Quantum memory as a perpetuum mobile? Stability v.s. reversibility of information processing
QQuantum memory as a perpetuum mobile?Stability v.s. reversibility of informationprocessing.
R. AlickiInstitute of Theoretical Physics and AstrophysicsUniversity of Gda´nsk, Polandand Weston Visiting ProfessorWeizmann Institute of Science, Rehovot, Israel
Abstract
It is argued using a Gedankenexperiment that a scalable quan-tum memory could be used as a perpetuum mobile of the secondkind and hence cannot be realized in Nature. The reasoning is basedon the assumption that the Landauer’s principle for measurements isa consequence of the second law of thermodynamics and not an in-dependent postulate. This implies a modification of the Landauer’sprinciple when applied for discrimination of equilibrium (metastable)states. While identification of the metastable state can be done at theinfinitesimally low cost, a change of such a state involves dissipationof energy proportional to its stability factor.
This note concerns the fundamental question in quantum information:
Isfault-tolerant Quantum Information Processing (QIP) feasible?
The extensively studied and well-developed theory of fault-tolerant quantumcomputation [1, 2, 3, 4] provides a positive answer, however its phenomeno-logical assumptions are doubtful and often criticized [5, 6, 7, 8]. On theother hand first principle models are very difficult to analyze and do notgive a complete and general answer yet. Therefore, one can ask an easierquestion:
Is a scalable quantum memory feasible?
In the recent years several models of quantum memory based on self-correctingspin systems have been proposed [9, 10, 11] and few of them rigorously an-alyzed [12, 13, 14, 15, 16]. Despite certain stability properties, proved or1 a r X i v : . [ qu a n t - ph ] A ug xpected for some of those models, there is no proof that any of the propos-als satisfies all the needed conditions for scalable quantum memory.A natural question arises: Can phenomenological thermodynamics providerestrictions or even no-go theorems for Quantum Memory, or generally forQIP ?
An attempt of [17] based on the KMS theory was not convincing for manyexperts because it was based on the mathematical structures related to ther-modynamical limit in terms of quasi-local algebras. Here another, moreheuristic, approach is presented involving the modified Landauer’s Principlefor quantum measurements and a
Gedankenexperiment on a system imple-menting quantum memory.The main problem with thermodynamical arguments is that the laws ofthermodynamics are usually formulated in a natural language and have acommon sense character. To apply them to some subtle problems one needsmore rigorous formulations, than those found in the most textbooks. This isparticularly important in the quantum theory, which often seems to be farfrom a ”common sense”.Another problem is the question of applicability of thermodynamics to QIP.There exist two points of view:1) Physical systems used for QIP are different from those considered inthermodynamics and therefore thermodynamical restrictions do not apply.2) Thermodynamics applies.The author of the present note shares the second opinion following thefamous statements :
But if your theory is found to be against the second law of thermodynamicsI can give you no hope; there is nothing for it but to collapse in deepesthumiliation. - Sir Arthur Stanley Eddington. (Thermodynamics)...is the only physical theory of universal content which Iam convinced that within the framework of applicability of its basic concepts,it will never be overthrown. - Albert Einstein.
The laws of thermodynamics possess a phenomenological and common sensecharacter. In particular the limits of their applicability lie beyond the scope2f phenomenological thermodynamics and need serious considerations basedon first principle microscopic models. For example one can take a possibleformulation of the
Zero-th Law : Any system coupled to a thermal bath relaxes to the thermodynamical equi-librium state at the bath’s temperature ,and the
Second Law : It is impossible to obtain a process such that the unique effect is the subtrac-tion of a positive heat from a reservoir and the production of a positive work.
