c) Demon ties up a weight on the same side of the molecule according to its measurement outcome. The partition will raise the weight up by the collisions of the molecule. (d) The expanding volume is in the quasistatic expansion. Note that if the demon ties the weight to the other side, the 2nd law will not be violated.. the molecule is transferred from the heat bath via the contact with the wall of chamber. Therefore the molecule can bounce in the chamber continually. In the second stage, the Maxwell demon inserts a thin, adiabatic and massless partition that divides the volume of chamber into two halves. Since the volume was divided, the molecule must bounce continually in one of two sections. Then the demon performs a measurement to observe the position of the molecule. According to the measurement outcome, the demon hangs a weight on the partition on the same side with the molecule, see Fig. 2.1 (c). In the last stage, the molecule collides with the partition repeatedly. During this collision, the occupied volume of molecule expands from a half to the. 15.

(21) whole chamber. The weight will be raised up for a distance gradually until the partition attaches to the bottom of the chamber. Then, the Maxwell demon draws the partition out and re-perform the next cycle again. The consequence implies that demon can extract amount of work by the cycle of Szilard engine. The cycle of the Szilard engine is summarized in Fig. 2.1. The cycle of Szilard engine violates the thermodynamic 2nd law apparently. However, it can be explained from two different points of view. First, the perfect energy conversion from heat to positive work reduces the entropy of the heat bath by the amount of k ln 2. Therefore, the same amount of the entropy must increase at somewhere for compensation, otherwise the 2nd law will be violated. On the other hand, any cycle of work extraction must be carried out between two heat baths of different temperatures. Szilard engine cycle extracts energy only by a single heat bath, that is in contradiction to Kelvin’s statement of the 2nd law. To find out the work extraction by the Szilard engine, the volume of heat bath must be large enough so that it keeps constant temperature T during the whole cycle of the Szilard engine. Therefore, the occupied volume is under isothermal condition so that the demon can extract work through the quasistatic expansion in the last stage. We denote the energy extraction from heat bath by Q and the positive work by W , and Q = W is assured under the isothermal expansion. The amount of extractable work from the heat bath is then ∫ V kT Wext = dV = kT ln 2 . (2.1) V V 2 To preserve the 2nd law, Szilard proposed that the measurement would increase the k ln 2 entropy at least. Although he did not propose any concrete method for performing the measurement, it seems a reasonable solution. Besides, other physicists also believed that the measurement is the key to resolve this paradox. In the cycle of Szilard engine, the work kT ln 2 is considered to be an fundamental value, that is one bit of work as the entity of the work substance is just a single molecule. This consideration is earlier than Shannon’s entropy by 20 years. Szilard engine also makes the Maxwell demon bridge the information and the physic, although the connection is weak here.. 16.

(22) 2.2 Landauer‘s principle To preserve the 2nd law, generations of physicists believed that the entropy is caused by performing the measurements. It was a temporary solution for decades. Until 1961, Landauer proposed that it is the erasure of information generating the entropy 8 . Landauer claimed that computation inevitably yields irreversible process, such as erasure. Landauer defined the erasure as a ’restore to one’ process, such process is equivalent to the irreversible one. He claimed that irreversible process is essential even the computation machine is a logically reversible device. Then, from the thermodynamics one can see that the irreversible process generates entropy. We can not avoid the irreversible process in computation. Take the binary reversible computer as an example, this machine use one or two variables for the inputs of logic gates. Suppose that there are N bits in our machine. After a machine cycle, the N binary elements undergo a function of the states. Since our computer is reversible, each binary element has appearing ONE and ZERO states with equal frequency in the computation cycle. This fact constrains the possible operations one can do, that is, we have to execute the operations whose truth table has equal numbers of ZERO’s and ONE’s, such as the XOR gate but not the AND and OR gates. Since one can not achieve the universal computations without AND and OR gates, thus it is not possible to perform the universal computations in a reversible way. Landauer then considered the connection between the logical irreversibility and the entropy generation. He performed the ’restore to one’ process by the thermalization. At the initial stage, an ensemble of states are prepared with the lower temperature and entropy than the ones of our computer. The binary computer has N bits for the input. Then, the computer is coupled to the ensemble for a period of time and reach the thermal equilibrium. By the entropy definition of statistical mechanic, S = k loge W , we set the entropy of computer before coupling is Si . The computer releases entropy to the ensemble of states during the thermalization process. When they reach the thermal equilibrium, the state of the computer are reduced to one of 2N possible states. Therefore, the ensemble increases the entropy by the amount of kN loge 2. In average, each bit increases k loge 2 entropy during the thermalization process. Based on the above reasoning Landauer delimited the lowest bound of the heat dissipation: During the erasure process, at least k ln 2 entropy for each 17.

