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Heat-to-work conversion by exploiting full or partial correlations of quantum particles levitin/arxiv/work March 8, 2022 Lev B Levitin and Tommaso Toffoli arXiv:1101.1325v2 [quant-ph] 7 Apr 2011 (levitin@bu.edu, tt@bu.edu) Electrical and Computer Engineering, Boston University, MA 02215 It is shown how information contained in the pairwise correlations (in general, partial) between atoms of a gas can be used to completely convert heat taken from a thermostat into mechanical work in a process of relaxation of the system to its thermal equilibrium state. Both classical correlations and quantum correlations (entanglement) are considered. The amount of heat converted into work is proportional to the entropy defect of the initial state of the system. For fully correlated particles, in the case of entanglement the amount of work obtained per particle is twice as large as in the case of classical correlations. However, in the case of entanglement, the amount of work does not depend on the degree of correlation, in contrast to the case of classical correlations. The results explicitly demonstrate the equivalence relation between information and work for the case of two-particle correlations. allowing it to relax to equilibrium are based on the information contained in one-particle distributions—as if all the particles comprising the system were independent. To the best of our knowledge, information stored in multi-particle distributions has never been used to extract mechanical work. Here we address the problem of using information contained in two-particle correlations for converting heat into work. Our analysis is based on two important assumptions: The connection between work and information was first pointed out by Szilard[1] in his analysis of Maxwell’s demon, and later by Landauer[2] and Bennett[3]. Using the setup of a “thought experiment” related to Gibbs’s paradox, it was shown[4, 5, 6] that using a system of two gases of N molecules each, with the molecules being in different (in general, nonorthogonal) quantum states ρ(1) and ρ(2) , it is possible to convert completely into mechanical work an amount of heat, taken from a thermostat, which is proportional to the entropy defect[7] of the system 1. Since a unitary transformation of the system’s state does not change its entropy, it can be performed without any energy dissipation; and W = 2N kT I0 , where W is work, T the temperature of the thermostat, and I0 the entropy defect (also called “quantum information” or “Levitin–Holevo bound”[8, 9]) of the system per molecule, which, in this particular case, is 2. Since orthogonal quantum states are perfectly distinguishable, there exist partitions which are permeable for one of such states but not for the other (a wellknown example of such “partitions” is a light polarizer). 2 I0 = − Tr ρ ln ρ + 1 1X Tr ρ(i) ln ρ(i) , ρ = (ρ(1) + ρ(2) ). 2 i=1 2 In order to avoid getting involved in definitional arguments—such as appear sometimes in the physics literature—concerning the distinction between work and heat, we shall consider an isothermal quasistatic process in which heat is taken from a thermostat and eventually transferred to another system in a form that is inequivocally mechanical work—namely, the lifting of a weight. We shall treat in separate sections the case when the two particles are classically correlated and that dealing with quantum correlations, and conclude with a discussion of the overall results. Entropy defect has the meaning of information (about the microstate of the system) associated with the selection of a subensemble (in this example, ρ(1) or ρ(2) ) from the total ensemble ρ. It was conjectured[4, 5, 6] that there exists a general equivalence relation between information and work; namely, that by having any information J about the state of a physical system, it is possible, by allowing the system to relax to its maximum-entropy state, to convert into mechanical work an amount of heat W = kT J without any entropy increase in the environment. (Of course, all information J is lost in the process, since the system reaches the thermal equilibrium state that has the same energy and volume as the initial state.) Since then, a number of papers (in particular, [10, 11, 12, 13, 14, 15, 16, 17]) has been devoted to various aspects of the connection between information and work. However, the general