In both cases one can ask the questions: How long does thermal relaxationor subtraction of heat can take? Are there any relations between the size ofthe systems and the time scale of those processes? Similarly, what should bea scale of produced work and how should it depend on the size of the system?As a simple example consider a ferromagnet consisting of N microscopicconstituents (”spins”) below the critical temperature. In principle, any polar-ized macroscopic state ultimately relax to the unique ”equilibrium state” forwhich the direction of magnetization (cid:126)M is completely unpredictable. Thestates with fixed direction of (cid:126)M are metastable with relaxation times in-creasing exponentially with N . For large N such metastable states pos-sess all expected features of equilibrium states and moreover the rigorousapproach involving thermodynamical limit treats them as true equilibriumstates corresponding to pure thermodynamical phases. To find an additionalrelation between admissible time scales of thermodynamic processes one canbe guided by the analogous problems in computer science. In the theory ofcomplexity the problem can be solved efficiently if the time needed for thesolution grows polynomially with the input size. This suggests the followingreformulation of the Zero-th Law : Any N -particle system coupled to a thermal bath relaxes to the (possiblynonunique) thermodynamical equilibrium state at the bath’s temperature withrelaxation time growing at most polynomially in N ,and the Second Law : It is impossible to obtain an effective process such that the unique effect is thesubtraction of a positive heat from a reservoir and the production of a positivework of the order of at least kT . The effective process means a process whichtakes at most polynomial time in the number of particles N . The study of applicability of thermodynamics to classical and quantum sys-tems reached a mature status quite recently with the development of fluctu-ation theorems [18]. The rough formulation of the fluctuation theorem is the3ollowing:
For a system consisting of N particles the probability of observing duringtime t an entropy production opposite to that dictated by the second law ofthermodynamics decreases exponentially with N t . Such a formulation has an immediate consequence for classical and quantumcomputing. Namely, one cannot hope that the efficiency of any computingscheme can follow from the hypothetical deviations from phenomenologicalthermodynamics.
Landauer’s Principle, first argued in 1961 by Rolf Landauer of IBM [19],holds that any logically irreversible manipulation of information, such as theerasure of a bit or the merging of two computation paths, must be accompaniedby a corresponding entropy increase in non-information bearing degrees offreedom of the information processing apparatus or its environment [20].Specifically, each bit of lost information will lead to the release of anamount kT ln 2 of heat, where k is the Boltzmann constant and T is theabsolute temperature of the circuit. On the other hand, if no information iserased, computation may in principle be achieved which is thermodynami-cally reversible, and require no release of heat. This has led to a considerableinterest in the study of reversible computing.The Landauer’s Principle has a direct relation to thermodynamics ofquantum measurement processes. Here, a quantum measurement is a projec-tive von Neumann measurement of an observable A = (cid:80) j a j P j such that foran initial state ρ of the system the final state after measurement is given by: ρ k = P k ρP k Tr( P k ρP k ) (1)when the outcome a k is recorded. In order to use the measurement’s out-come for the system’s control one has to assume that after a measurementthe system remains in the corresponding state for the time at least of theorder O (1).Consider a 2-level quantum system with a trivial initial Hamiltonian H ( t ) =0 coupled to a heat bath at the temperature T . One can design a cyclic pro-cedure of extracting work from a bath consisting of the following steps [22]:4) Measurement on the system at the initial state ρ ( t ) = I/ σ z which yields the outcome s = ± ρ ( t ) = | s (cid:105)(cid:104) s | .ii) Fast (in comparison to the thermal relaxation time) switching on an exter-nal field producing the Hamiltonian H s ( t ) = ( sE/ sI − σ z ) which increasesthe energy of the state | − s (cid:105) by E >> kT and does not change the energyof the state | s (cid:105) .iii) Slow (again in comparison to the thermal relaxation time) switching offthe external fields such that H ( t ) = 0. Reseting of the measuring device.One can compute the balance of work W ( t ), heat Q ( t ) and internal energy E ( t ) during the full cycle t → t → t using the basic definitions discussedin [23, 22] and recalled in the Appendix I E = Tr( ρH ) , dW = Tr( ρ dH ) , dQ = Tr( dρ H ) . (2)In the step i) the state of the system evolves from the complete mixture ρ ( t ) = I/ | s (cid:105)(cid:104) s | . No work is performed on the system.As the energy levels remain degenerated there is no heat exchange but thedecrease of entropy is compensated by the entropy increase in the measure-ment device and the environment. In the step ii) the state remains the sameand again no work is performed. Similarly, no heat is exchanged and theentropy remains the same. During the step iii) the system equilibrates atany moment and the work W ≤ kT ln 2 is adiabatically extracted from thebath and the entropy grows to its maximal value. The system ends the cycleagain in the state ρ ( t ) = I/