(23) bit is released to the environment. This is called the Landauer’s principle. This value is also called the Landauer’s limit. Then Landauer derived the lower bound of energy dissipation in erasing a bit into the standard state. Except for erasure, both quantum and classical computers can operate without energy dissipation in principle. If the memory space are large enough for the whole computation process, then we do not worry about the energy consumption. Landauer’s principle points out that the keystone of the energy dissipation in the computation process. But it is still not clear if the measurement would cause the heat dissipation or not. Although Landauer’s principle connect the information theory to the thermodynamics, it is still not enough to resolve the paradox raised by the Maxwell demon.. 2.3 Resolution of the Maxwell demon paradox C.H. Benett related the Maxwell demon to Landauer’s principle and led to a substantial conclusion for the 2nd law of thermodynamics in 1982 4 . In 4 , Bennett showed that Maxwell demon can not violate the 2nd law in the cycle of Szilard engine. Since demon has to perform a measurement about the position of molecule, and it has to take down the measurement outcome to its memory. Before the next cycle of Szilard engine, demon has to erase its memory. Then the energy consumption is produced by the erasure process, not by the measurement. Thus demon can not extract work from the cycle of Szilard and thus the 2nd law is not violated. Measurement is the process that copies the information from the molecule to the memory apparatus. It only transforms the state of memory bit from initial standard state to the target state. This transformation is different from the overwriting, the initial states of the memory apparatus can be retraced. Therefore, recording information can be carried out reversibly without entropy generation. Furthermore, Bennett attributes the enhance of entropy to the physically irreversible process. He proposed that it is possible to erase the intermediate logical states by logically retracing at every step 3 : First, he considered a reversible Turing machine with three tapes: working, history and output tapes. The history tape and output tape are set to the blank initially, and the working tape to the state of the input. The. 18.

(24) machine cycle of the reversible Turing machine has three stages. The first stage is the forward computation for the Turing machine to transform the input values to the ones ready for the output on the working tape and to save the intermediate states to the history tape. The second stage is the copy operation: the Turing machine copies the computation outcome from the working tape to the output tape. Copying is a reversible process as the measurement. Thus it is an isentropic process during the second stage. After the copy operation, the output tape has the same information as on the working tape. The third stage is the clean-up operation. Since the forward computation is reversible, it is possible for the Turing machine to retrace the states of working tape and history tape from final states. Turing machine then leaves the output tape alone, and perform the backward computation on the working tape and the history tape. This operation can transform history tape to blank, and transform the output values to initial input values on the working tape. The backward computation is to clean up the information of working tape and history tape in an reversible way. After the three stages of operations, only the information of input value and output value are kept. Note that we can only clean up the intermediate logical state by reversible process, we can not set the physical state to its initial value by this method. Therefore, there must exist entropy generation in the erasure process of the physical states. If we do not want the release of the entropy to the environment, the only way is to perform the external work on our physical system. The external work can fix the states of our machine to the standard state. Thus Bennett pointed out that we need to do work on the system to erase the information. Back to the cycle of the Szilard engine, the Maxwell demon must perform extra work to fix its physical memory into the standard state. Such manipulation counteracts the energy extraction from the heat bath, therefore we can not get any extra energy from the cycle of the Szilard engine. This extra work prevents from the violation of he 2nd law. Although Bennett preserves the 2nd law by the same reason as Szilard, i.e., the demon causes the entropy generation, but with totally different perspectives. The exquisite works of Bennett and Landauer ingeniously link thermodynamics and information theory via the Maxwell demon. It also implies the boundless power of the 2nd law in the field of information theory.. 19.

