Bounds on the entropy generated when timing information is extracted from microscopic systems Dominik Janzing∗ and Thomas Beth Institut fu¨r Algorithmen und Kognitive Systeme, Universit¨at Karlsruhe, Am Fasanengarten 5, D–76131 Karlsruhe, Germany WeconsiderHamiltonianquantumsystemswithenergybandwidth∆E andshowthateachmea- surementthatdeterminesthetimeuptoanerror∆tgeneratesatleasttheentropy(~/(∆t∆E))2/2. Our result describes quantitatively to what extent all timing information is quantum information in systems with limited energy. It provides a lower bound on the dissipated energy when timing information of microscopic systems is converted to classical information. This is relevant for low power computation since it shows the amount of heat generated whenever a band limited signal 3 controls a classical bit switch. 0 Our result provides a general bound on the information-disturbance trade-off for von-Neumann 0 measurementsthatdistinguishstatesontheorbitsofcontinuousunitaryone-parametergroupswith 2 boundedspectrum. Incontrast,informationgainwithoutdisturbanceispossibleforsomecompletely positivesemi-groups. Thisshowsthatreadoutoftiminginformationcanbepossiblewithoutentropy n generation if the autonomous dynamical evolution of the “clock” is dissipative itself. a J 3 I. TIMING INFORMATION GAIN WITHOUT pφ φ +q φ⊥ φ⊥ , | ih | | ih | 2 DISTURBANCE? i.e.,themeasurementcausesanentropyincreaseof∆S = 1 plnp qlnq. v −The g−eneral condition under which information about 5 Listening to “folklore versions” of quantum mechan- unknownquantumstatescanbegainedwithoutdisturb- 2 ics one may consider it as a key statement of quantum ing them is well-known and reads as follows [1]: 1 theory that there is no measurement without disturb- Let ρ be the unknown density matrix of a system. By 1 ing the measured system. However, the fact that this is 0 priorinformationoneknowsthatρ isanelementofaset not true is well-known in modern quantum information 3 Γ of possible states. Then one can get some information theory andis, insome sense, the reasonwhy classicalin- 0 on ρ if and only if there is a projection Q commuting formation exists at all although our world is quantum. / with all matrices in Γ such that the value tr(Qρ) is not h Consideratwo-levelsystem,i.e.,aquantumsystemwith p Hilbert space C2 and denote its upper or lower state by the same for all ρ∈Γ. - As noted in [2], this can never be the case if the set Γ t 1 and 0 , respectively. Assume that we know by prior n | i | i is the orbit(ρt)t∈R of a Hamiltonian system evolvingac- informationthatthesystemisnotinaquantumsuperpo- a cordingto ρ =exp( iHt)ρexp(iHt). This holdsevenif t u sitionbutonlyinoneofthetwostates|ji. Thenthemea- ρ andH acton anin−finite dimensionalHilbert space. In q surement with projections P0 := |0ih0| and P1 := |1ih1| this sense, timing information is always to some extent : show which state is present without disturbing it at all. v quantum information that cannot be read out without Here we have used the two-level system as classical bit. i state disturbance. It can only become classical informa- X The situation changes if the two-level system is used as tion if either (1) prior information tells us that the time r quantum bit (“qubit”) and is prepared in a quantum su- a perposition ψ := c 0 +c 1 where the complex coef- t is an element of some discrete set {t1,t2,...} (see [2]) ficients c an|dic wi0t|hic 21+| ic 2 = 1 are unknown to or (2) in the limit of infinite system energy [3,2]. 0 1 | 0| | 1| At first sight the statement that classical timing in- the person who measures. Then any vonNeumann mea- formation can only exist in one of these two cases seems surement with projections φ φ and φ⊥ φ⊥ will on | ih | | ih | to be disproved by the following dissipative “quantum the one hand only provide some information about ψ | i clock”: and will on the other hand disturb the unknown state Let ρ := 1 1 be the upper state of a two-level sys- ψ since it “collapses” to the state ψ with probabil- 0 | ih | |ityip := ψ φ 2 and to the orthogon|alistate φ⊥ with tem. Let the system’s time evolution for positive t be probabili|thyq|:=i| ψ φ⊥ 2. Fromthepointofv|iewiofthe described by the Bloch relaxation(see e.g. [4]) |h | i| person who has prepared