equivalence relation remains up to now unproven. (Part of the problem, in our opinion, is the still existing fuzziness of distinction between “mechanical work” and “heat.”) Moreover, all real-life examples of getting work from a non-equilibrium system (e.g., chemical reactions, separated electrical charges, compressed gas) by 1 The case of classical correlations Consider a gas of molecules that consist of two different atoms, A and B. To each atom is associated a 2-dimensional Hilbert space of states with basis vectors |0i and |1i (a qubit[18]). The thermal equilibrium (maximum-entropy) state of a gas molecule is described by a density matrix ρ= 1 ρ1 = 12 (|00i h00| + |11i h11|), 1 (ρ1 + ρ2 ), with 2 ρ2 = 12 (|01i h01| + |10i h10|). States ρ1 and ρ2 are orthogonal, and correspond to (classically) maximally correlated atoms A and B. Each of these states has an entropy defect (in nats) Ic = ln 2 per pair, or Jc = N ln 2 for a gas of N molecules. Note that the oneparticle probabilities of states |0i and |1i of both atoms are the same in states ρ1 and ρ2 as at equilibrium. Therefore, it is impossible to distinguish between this state and the equilibrium state on the basis of one-particle measurements. Without loss of generality, assume we know that the gas is in state ρ1 (let us call it gas 1). Let the gas occupy a vessel of volume 2V divided by a partition into two equal parts of volume V , each occupied by N2 molecules, and being in thermal contact with a thermostat at temperature T . To the gas on the right side of the partition apply a unitary transformation U1 = IA ⊗ UB , where IA is the identity operator in the Hilbert space of A and UB the operator that interchanges states |0i and |1i of B—that is, the Pauli matrix σx . Transformation U1 converts gas 1 into a gas with density matrix ρ2 (gas 2). Now, replace the partition by two movable semipermeable partitions such that the partition that faces gas 1 is permeable to gas 1 but not to gas 2, and vice versa for the other partition. Since the total pressure of the mixed gases is larger than the pressure of one of the gases, the two partitions will be pushed apart—and we can use the setup shown in Fig. 1 to lift weights m1 and m2 . initial information about the location of each pair of atoms, contained in the correlation of states of the two atoms in a molecule, is now erased, and this increase in entropy exactly compensates for the decrease of entropy of the thermostat. Consider now the case of partial classical correlations, when each molecule is in a separable state ρ1p = p2 (|00i h00| + |11i h11|) + 1−p 2 (|01i h01| + |10i h10|), where 12 < p ≤ 1. (If p = 12 , the system is in the maximumentropy—i.e., equilibrium—state, and no correlation exists.) The complementary state is ρ2p = 1−p 2 (|00i h00| + |11i h11|) + 21 (|01i h01| + |10i h10|), so that the equilibrium state is ρ = 21 (ρ1p + ρ2p ). Note that the marginal probabilities for each atom to be in state |0i or |1i are equal; thus, the deviation from equilibrium is entirely due to correlations. The entropy defect (per molecule) of the system is now 2 Icp = − Tr ρ ln ρ + 1X Tr ρip ln ρip = ln 2 − h(p), 2 i=1 where h(p) = −p ln p− (1 − p) ln(1 − p) is the binary entropy function. Suppose the gas is in state ρ1p (gas 1p). Using the same gas 1 gas 2 experimental setup as in the case of maximally correlated .❡ .❡ atoms, we apply the unitary transformation U1 to the gas ρ1 ρ ρ2 on the right side of the partition. This transforms gas 1p ✇m1 m2 ✇ into gas 2p with density matrix ρ2p . Now let us use exactly the same movable semipermeable Figure 1: Scheme for obtaining work by mixing gases. The partitions as before. Note that the partition that faces gas excess pressure of the gases between the semipermeable par1p performs in fact a measurement over the state ρ1p , which titions pushes them apart and lifts weights m1 and m2 . results either in state ρ1 (with probability p) or ρ2 (with Consider the moment when the right partition has ad- probability 1 − p). Similarly, the partition that faces gas vanced to the right by a volume V1 and the left partition 2p produces molecules in state ρ2 (with probability p), or to the left by a volume V2 . Gas 1 will then fill the volume in state ρ1 (with probability 1 − p). As a result, the gas V + V1 and gas 2 the volume V + V2 . Let us assume that between the partitions has the maximum-entropy density the gases are ideal and rarefied. Then