(25) 2.4 Information erasure in a gas model We will review the erasure process in a gas model with energy consumption in this subsection 12 . By the definition of Landauer, erasure is a “restoreto-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 chamber 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 ∫ V Werasure = P dV (2.2) ∫. V 2. V. = V 2. kT dV V. = kT ln 2 .. (2.3) (2.4). 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 Landauer’s principle that k ln 2 is the minima entropy dissipation. 20.

(26) 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.. 21.

(27) Chapter 3 Information and thermodynamic There are verifications of Landauer‘s principle see ( 18 and 13 ), and even in the experiments 2 . These verifications indicate the deep link between information 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 bound 7,10 . 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 work 14,15 .. 3.1. Holevo Bound in a thermodynamic cycle. In the following we will review the works 7,10 . The Fig 3.1 shows the thermodynamic 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 22.

(28) 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. Therefore, 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, p1 V and p2 V 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 p1 N of particles in the left chamber and 23.

(29) Figure 3.2: Extract work in a gas chamber with semi-permeable movable membrane partitions: A set of semi-permeable membranes are initially inserted 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, respectively. Then we can raise the weights up by the expansion of the gas molecules. p2 N 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 24.

(30) 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 ∫. V. Wψ1 = p1 V. p1 N KT dV = −p1 N KT ln(p1 ) . V. (3.2). Similarly, the extractable work for the molecules in the state |ψ1 ⟩ is Wψ2 = −p2 N KT ln(p2 ). Thus the total extractable work in the complete expansion is Wtotal = Wψ1 + Wψ2 = −N KT (p1 N KT ln(p1 ) + p2 N KT ln(p2 )) = H(A) .. (3.3) (3.4) (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 distinguish 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 |Lef t⟩ 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 25.

(31) 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 ∫ V p(e1 )p(ψ2 |e1 )N KT = dV V p(ψ2 |e1 ) = −p(e1 )p(ψ2 |e1 )N KT ln p(ψ2 |e1 ) .. (3.6) (3.7) (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 ) , Wp(ψ1 |e1 ) = −p(e1 )p(ψ1 |e1 )N KT ln p(ψ1 |e1 ) , Wp(ψ1 |e2 ) = −p(e2 )p(ψ1 |e2 )N KT ln p(ψ1 |e2 ) .. (3.9) (3.10) (3.11). Therefore, the total extractable work for such an imperfect demon is Wmargin = Wp(ψ1 |e2 ) + Wp(ψ2 |e1 ) + Wp(ψ1 |e1 ) + Wp(ψ2 |e2 ) = H(A|B) .. (3.12) (3.13) (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 26.

(32) 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 from 10 . 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 in 10 : 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 27.

(33) vessel. Then the demon push the membrane M2 but with M1 fixed. 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). 3.2. Quantum thermodynamic and Maxwell demon. 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 works 14,15 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. 16 Consider 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 ˆ is the end. We set ρ as the density matrix of our potential well, and H Hamiltonian. Then the expectation value of Hamiltonian yields the internal. 28.

(34) energy ˆ , U = T r(ρH) ˆ + T r(Hdρ) ˆ ⇒ dU = T r(ρdH) .. (3.17) (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 ˆ = dR Q ˆ + dR W ˆ . dR H. (3.19). ˆ and the second term is the The first term is the off-diagonal part of dR H, diagonal one. We then express the second term in the expansion of the basis, i.e., ∑ ˆ = dR W dEm |m⟩ ⟨m| (3.20) m. ˆ where dEm = ⟨m|dR H|m⟩ with Em the eigen-energy of |m⟩. Then the variation of work done during the process is given by ∑ ˆ)= dW = T r(ρdR W Pm dEm .. (3.21). m. 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 ∑ ∑ = Em dPm + Pm dEm − dW , m. =. ∑. (3.22) (3.23). m. Em dPm .. (3.24). m. 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 29.

(35) 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 ∑ ˆ = (En − E0 ) |n⟩ ⟨n| = ∆ |1⟩ ⟨1| (3.25) H n=0. 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 ˆ A = ∆A |1⟩ ⟨1|. H A 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). |1⟩ ⟨1| +. (3.27). D. ρ =. PD1. PD0. |0⟩ ⟨0| .. Figure 3.5: The quantum heat engine of two-level systems.. 30.