the state ψ and does not no- ρ :=exp( λt)1 1 +(1 exp( λt))0 0 . tice the measured result (“non-selec|tivie operation”), the t − | ih | − − | ih | measurement process changes the density matrix of the Since all the states ρ commute with the projections P t 0 system from ψ ψ to and P one can certainly gain some information about 1 | ih | ∗ Electronic address: [email protected] 1 t by the measurement with projections (P ,P ). How- if the unselected state is considered, i.e., the outcome is 0 1 ever, this situation is actually the infinite energy limit ignored. since semi-group dynamics of this form is generated by The post-measurement state γ˜ coincides with γ if and couplingthesystemtoaheatbathofinfinitesizeandin- only if γ commutes with all projections P . Now we will j finite energy spectrum [5]. Of course the fact that well- compare the von-Neumann entropy of γ and γ˜. For any known derivations of relaxation dynamics require heat state γ the von-Neumann entropy is defined as baths with infinite energy spectrum does not prove our claimthatthisisnecessarilythecase. Thisclaimisrather S(γ):= tr(γlnγ). − an implication of Theorem 1 in Section II. The paper is organized as follows. In Section II we ThefollowingLemmashowsthatthemeasurementcan show quantitatively to what extent information on the never decrease the entropy: (non-discrete) time is always quantum information as long as the clock is a system with limited energy band- Lemma 1 Let γ be an arbitrary density matrix on a width∆E. Explicitly,weprovealowerboundontheen- Hilbert space and (P ) be a family of orthogonal projec- tropyincreaseintheclockcausedbyvon-Neumannmea- j tions with P =1. Set γ˜ := P γP Then we have surements, i.e., measurements that are described by an ⊕ j j j j P orthogonal family of projections where the state change ∆S :=S(γ˜) S(γ)=K(γ γ˜) of the system is described by the projection postulate. − || Generalizedmeasurementproceduresareconsideredin where K(γ γ˜) is the Kullback-Leibler distance (or Rela- SectionIII. Theydonotnecessarilyincreasethe entropy || tive Entropy) between γ˜ and γ. It is defined by [6] ofthe measuredclocksincethe processcaninclude some kind of cooling mechanisms but will lead to an increase K(γ γ˜):=tr(γlnγ) tr(γlnγ˜). (1) of entropy of the total system which includes the clock’s || − environment. This has implications for low power com- putation since it shows that the distribution of timing informationinherentinamicroscopicclockproducesnec- Proof: According to the definition of entropy it is suffi- essarily some phenomenological entropy. cient to show the equation tr(γlnγ˜) = tr(γ˜lnγ˜). The In Section IV we discuss whether our bound is tight. second term on the right in eq. (1) is equal to In Section V we shall show that the result of Section III can be applied to the situation that a clock with limited energy controls the switching process of a classical bit. tr( P γln( P γP )=tr( P γln( P γP )P ) i j j i j j i Hereaclassicalbitisunderstoodasatwostatequantum Xi Xj Xi Xj systemon which decoherencetakes place on a time scale that is in the same order as the switching time or on a Note that ln( P γP ) commutes with each P . Hence j j j i smaller scale. For low power computation, this proves we get P which amount of dissipation is required whenever the autonomous dynamics of a microscopic device controls tr( P γln( P γP )P )= i j j i a classical output. In Section VI we apply the bound to X X i j other one-parameter groups. tr( P γP ln( P γP ))= i i j j X X i j II. ENTROPY INCREASE OF A QUANTUM tr(γ˜lnγ˜) CLOCK CAUSED BY VON-NEUMANN MEASUREMENTS This completes the proof. 