the pressures of the matrix ρ. Initially, the total pressure of the mixed gases between gases to the left, to the right, and in between the partitions the partitions is larger than the pressures of the gases to the are, respectively, left or to the right of both partitions. Hence the partitions (N/2)kT (N/2)kT are pushed apart as in the previous case, and the gases can P1 = , P2 = , V + V1 V + V2 produce work by lifting weights (Fig. 2). Note, however, N kT  1 1  that the compositions of the gases beyond the partitions Pm = . + 2 V + V1 V + V2 are changing in the process, owing to the filtering action of ′ The work produced by the gas by lifting masses of appro- the partitions (we denote changing mixtures by gases (1p) ′ and (2p) ). priate weights in this quasistatic isothermal process is Z V Z V Wc = (Pm − P2 )dV1 + (Pm − P1 )dV2 = N kT ln 2. ρ gas (1p)′ gas (2p)′ .❡ 0 0 .❡ V1 +V2 Note that the total energy of the system has not changed ✇m1 m2 ✇ in the process. Thus, the amount of heat converted into mechanical work is proportional to the entropy defect of Figure 2: Scheme for converting heat to work by mixing the system, partially correlated gases. Wc = N kT Ic = kT Jc . (1) As before, let the right partition be moved to the right over a volume V1 , and the left partition to the left over a By the end of the process the entire vessel is occupied by a mixture of gas 1 and gas 2 with density matrix ρ. The 2 volume V2 . The pressures of the gases to the left, the right, and UΨ analogously permutes Ψ+ and Ψ− . Transformation and in between the partitions then become, respectively, U2 converts ρ(1) into a mixed state N kT p 1−p
+ , 2 V + V1 V − V2 p 1−p
N kT + , = 2 V + V2 V − V1 pN kT 1 1
= + . 2 V + V1 V + V2 ρ(2) = P1p = P2p Pmp P1p = Pmp = P2p . Hence, the total work produced by the gas from the heat taken from the thermostat is 0 (Pmp − P2p )dV1 + Z 0 Now assume that the gas molecules are initially in a pure partially entangled state ψ = a |00i + b |01i + c |10i + d |11i , (2p−1)V (Pmp − P1p )dV2 (3) Since the entropy defect of the entangled states is two times as large as that of classically correlated states, the amount of work Wq is also larger than Wc by a factor of 2. V1max = V2max = (2p − 1)V . (2p−1)V Ψ− ). Wq = W1 + W2 = 2N kT ln 2 = N kT Iq = kT Jq . Solving these equations, one obtains Z Φ− + Ψ − Note that states ρ(1) and ρ(2) are orthogonal, and thus perfectly distinguishable. Employing transformation U2 and using the same setup as before, we can now repeat the mixing procedure. This will yield additional work W2 = N k ln 2, bringing the system to the equilibrium state ρ = 21 (ρ(1) + ρ(2) ). Hence, the total amount of heat converted into work is The partitions will stop moving when the pressures on the two sides of the partitions become equal, i.e., Wcp = 1 ( Φ− 2 (2) where = N kT (ln 2 − h(p)) = N kT Icp. |a|2 = |d|2 = p2 , |b|2 = |c|2 = Note that by the end of the process all three parts of the entire vessel are occupied by gas in the equilibrium state ρ. Thus, the initial information (equal to the entropy defect) is expended in the conversion of heat into work. 1−p 2 , c = b∗ , d = −a∗ . (4) Conditions (4) ensure that the one-particle density matrices are the same as at equilibrium. (Entanglement vanishes when p = 21 .) State ψ displays the same partial two-particle correlation as the mixed state ρ1p . The important difference is that in this case the correlation is entirely due to the 2 The case of quantum correlations entanglement. Thus, the two-particle system is in a pure state and its entropy defect is Iq = 2 ln 2, independently of (entanglement) the parameter p. This leads to the paradoxical fact that the amount of work that can be obtained by the use of a Consider now the case where atoms forming a molecule are gas in a partially entangled state is the same as in the case in a maximally entangled state. There are four such states of maximally entangled state, namely, + − + of two qubits, namely, the Bell states[8] Φ , Φ , Ψ , and − Ψ , defined as follows Wq = 2N kT ln 2 = N kT Iq . (5) 1 1 Φ± = √ (|00i ± |11i), Ψ± = √ (|01i ± |10i). Indeed, state ψ can be transformed by an appropriate uni2 2 tary operator into state Φ+ , and the procedure described These four states are orthogonal and form a basis in the above can be applied to convert to work an amount of heat 4-dimensional tensor-product Hilbert space of those two equal to Wq . atoms. The entropy defect of the system in one of those Expressions (1), (2), (3), and (4) present results related states is Iq = 2 ln 2 per pair, or Jq = 2N ln 2 for the whole to the two extreme cases of entirely “classical” and entirely gas (as before, we assume the gas to be ideal and the states “quantum” correlations. One could consider a more general of the molecules, independent). case of a mixed non-separable state, where the correlation Suppose that the gas is initially in state Φ+ . Then, by use is partially “quantum” (entanglement) and partially “clasof the same transformation U1 as in the previous section, we sical.” The amount of work in this case should fall between can convert half of the gas molecules into state Ψ+ . Using the values given by (2) and (3). the same setup as in Fig. 1, we obtain work W1 = N kT ln 2; the final state of the gas will be a mixture 3 ρ(1) = 1 ( Φ+ 2 Φ+ + Ψ + Ψ+ ). Conclusions The above results demonstrate the equivalence relation between information and work for the case when the inforNow we can use a unitary transformation U2 = UΦ ⊗UΨ , mation is contained only in two-particle probability distriwhere UΦ is the unitary transformation that permutes Φ+ butions, and cannot be extracted from a mere one-particle and Φ− in the subspace spanned by these orthogonal states, distribution. 3 The situation with the equivalence between information and work in general remains unclear and, in a way, paradoxical. On one hand, many researchers are strongly convinced of it and express it in a very firm way, e.g.,“Information has an energetic value: It can be converted into work” (Zureck[19]; but cf. [17]). (In fact, it is not information that can be converted into work, but heat that can be converted into work by use of information.) On the other hand, in our opinion, one still lacks a convincing general proof that any kind of information can be equally successfully used to convert heat into any kind of work, so that the general statement remains a sort of “folk theorem.” We believe that the root of this problem is the fact that, in spite of many unquestionable examples of “work” known to physicists, there is no general rigorous definition that would distinguish between these two forms of energy transfer—heat and work. Informally speaking, work is an “informed” transfer of energy, i.e., a transfer such that we know exactly the change of the state of each degree of freedom, resulting from this transfer; while heat is energy transfered in such a way that we have no such knowledge. From that viewpoint, the equivalence between information and work becomes indeed a tautology. However, a rigorous formalization of these ideas has not yet been presented. 4 [8] Bennett, Charles H, and Peter Shor, “Quantum information theory,” IEEE Trans. Info. Theory 44 (1998), 2724– 2742. [9] Peres, Asher, Quantum Theory, Concepts and Methods, Kluwer 1993. [10] Alicki, Robert, Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki, “Thermodynamics of quantum informational systems—Hamiltonian description,” Open Syst. Info. Dyn. 11 (2004), 205–217 [11] Dahlsten, Oscar, Renato Renner, Elizabeth Rieper, and Vladko Vedral, “The work value of information,” arXiv:quant-ph/0908.0424 (2009). [12] Feldmann, Tova, and Ronnie Kosloff, “Quantum fourstroke engine: Thermodynamic observables in a model with intrinsic friction,” Phys. Rev. E 68 (2003), 016101. [13] Horodecki, Michal, Jonathan Oppenheim, and Ryszard Horodecki, “Are the laws of entanglement theory thermodynamical?” Phys. Rev. Lett. 89 (2002), 240403. [14] Lloyd, Seth, “Quantum-mechanical Maxwell’s demon,” Phys. Rev. A 56 (1997), 3374–3382. [15] Oppenheim, Jonathan, Michal Horodecki, Pawel Horodecki, and Ryszard Horodecki, “A thermodynamic approach to quantifying quantum correlations,” PRL 89 (2002), 180402. [16] Scully, Marlan, “Extracting work from a single thermal bath via quantum negentropy,” Phys. Rev. Lett. 87 (2001), 220601. Acknowledgments We wish to thank Charles Bennett for bringing to our attention a number of related papers, and to Oscar Dahlsten and Michal Horodecki for updated references. [17] Zureck, Wojciech, “Quantum discord and Maxwell’s demons,” Phys. Rev. A 67 (2003), 012320. [18] Schumacher, Benjamin, “Quantum coding,” Phys. Rev. A 51 (1995), 2738–2747, [19] Zureck, Wojciech, “Quantum discord and Maxwell’s demons,” arXiv:quant-ph/0202123 (2002). 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