(36) Therefore, the initial joint state is 1,1 1,0 ρSD (1) = PS,D |1, 1⟩ ⟨1, 1| + PS,D |1, 0⟩ ⟨1, 0| 0,1 0,0 +PS,D |0, 1⟩ ⟨0, 1| + PS,D |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 ˆ A→B : operations by the interaction Hamiltonian H ˆ A→B = g (1 + σ A ) ⊗ σ B . H (3.29) Z X 4 ρS (1). CEV. .. ρ (1) . D. ρS (3) ρD (3). 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 ˆ A→B , the demons perform a second operation. After coupling ρD to ρS by H measurement via the C-NOT operation, and take down the outcome to its memory ρD , this then results a new state as 1,1 1,0 ρSD (2) = PS,D |1, 0⟩ ⟨1, 0| + PS,D |1, 1⟩ ⟨1, 1| 0,1 0,0 +PS,D |0, 1⟩ ⟨0, 1| + PS,D |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 : ˆ D ] − TrD [TrS (ρSD )(1)H ˆD] , WD = TrD [TrS (ρSD )(2)H =. ∆D (PD1. −. 1,0 PS,D. −. 0,1 PS,D ). 31. (3.31) (3.32).

(37) 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⟩ → |e 1⟩ = cos θ |1⟩ + sin θ |0⟩ |0⟩ → |e 0⟩ = − sin θ |1⟩ + cos θ |0⟩ .. (3.33) (3.34). Thus the feedback action of the demon to the system ρS yields the final state as follows: 1,1 1,0 e ρSD (3) = PS,D |1, 0⟩ ⟨1, 0| + PS,D |1, 1⟩ ⟨e 1, 1| 0,1 e 0,0 +PS,D |0, 1⟩ ⟨e 0, 1| + PS,D |0, 0⟩ ⟨0, 0| .. (3.35). Similarly, the work done by the system ρS is ˆ S ] − TrS [TrD (ρSD )(3)H ˆS] , WS = TrS [TrD (ρSD )(2)H 1,1 1,0 0,1 = ∆S (PS1 − PS,D − PS,D | ⟨e 1|1⟩ |2 − PS,D | ⟨e 0|1⟩ |2 ) .. (3.36) (3.37). When the two operations are accomplished, then ρS and ρD are decoupled. 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: ˆ S ] − TrS [TrD (ρSD )(3)H ˆS] Qin = TrS [TrD (ρSD )(1)H 1,0 1,0 0,1 = ∆S (PS,D − PS,D | ⟨e 1|1⟩ |2 − PS,D | ⟨e 0|1⟩ |2 ). (3.38) (3.39). for the heat absorbed by ρS , and ˆ D ] − TrD [TrS (ρSD )(1)H ˆD] Qout = TrD [TrS (ρSD )(3)H =. 1,0 ∆D (PS,D. +. 0,1 PS,D. −. PD1 ). (3.40) (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 , =WS − WD ,. (3.42) (3.43). 1,1 1,0 0,1 − PS,D | ⟨e 1|1⟩ |2 − PS,D | ⟨e 0|1⟩ |2 ) =∆S (PS1 − PS,D 1,0 0,1 − PS,D ). − ∆D (PD1 − PS,D. 32. (3.44).

(38) 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.. 33.

(39) 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 34.

(40) |1⟩A. .. |1⟩B |0⟩B. |0⟩A .. Figure 4.1: Alice and Bob are the two-level systems with different eigenspectrum. 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 decide 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 in 7 , 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. 35.

(41) not change. Therefore, we can prepare two different potential wells. Each hosts a twolevel 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.. 4.2. Circuit with memory bit. Fig. 4.4 shows a quantum circuit for the demon with a physical memory bit. The design of this circuit follows the discussion in last subsection. Initially, we prepared the two-level systems ρA and ρB as follows: ρA (1) = P (ψ1 ) |ψ1 ⟩ ⟨ψ1 | + P (ψ2 ) |ψ2 ⟩ ⟨ψ2 | ,. ρB (1) = |ψ1 ⟩ ⟨ψ1 | .. (4.1). For this initial state, the information in the chamber A is S(A) = −kB T (P (ψ1 ) ln P (ψ1 ) + P (ψ2 ) ln P (ψ2 )) ,. 36. (4.2).