2. To avoid the conflict with the Second Lawwe have to conclude that the following Landauer’s principle for measurement(LPM) holds :
A completion of a binary measurement, including reseting of a measuringdevice needs at least kT ln 2 of work. Remarks
It is often claimed that the work (at least kT ln 2 ) needed toperform a binary measurement is actually used to erase a bit of informationin a ”memory” of a measuring device [20]. Analogically, one can estimateby ∼ kT a minimal energy cost of any irreversible elementary gate. To sup-port this picture microscopic models of erasure have been proposed involvingcertain entropy-energy balance [24]. However, as shown in the AppendixII this argument is generally not convincing. Therefore, one is left withphenomenological arguments as presented above which do not depend on thedetailed model of the measurement procedure . | ± (cid:105) are equilibrium ones in the sense of the definition of above, i.e. theirrelaxation times to the unique Gibbs state are exponentially long .Therefore, one faces the following alternative :1) The LPM is valid for all measurements, and therefore is not a consequenceof the Second Law, but must be added as an independent additional postu-late.2) The LPM is a consequence of the Second Law and therefore does not needto hold for equilibrium states.The second possibility is much more likely as the laws of thermodynamicsseem to provide ultimate, model independent limitations on physical pro-cesses. It seems that the task which does not violate the laws of thermody-namics can be realized in principle by a certain physical process. Therefore,one can formulate the plausible hypothesis which does not violate the lawsof thermodynamics: Hypothesis I
Equilibrium (metastable) states can be distinguished (measured), with the er-ror decreasing exponentially with the size of the system, at the arbitrarily lowenergy cost.
Hypothesis I makes distinction between the information encoded in re-laxing states or metastable states. Only for the former the associated infor-mation gain has a physical meaning and should be included into the thermo-dynamical entropy balance. This information is unstable and therefore mustbe recorded while the stable information need not. The crossover betweenthem is described by the error behavior. For smaller systems the error growsand the ”cost-free” acquiring of information must be replaced by the ”costly”recording procedure. A multitude of metastable states for a glassy system must be also treated as anequilibrium state, because the determination of an energy landscape is a computationallyhard problem what makes impossible to design a cyclic process of work extraction from aheat bath. Perpetuum mobile based on quantum mem-ory
A scalable quantum memory for a single qubit is a system which consists of N microscopic constituents (e.g. atoms, spins,..) interacting with a heat bathat the temperature T >
0. The system is designed in such a way that theinformation bearing degrees of freedom form a quantum subsystem (encodedqubit) described by 2-dimensional Hilbert space spanned by the vectors ofthe form | ψ (cid:105) ≡ | ψ (cid:105) ⊗ | ω R (cid:105) , with | ω R (cid:105) being a purification of the fixed thermalequilibrium state for all other degrees of freedom.Under the Hypothesis I , the minimal needed assumptions concerning op-erations on the single-qubit quantum memory are the following:I) The eigenstates of σ z and σ x are equilibrium ones (i.e. metastable withlife-times exponentially long in N ).II) One can perform effectively and ”cost-free” measurements of the observ-able σ z ,III) The observables σ x , σ y and σ z can be implemented effectively to con-struct an interaction Hamiltonian with a relaxing qubit described by Pauliobservables X, Y, Z H int = σ x ⊗ X + σ y ⊗ Y + σ z ⊗ Z. (3)Here again ”effectively” means that one needs time at most polynomial in N . One can design now the following cyclic process which effectively extractswork from a heat bath using such quantum memory. The process consistsof:A) measuring the memory observable σ z ,B) switching on the coupling Hamiltonian (3), performing a SWAP operation[21], and transferring the post-measurement memory state to the relaxingqubit,C) extracting kT ln 2 work from the bath using the knowledge of the state ofthe relaxing qubit and applying the procedure described in Section 3.In this process one uses the memory to reset the relaxing qubit into aknown state without spending work. Therefore, the net effect of this cyclicprocess is a subtraction of heat from the bath and the production of workwhat violates the Second Law of Thermodynamics. Remark