(42) OFF. |0⟩A |0⟩B. .. ON. |0⟩A. |1⟩A. |1⟩A. |0⟩B. |0⟩B. |1⟩B. Figure 4.3: Use the C-NOT gate to mimic the semi-permeable membrane. Left: the corresponding C-NOT gate for the OFF (opaque) case. Right: the corresponding C-NOT gate for the ON (transparent) case. Flipping the target bit will increase the entropy of the target ensemble.. and in the chamber B is S(B) = 0. This preparation is similar to the initial state for the gas model 7 . Therefore, the goal of this circuit is to swap the information of A and B. The design of this circuit follows the discussion in last subsection. In the first and the second parts of the circuit Fig. 4.4, the demon reads the states of the molecules and take the information down to its memory. For simplicity, we let the system ρA emits two bits at a time. The first bit of ρA is used to join the circuit operation, the second bit becomes the memory bit of demon instead. In the other words, ρA ’hands over’ the memory bit to the demon. The arrow from ρA to M is the “hand” over this process. In the third part of circuit, then demon changes the entropy of each potential well. Z. ρA i. ρB i. ρA f. M. ρM. .. ρB f. Figure 4.4: The circuit for a quantum demon with a physical memory bit. We set ground state as the ON state and labelled as ◦, set excited state as the ON state and labelled as •. To avoid confusion, note that in the following discussion of the first C37.

(43) NOT operation, denoted by the path A, we will require the same shape of the potential wells for A and B so that they have the same eigen-spectrum, Â = H ˆ B = ∆A |1⟩ ⟨1|. However, for the remaining operations, i.e., the i.e., H 2nd C-NOT and Control-Z denoted by the path B, we will deform the potential well of B so that ∆B ̸= ∆A . First, we perform the thermodynamic path A in our circuit. Initially, the density matrix of the total system (including ρA , ρB and ρM ) is given by ρAM B (1) = P (ψ1 ) |ψ1 ψ1 ψ1 ⟩ ⟨ψ1 ψ1 ψ1 | + P (ψ2 ) |ψ2 ψ2 ψ1 ⟩ ⟨ψ2 ψ2 ψ1 | .. (4.3). Note that ρA hands over one of her bits to ρM , thus the reduced density matrix of ρA and ρM are the same. We also can say that ρA has the same information amount as ρM . Next, we perform the C-NOT operation to ρB , and the state |ψ1 ⟩ of ρM is set to the ON state. After that, we arrive the new state as follows: ρAM B (2) = P (ψ1 ) |ψ1 ψ1 ψ2 ⟩ ⟨ψ1 ψ1 ψ2 | + P (ψ2 ) |ψ2 ψ2 ψ1 ⟩ ⟨ψ2 ψ2 ψ1 | .. (4.4). In the third step, the demon decreases the entropy of ρA also by the C-NOT operation, and arrive ρAM B (3) = P (ψ1 ) |ψ1 ψ1 ψ2 ⟩ ⟨ψ1 ψ1 ψ2 | + P (ψ2 ) |ψ1 ψ2 ψ1 ⟩ ⟨ψ1 ψ2 ψ1 | .. (4.5). Then we get ρ(3)A and ρ(3)B by taking the partial trace, viz, ρA := TrB,M (ρAM B ) and ρB := TrA,M (ρAM B ). (4.6). ⇒ ρ (3) = |ψ1 ⟩ ⟨ψ1 | ,. (4.7). A. ρB (3) = P (ψ1 ) |ψ2 ⟩ ⟨ψ2 | + P (ψ2 ) |ψ1 ⟩ ⟨ψ1 | .. (4.8). In the C-NOT operations of step 2 and step 3, the work done by ρA and ρ are B. ˆ A ) − Tr(ρA (3)H ˆ A ) = P (ψ2 )∆A , WA = Tr(ρA (1)H ˆ B ) − Tr(ρB (3)H ˆ B ) = −P (ψ1 )∆A , WB = Tr(ρB (1)H. (4.9) (4.10). respectively. Thus, the total work extraction from ρA and ρB is WpathA = WA + WB , = (P (ψ2 ) − P (ψ1 ))∆A . 38. (4.11) (4.12).

(44) Then we perform the thermodynamic path B in the quantum circuit. The difference between the paths A and B is the choice of the operation basis. The path B performs the operation with the basis {e1 , e2 }. The connection between the two basis set is by a θ-rotation: |ψ2 ⟩ = sin θ |e1 ⟩ + cos θ |e2 ⟩ , |ψ1 ⟩ = cos θ |e1 ⟩ − sin θ |e2 ⟩ .. (4.13) (4.14). Now we start the operation of circuit for the path B. Since the initial entropy S(ρB (1)) = 0 so that ρB (1) = |e1 ⟩ ⟨e1 |. Then the density matrix of the initial joint state is ρAM B (1) = P (ψ1 ) |ψ1 ψ1 e1 ⟩ ⟨ψ1 ψ1 e1 | + P (ψ2 ) |ψ2 ψ2 e1 ⟩ ⟨ψ2 ψ2 e1 | .. (4.15). Since the demon executes the C-NOT operation with the basis {e1 , e2 }, thus the outcome depends on the probabilities of projection ⟨ψi |ej ⟩. Therefore we expand the state of memory bit |ψi ⟩ by {e1 , e2 } basis to yield ρAM B (1) = P (ψ1 )[⟨ψ1 |e1 ⟩ |ψ1 e1 e1 ⟩ + ⟨ψ1 |e2 ⟩ |ψ1 e2 e1 ⟩] [⟨ψ1 e1 e1 | ⟨e1 |ψ1 ⟩ + ⟨ψ1 e2 e1 | ⟨e2 |ψ1 ⟩] +P (ψ2 )[⟨ψ2 |e1 ⟩ |ψ2 e1 e1 ⟩ + ⟨ψ2 |e2 ⟩ |ψ2 e2 e1 ⟩] [⟨ψ2 e1 e1 | ⟨e1 |ψ2 ⟩ + ⟨ψ2 e2 e1 | ⟨e2 |ψ2 ⟩] .. (4.16). We then perform the first C-NOT operation to ρB with |e1 ⟩ of ρM as ON. After that the density matrix of the joint state becomes ρAM B (2) = P (ψ1 )[⟨ψ1 |e1 ⟩ |ψ1 e1 e2 ⟩ + ⟨ψ1 |e2 ⟩ |ψ1 e2 e1 ⟩] [⟨ψ1 e1 e2 | ⟨e1 |ψ1 ⟩ + ⟨ψ1 e2 e1 | ⟨e2 |ψ1 ⟩] +P (ψ2 )[⟨ψ2 |e1 ⟩ |ψ2 e1 e2 ⟩ + ⟨ψ2 |e2 ⟩ |ψ2 e2 e1 ⟩] [⟨ψ2 e1 e2 | ⟨e1 |ψ2 ⟩ + ⟨ψ2 e2 e1 | ⟨e2 |ψ2 ⟩] .. (4.17). The next step is to decrease the entropy of ρA by acting with a sequential C-NOT and Control-Z operation. It turns out that the input state and the output state of this sequential operation differs at most only by a sign and a flip of the target bit. For example, consider the state |e2 ψ2 ⟩ under this. 39.