The example of above does not contradict the known results oncerning the existence of metastable encoded qubit observables is some spinmodels like 4D-Kitaev model [14]. Namely, those observables are not given interms of self-adjoint operators which can be effectively implemented and usedas ingredients of the interaction Hamiltonian (3), but are defined in termof measurement procedures accompanied by certain computational classicalalgorithms. The conclusion from the
Gedankenexperiment discussed above raises a ques-tion which of the assumptions I, II, III concerning the properties of quantummemory are internally inconsistent. It seems that there exists a fundamentalconflict between the stability of states and possibility of applying reversibleoperations (gates) changing such states. Indeed, in the scheme discussedabove one uses a reversible, unitary SWAP gate applied to stable statesof the memory. Our experience based on the 4D-Kitaev models suggeststhat there is a common mechanism of state stabilization in the classicaland quantum domains. It involves free-energy barriers separating differentmetastable states which have to be overcame in the process of performinggate. Such an operation costs work which is then dissipated into environ-ment. Therefore, the gates performed on stabilized states must be irreversible(non-Hamiltonians) transformations. This does not impair classical digitalinformation processing where all practically used gates are irreversible but isa serious obstacle in the case of quantum information. Notice, that also clas-sical reversible computation would be sensitive to chaos what implies thatthe classical analog computation could not overpower the digital one.One can make the statements of above more precise assuming that thework W g invested in the irreversible gate is of the order of the free-energybarrier F N protecting information carrying states. Here, N denotes the num-ber of microscopic constituents of the information carrier (atoms, electrons,spins, etc.) and typically F N ∼ N . The same factor F N determines thestability of protected states with respect to thermal noise characterized bytheir life-time τ N (cid:39) τ exp (cid:16) F N kT (cid:17) , N >> τ is a typical microscopic relaxation time scale. One can formulatethe following Hypothesis II which gives a more realistic estimation of the8hermodynamical cost of irreversible information processing than the stan-dard one based on the Landauer’s Principle.
Hypothesis II
In order to perform a gate on the protected state one needs the amount ofwork W g (cid:39) kT ln (cid:16) τ N τ (cid:17) (cid:39) kT N. (5) which is dissipated into environment .Actually, the zero-temperature analog of the formula (5) ( kT is replacedby quantum fluctuations) is derived in the Appendix III for a spin-bosonmodel and the presented derivation can be easily generalized to finite tem-peratures. Therefore, the Hypothesis II is in fact a plausible conjecture that(5) is valid also for more sophisticated models. The discussion of
Gedankenexperiment shows that a scalable quantum mem-ory could be used as a perpetuum mobile of the second kind and hence cannotbe realized in Nature. The fundamental assumptions behind the analysis ofthis model is that the properly formulated laws of thermodynamics are validand the physical processes which are not forbidden by them can be realized.Those general principles suggest the alternative hypothesis concerning thethermodynamical cost of acquiring information and performing operationsfor the case of stabilized information carriers. The conflict between stabilityof information and reversibility of gates does not restrict the irreversible clas-sical digital information processing but suggests unfeasibility of large scalequantum information one. The heuristic arguments presented in this papercan be generally accepted only if there are supported by a large body of in-dependent theoretical and experimental evidence. Therefore, the analysis ofmicroscopic models of the candidates for quantum memory is still importantand will be for sure continued in the near future.
Acknowledgments
The author thanks Hector Bombin, Micha(cid:32)l Horodecki,Charles Bennett, Daniel Lidar and Stanis(cid:32)law Kryszewski for discussions. Thesupport by the Polish Ministry of Science and Higher Education, grant NN202208238 is acknowledged.