(45) sequential operation: |e2 ψ2 ⟩ = sin θ |e2 e1 ⟩ + cos θ |e2 e2 ⟩ C-NOT operation =⇒ sin θ |e2 e2 ⟩ + cos θ |e2 e1 ⟩ Control-Z operation =⇒ − sin θ |e2 e2 ⟩ + cos θ |e2 e1 ⟩ = |e2 ⟩ (cos θ |e1 ⟩ − sin θ |e2 ⟩) = |e2 ψ1 ⟩. (4.18) (4.19) (4.20) (4.21) (4.22). By the similar process, we can derive the other maps for the above sequential operation: |e2 ψ1 ⟩ −→ − |e2 ψ2 ⟩ , |e1 ψ2 ⟩ −→ |e1 ψ2 ⟩ , |e1 ψ1 ⟩ −→ |e1 ψ1 ⟩ .. (4.23) (4.24) (4.25). Obviously, this sequential operation is a C-NOT operation with control and target bits in different basis. After the above sequential C-NOT operation, by using the above maps we arrive the new state characterized by ρAM B (3) = P (ψ1 )[⟨ψ1 |e1 ⟩ |ψ1 e1 e2 ⟩ − ⟨ψ1 |e2 ⟩ |ψ2 e2 e1 ⟩] [⟨ψ1 e1 e2 | ⟨e1 |ψ1 ⟩ − ⟨ψ2 e2 e1 | ⟨e2 |ψ1 ⟩] +P (ψ2 )[⟨ψ2 |e1 ⟩ |ψ2 e1 e2 ⟩ + ⟨ψ2 |e2 ⟩ |ψ1 e2 e1 ⟩] [⟨ψ2 e1 e2 | ⟨e1 |ψ2 ⟩ + ⟨ψ1 e2 e1 | ⟨e2 |ψ2 ⟩] .. (4.26). Based on (4.26), we can get the reduced density matrix of each subsystem by taking the partial traces: ρA (3) = (P (ψ1 )P (e1 |ψ1 ) + P (ψ2 )P (e2 |ψ2 )) |ψ1 ⟩ ⟨ψ1 | + (P (ψ1 )P (e2 |ψ1 ) + P (ψ2 )P (e1 |ψ2 )) |ψ2 ⟩ ⟨ψ2 | ,. (4.27). ρ (3) = P (e1 ) |e1 ⟩ ⟨e1 | + P (e2 ) |e2 ⟩ ⟨e2 | ,. (4.28). ρB (3) = P (e2 ) |e1 ⟩ ⟨e1 | + P (e1 ) |e2 ⟩ ⟨e2 | .. (4.29). M. 40.

(46) Now we can evaluate the work done by each subsystem as follows: ˆ A ) − TrA (ρA (3)H ˆ A) WA = TrA (ρA (1)H = (P (ψ2 )P (e2 |ψ2 ) − P (ψ1 )P (e2 |ψ1 ))∆A ˆ M ) − TrM (ρM (3)H ˆM ) WM = TrM (ρM (1)H. (4.30) (4.31). = ∆M (P (e2 ) − P (ψ1 ) | ⟨e2 |ψ1 ⟩ | −P (ψ2 ) | ⟨e2 |ψ2 ⟩ | ) = 0 ˆ B ) − TrB (ρB (3)H ˆ B ) = −P (e1 )∆B WB = TrB (ρB (1)H 2. 2. (4.32) (4.33) (4.34). Note that the work done by ρM is zero as there is no operation performed on ρM . Thus, the total work done in the path B is WpathB = WA + WM + WB = (P (ψ2 )P (e2 |ψ2 ) − P (ψ1 )P (e2 |ψ1 ))∆A − P (e1 )∆B .. (4.35) (4.36). Moreover, it is easy to see that the entropy S(ρA (3)) is not zero after the process of the path B. This is the same as in the gas model. In subsection 4.4 we will discuss the counterpart of the path C in the context of circuit model, and obtain the constraint from the 2nd law of thermodynamics, which is then used to test the Holevo bound.. 4.3 Circuit without the memory bit In contrast to the quantum circuit discussed in the previous subsection, we now will consider another circuit mimic the demon without using the memory bit. This circuit also imitates the thermodynamic paths of the gas model in 7 . This circuit can be seen as a mighty demon that can reach the final state of 7 without using memory bit and the irreversible operation. We will come back to this issue in the next subsection. The quantum circuit for our purpose is shown in Fig. 4.5. The preparation of the initial state is the same as in the previous subsection. We first imitate the thermodynamic the path A. The feature of this path is that the demon has the ability to distinguish the states |ψ1 ⟩ and |ψ2 ⟩ unambiguously. Thus we also set the basis of ρA and ρB to be the same. Initially, these two subsystems are decoupled, and their density matrices and. 41.

(47) the joint one are as follows: ρA (1) = P (ψ1 ) |ψ1 ⟩ ⟨ψ1 | + P (ψ2 ) |ψ2 ⟩ ⟨ψ2 |. (4.37). ρ (1) = |ψ1 ⟩ ⟨ψ1 |. (4.38). B. ⇒ ρAB (1) = P (ψ1 ) |ψ1 ψ1 ⟩ ⟨ψ1 ψ1 | + P (ψ2 ) |ψ2 ψ1 ⟩ ⟨ψ2 ψ1 |. (4.39). The next step is to manipulate our working substance, i.e, the two-level systems by the circuit in Fig. 4.5. This circuit contains two steps. ρA (1) ρB (1). ρA (3) .. ρB (3). Figure 4.5: The quantum circuit without using the physical memory bit. It can achieve the thermodynamic cycle of the gas model in 7 . The key feature of this model is of no need to include the irreversible operation. After the first C-NOT operation, the joint density matrix ρAB (2) becomes ρAB (2) = P (ψ1 ) |ψ1 ψ2 ⟩ ⟨ψ1 ψ2 | + P (ψ2 ) |ψ2 ψ1 ⟩ ⟨ψ2 ψ1 | .. (4.40). The next C-NOT operation tries to decrease the entropy S(ρA ) and it yields the final state characterized by ρAB (3) = P (ψ1 ) |ψ1 ψ2 ⟩ ⟨ψ1 ψ2 | + P (ψ2 ) |ψ1 ψ1 ⟩ ⟨ψ1 ψ1 | .. (4.41). After taking the partial trace, we obtain the reduced density matrrices of subsystems: ρA (3) = |ψ1 ⟩ ⟨ψ1 | ,. (4.42). ρ (3) = P (ψ1 ) |ψ2 ⟩ ⟨ψ2 | + P (ψ2 ) |ψ1 ⟩ ⟨ψ1 | .. (4.43). B. Obviously, the informations of A and B swap and thus S(ρA (3)) = 0. It is in analogy to the situation of the gas model. Now use the circuit to consider a little more complicated situation: ρA uses different basis, i.e., {e1 , e2 } from the one, i.e., {ψ1 , ψ2 } used by ρB and 42.

(48) the demon (the C-NOT gates). These two bases are related by the θ rotation as usual. Then, the initial joint state is ρAB (1) = P (ψ1 ) |ψ1 e1 ⟩ ⟨ψ1 e1 | + P (ψ2 ) |ψ2 e1 ⟩ ⟨ψ2 e1 | .. (4.44). After the first C-NOT operation (under the condition that |e1 ⟩ of |ρA ⟩ is set to the ON state.), we arrive ρAB (2) = P (ψ1 )(⟨ψ1 |e1 ⟩ |e1 e2 ⟩ + ⟨ψ1 |e2 ⟩ |e2 e1 ⟩) (⟨e1 e2 | ⟨e1 |ψ1 ⟩ + ⟨e2 e1 | ⟨e2 |ψ1 ⟩) +P (ψ2 )(⟨ψ2 |e1 ⟩ |e1 e2 ⟩ + ⟨ψ2 |e2 ⟩ |e2 e1 ⟩) (⟨e1 e2 | ⟨e1 |ψ2 ⟩ + ⟨e2 e1 | ⟨e2 |ψ2 ⟩) .. (4.45). Similarly, after the C-NOT operation (under the condition that |e1 ⟩ of ρB is set to the ON state), we obtain ρAB (3) = P (ψ1 )(⟨ψ1 |e1 ⟩ |e1 e2 ⟩ + ⟨ψ1 |e2 ⟩ |e1 e1 ⟩) (⟨e1 e2 | ⟨e1 |ψ1 ⟩ + ⟨e1 e1 | ⟨e2 |ψ1 ⟩) +P (ψ2 )(⟨ψ2 |e1 ⟩ |e1 e2 ⟩ + ⟨ψ2 |e2 ⟩ |e1 e1 ⟩) (⟨e1 e2 | ⟨e1 |ψ2 ⟩ + ⟨e1 e1 | ⟨e2 |ψ2 ⟩) .. (4.46). Thus, the reduced density matrices of the final state are ρA (3) = |e1 ⟩ ⟨e1 | ,. (4.47). ρB (3) = P (e1 ) |e2 ⟩ ⟨e2 | + P (e2 ) |e1 ⟩ ⟨e1 | .. (4.48) (4.49). Again, we see S(ρA (3)) = 0 but the information of A is not transferred to B in the exact way. Compared to the final state of the circuit model in last subsection, i.e., with 1 physical memory bit, in the current model S(ρA (3)) = 0 holds for any basis. This implies that one does not need to invoke the irreversible process to decrease S(ρA (3)) to zero. This is similar to the mighty demon of the Szilard engine. Note that, this is not the case for the model considered in last subsection.. 43.