Appendix I. Markovian model reproducing the laws of thermo-dynamics ρ ( t ) satisfies the followingMarkovian Master Equation ddt ρ ( t ) = − i [ H ( t ) , ρ ( t )] + (cid:88) j L j ( t ) ρ ( t ) , (6)where for any 0 ≤ t ≤ ∞ L j ( t ) is a generator of a completely positivedynamical semigroup which satisfies L j ( t ) ρ eqj ( t ) = 0 , ρ eqj ( t ) = e − β j H ( t ) Tr e − β j H ( t ) . (7)Here β j = 1 /kT j is the inverse temperature of the j -th heat bath. Theequation of motion (6)(7) can be derived from a microscopic Hamiltonianmodel using the weak coupling assumption and for slowly varying externalfields [25].The First Law of thermodynamics becomes now the definition of work W performed on a system and heat Q absorbed by the system with the obviousdefinition of the internal energy EE ( t ) = Tr (cid:0) ρ ( t ) H ( t ) (cid:1) , ddt W ( t ) = Tr (cid:0) ρ ( t ) dH ( t ) dt (cid:1) , (8) ddt Q ( t ) = Tr (cid:0) dρ ( t ) dt H ( t ) (cid:1) = (cid:88) j Tr (cid:0) H ( t ) L j ( t ) ρ ( t ) (cid:1) ≡ (cid:88) j ddt Q j ( t ) . (9)where Q j is the heat absorbed by the system from the j -th bath.Defining the entropy as S ( t ) = − k Tr (cid:0) ρ ( t ) ln ρ ( t ) (cid:1) one obtains the SecondLaw ddt S ( t ) − (cid:88) j T j ddt Q j ( t ) = (cid:88) j σ j ( t ) ≥ j -baths is given by σ j ( t ) = k Tr (cid:0) L j ( t ) ρ ( t )[ln ρ ( t ) − ln ρ eqj ( t )] (cid:1) ≥ { s L j ( t ) } , s ≥ Appendix II. Argument based on energy and entropy balance ρ in ⊗ ω ( β ) → ρ fin ⊗ ω (cid:48) (12)where ω ( β ) is a Gibbs state of a bath at the inverse temperature β and ω (cid:48) isa final state of a bath, not necessarily given by another Gibbs state.Using the definitions and notation ω ( β ) = e − βH bath Z ( β ) , E ( β ) = Tr (cid:0) ω ( β ) H bath (cid:1) , S ( β ) = − k Tr (cid:0) ω ( β ) ln ω ( β ) (cid:1) (13)one can easily compute ddβ S ( β ) = kβ ddβ E ( β ) . (14)From (12) and the fact that the total system is an isolated Hamiltonian one,the entropy balance follows S ( ρ fin ) − S ( ρ in ) = S ( β ) − S ( β (cid:48) ) (cid:39) kβ (cid:0) E ( β ) − E ( β (cid:48) ) (cid:1) (15)where β (cid:48) is the inverse temperature of the Gibbs state such that S ( β (cid:48) ) = − k Tr( ω (cid:48) ln ω (cid:48) ). In the last equality in (15)one uses the fact that a heat bathis a large physical system and its interaction with a small open system canonly infinitesimally change bath’s intensive parameters what implies β (cid:48) (cid:39) β .As the Gibbs state minimizes internal energy under the condition of a fixedentropy the energy gain of a bath satisfies∆ E = Tr( ω (cid:48) H bath ) − E ( β ) ≥ E ( β (cid:48) ) − E ( β ) . (16)Defining ∆ S = S ( ρ fin ) − S ( ρ in ) = S ( β ) − S ( β (cid:48) ) (17)and using (15),(16) one obtains the inequality∆ E + T ∆ S ≥ . (18)One can apply now the scheme of above to a model of reseting a single bit ofinformation in a memory of measuring device. The bit is supported by two11egenerated eigenstates of the memory | (cid:105) , | (cid:105) . The initial state encodes anunknown bit what corresponds to the initial state ρ in = 1 / | (cid:105)(cid:104) | + | (cid:105)(cid:104) | )with the entropy k ln 2, and the final state is a fixed reference state, say | (cid:105) ,with the entropy equal to 0. Therefore, using (18) one obtains the lowerbound for the increase of the bath’s internal energy∆ E ≥ kT ln 2 . (19)As the process is cyclic in the sense that the external time-dependent controlfields switched on at the beginning of the process are switched off at itsend, and the energy of the 2-level system is not changed one can attribute∆ E to the amount of work performed by the external forces and dissipatedinto the bath’s degrees of freedom. Hence, the Landauer’s principle for bit’serasure seems to be justified on the microscopic basis. Moreover, as an actualmeasurement which transforms the initial reference state | (cid:105) into | (cid:105) or | (cid:105) does not change the entropy this part of a cyclic measurement process doesnot need work and hence the LPM seems to be valid as well.Unfortunately, the arguments of above are not convincing. The mainproblem is the entropy balance based on the assumption of the exact prod-uct structure for initial and final states (12). Indeed, a weak coupling to aheath bath suggests an approximative product state structure but due to the discontinuity of entropy in the limit of large systems we cannot use it for theestimation of entropy. This fact is expressed in terms of Fannes inequality[26] for two close density matrices of a system with D -dimensional Hilbertspace and (cid:107) · (cid:107) denoting the trace norm | S ( ρ ) − S ( ρ (cid:48) ) | ≤ (cid:107) ρ − ρ (cid:48) (cid:107) ln D − (cid:107) ρ − ρ (cid:48) (cid:107) ln( (cid:107) ρ − ρ (cid:48) (cid:107) ) . (20)To illustrate this problem one can consider the model discussed in the Ap-pendix I. The validity of the Markovian approximation means that the stateof the total system is well-approximated by the product ρ ( t ) ⊗ ω B where ρ ( t )is a solution of the Master equation (6), and ω B is a fixed stationary state ofthe bath. Obviously, this product form is not consistent with the constantentropy of the total Hamiltonian system. The missing entropy is hidden inthe small correction terms describing the residual system - bath correlationsand local perturbations of the bath’s state which practically do not influencethe values of measured observables. Therefore, to obtain a proper entropy,heat and work balance one has to use their definitions as presented in Ap-pendix I and needs an equation of motion for a system like that obtained inthe Markovian limit (6). 12 ppendix III. A quantum model of stable information carrier The model represents a system with two degenerate ground states whichare ”macroscopically distinguishable” and stable and hence can be used asa single bit memory, in particular, as an element of a quantum measure-ment apparatus which records the value of a dichotomic observable. Thesystem consists of a spin-1/2 described by the Pauli matrices coupled to a”macroscopic” system defined in terms of bosonic fields a ( ω ) , a + ( ω ) satisfyingcanonical commutation relations[ a ( ω ) , a + ( ω (cid:48) )] = δ ( ω − ω (cid:48) ) , ω, ω (cid:48) ∈ [0 , ∞ ) (21)The system Hamiltonian is a simple version of the spin-boson Hamiltonianstudied, for example, in [27] H = H + σ z ⊗ (cid:90) ∞ dω ω (cid:0) ¯ g ( ω ) a ( ω ) + g ( ω ) a + ( ω ) (cid:1) (22)with H = (cid:90) ∞ dω ω a + ( ω ) a ( ω ) . (23)The unitary ”dressing” operator defined by U d = exp (cid:110) σ z ⊗ (cid:90) ∞ dω (¯ g ( ω ) a ( ω ) − g ( ω ) a + ( ω )) (cid:111) (24)diagonalizes the system Hamiltonian, i.e. U d HU † d = H and therefore makesthe model exactly solvable. In particular one can find two degenerate groundstates of H H | ψ ± (cid:105) = − E g | ψ ± (cid:105) (25)where | ψ ± (cid:105) = |±(cid:105) ⊗ | [ ± g ] (cid:105) , σ z |±(cid:105) = ±|±(cid:105) (26)and | [ f ] (cid:105) denotes the field coherent state obtained from the vacuum state | Ω (cid:105) by the action of the Weyl operator W [ f ], i.e. | [ f ] (cid:105) = W [ f ] | Ω (cid:105) , W [ f ] = exp (cid:110)(cid:90) ∞ dω ( ¯ f ( ω ) a ( ω ) − f ( ω ) a + ( ω ) (cid:111) . (27)The ”classicality” or ”macroscopicality” of the field states | [ ± g ] (cid:105) is charac-terized by the averaged number of bosons N and the energy of ground states − E g = (cid:90) ∞ | g ( ω ) | dω, E g = (cid:90) ∞ ω | g ( ω ) | dω . (7)For N >> ± g are those macroscopic ”pointer states” which allow di-rectly, without any cost, to determine which of the two ground states isoccupied by the system. The error of this distinguishing process is exponen-tially small in N and given by the overlap of coherent states (cid:15) = |(cid:104) [ g ] | [ − g ] (cid:105)| = e − N . (28)The spin-boson system can be used as a carrier of a bit or an element ofthe quantum measurement device which records the measurement result. Inboth cases one applies a fast NOT (or CNOT) gate on the spin part, whichproduces a new spin-boson state from one of the initial ground states, say | ψ + (cid:105) | ψ (cid:48) + (cid:105) = σ x | ψ + (cid:105) = |−(cid:105) ⊗ | [ g ] (cid:105) (29)with the averaged energy (cid:104) ψ (cid:48) + | H | ψ (cid:48) + (cid:105) = 3 E g . The difference between thisenergy and the ground state one (energy barrier) is equal to the work W g =4 E g needed to implement a NOT gate by a suitable time-dependent Hamil-tonian. One can exactly compute the subsequent evolution of the spin-bosonstate | ψ (cid:48) + ( t ) (cid:105) = e − iHt | ψ (cid:48) + (cid:105) = e iα ( t ) |−(cid:105) ⊗ W [2 g t − g ] | Ω (cid:105) (30)where α ( t ) is an irrelevant phase and g t ( ω ) = e − iωt g ( ω ) is a traveling wave.For long t → ∞ the classical traveling wave 2 g t becomes orthogonal to theinitial bounded field − g and the state possesses a product structure | ψ (cid:48) + ( t ) (cid:105) ∼ | ψ − (cid:105) ⊗ | ψ [2 g t ] (cid:105) . (31)The component | ψ [2 g t ] (cid:105) carries the energy 4 E g to infinity describing its dissi-pation into environment. The total state | ψ (cid:48) + ( t ) (cid:105) becomes indistinguishablefrom the ground state | ψ − (cid:105) by any local measurement. Hence, one obtainsan irreversible NOT gate performed on the stable states encoding a bit ofinformation and the thermodynamical cost of the gate is 4 E g of work. No-tice, that when the system is applied to record a measured state one does This tensor product structure corresponds to the property of Fock spaces F ( H (cid:48) ⊕H (cid:48)(cid:48) ) = F ( H (cid:48) ) ⊗ F ( H (cid:48)(cid:48) ) with the corresponding identification of vacuum states | (cid:105) ≡ | (cid:48) (cid:105) ⊗ | (cid:48)(cid:48) (cid:105) . change of the spin-boson ground state could be an indicator of a one of two statesof the measured system.It is instructive to see how the dissipation mechanism of above preventsthe encoding of a qubit state in our memory device. Consider a fast SWAPgate between a certain qubit at the initial state α | (cid:105) + β | (cid:105) and the spin degreeof freedom of our memory at the initial total memory state | ψ + (cid:105) = | + (cid:105)⊗| [ g ] (cid:105) .The state of the total system, just after SWAP gate, is given by (cid:0) α |−(cid:105) + β | + (cid:105) (cid:1) ⊗ | [ g ] |(cid:105) ⊗ | (cid:105) . (32)For long enough times the total state evolves into (cid:104) α |−(cid:105) ⊗ | [ − g ] (cid:105) ⊗ | [2 g t ] (cid:105) + β | + (cid:105) ⊗ | [ g ] (cid:105) ⊗ | Ω (cid:48)(cid:105) (cid:105) ⊗ | (cid:105) (33)where one uses the fact that lim t →∞ (cid:104) g | g t (cid:105) = 0 and | Ω (cid:48)(cid:105) denotes the vacuumstate of the field degrees of freedom which represent the traveling waves withsupport far away from the localized states of the memory. Those travelingwaves form the environment of the memory which can be traced out to givethe following asymptotic state of the memory itself ρ M = | α | | ψ − (cid:105)(cid:104) ψ − | + | β | | ψ + (cid:105)(cid:104) ψ + | + (cid:104) α ¯ β (cid:104) Ω (cid:48)| [2 g t ] (cid:105)| ψ − (cid:105)(cid:104) ψ + | + h . c (cid:105) . (34)As the off-diagonal terms are exponentially small ( |(cid:104) Ω (cid:48)| [2 g t ] (cid:105)| = e − N ) thestate (34) is a mixture of two stable memory states. It means that thedecoherence process accompanying the dissipation of 4 E g of energy destroysquantum coherence of the swapped qubit state.The similar analysis of the measurement/encoding process in the case of finitetemperature will be presented in the forthcoming paper. References [1] Knill, E., Laflamme, R., and ˙Zurek W. H.,
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