2 Since the entropy increase is the Kullback-Leibler dis- For the moment we will restrict our attention to von- tance between the pre- and the post-measurement state Neumann measurements. For technical reasons we will wecanobtainalowerboundon∆S intermsofthetrace- assume them to have a finite set of possible outcomes. norm distance between them: Hence the measurement is described by a family (P ) of j mutual orthogonal projections on the system’s Hilbert space that may be infinite dimensional. The state of Lemma 2 For the entropy increase ∆S we obtain the H the system is described by a density matrix, i.e., a pos- lower bound itive operator with trace 1 acting on . According to H 1 the projectionpostulate anystateγ changesto the post- ∆S γ˜ γ 2 measurement state ≥ 2k − k1 γ˜ := PjγPj, where kak1 :=tr(√a†a) is thetrace-norm of an arbitrary X matrix a. j 2 The proof is immediate using where s (t) = 1 if p˙ (t) 0 and s (t) = 1 if p˙ (t) < 0. j j j j ≥ − Note that we have γ˜ γ 2 K(γ γ˜) k − k1 || ≥ 2 p˙(t) 1 = sj(t)p˙j(t)=tr(i[H,R(t)]ρt). k k X j (see [6]). Now we consider the states ρt on the orbit of the time Furthermore easy computation shows that evolution and show that the norm distance between ρt tr(i[H,R(t)]ρ˜t)=0. We conclude andρ˜ islargeateverymomentwheretheoutcomeprob- t abilities of the measurement (P ) change quickly, i.e., p˙(t) =tr(i[H,R(t)](ρ ρ˜)) 2 H ρ ρ˜ , j 1 t t t t 1 k k − ≤ k kk − k where the values tr(i[H,P ]ρ ) are large. This is the es- j t sential statement that is used to prove the information- since the operator norm of R(t) is 1. We assume H = k k disturbance trade-off relation. But first we have to in- ∆E/2 without loss of generality and have troduce the energy bandwidth of a system. For Hamil- p˙(t) tonians with discrete eigenvalues it is just the difference ρ ρ˜ k k1 . t t 1 between the greatest and smallest eigenvalues of the en- k − k ≥ ∆E ergy states with non-zero occupation probability. The 2 generalization to Hamiltonians with continuous and dis- crete parts in the spectrum reads: Now we are able to prove our main theorem. Definition 1 Let H be the Hamiltonian of a quantum Theorem 1 Let the energy bandwidth ∆E of a quantum system and (QE)E∈R be the spectral family correspond- system be less than ∞. Assume that the true time t is ing to H, i.e., Q projects onto the Hilbert space cor- in the interval [0,T] for arbitrary T > 0. Let the prior E responding to energy values not greater than E. Then probability for t be the uniform distribution on [0,T]. Let f(E) := tr(QEρt) is the distribution function of a (Pj) be the family of projections corresponding to a von- (time-independent) probability measure µ which is called Neumann measurement. Assume that the measurement the spectral measure of ρ corresponding to H. Let has the time resolution ∆t in the following sense: For t [E ,E ]bethesmallestintervalsupportingthisspec- each t [0,T ∆t] there is a decision rule based on the min max ∈ − tral measure. Then ∆E := Emax Emin is the energy measurement outcome that decides whether the state ρt − bandwidth of the system in the state ρt. or ρt+∆t is present with error probability at most 1/4. Thenthemeanentropyincrease∆S (averagedoverthe By rescaling the Hamiltonian it is easy to see that interval [0,T]) caused by the measurement is at least the time evolution of the state ρ can equivalently be de- scribed by 1 ~ 2 . 2(cid:16)∆t∆E(cid:17) 1 H′ :=(Q Q )H (E E )1, Emax − Emin − 2 max− min Note that our definition of time resolution is not the since exp( iH′t)ρexp(iH′t) = exp( iHt)ρexp(iHt). usualonesinceitdoesnotrequirethatthe measurement Clearly, H−′ ∆E/2. The energy s−pread is decisive distinguishes between ρ and ρ , for instance. How- k k ≤ t t+2∆t for our lower bound on the trace-normdistance between ever, it is exactly the definition of time resolution that ρt and ρ˜t. we need in the proof. Lemma 3 Let p (t):=tr(P ρ ) be the probability of the Proof: Each decision rule distinguishing between ρ and j j t t measurement outcome j at time t. Let ∆E be the energy ρ that is based on the measurement outcome is of t+∆t bandwidth of ρ . Set the following form: If the outcome is j it decides for t t with probabilityq . If the true state is ρ it decides for t j t d withprobability q p (t) andif the true state isρ p˙(t) 1 := pj(t) . j j j t+∆t k k X|dt | it decides erroneoPusly for t with probability j t+∆t Then we have q p (t+∆t)= q p (t)+ q p˙ (t′)dt′. j j j j jZ j p˙(t) 1 Xj Xj Xj t ρ ρ˜ k k . t t 1 k − k ≥ ∆E Thedifferencebetweentheprobabilitytodecidecorrectly fort andto decide erroneouslyfort is atleast1/2by as- sumption. Hence Proof: For a specific moment t define the operator R(t) by t+∆t q p (t′)dt′ 1/2. |Z j j |≥ R(t):= sj(t)Pj t Xj 3 Notethat2q 1isintheinterval[ 1,1]forallj. There- (see [7] for the most general description of measure- j − − fore ments). Ifthesetofpossibleoutcomesisfinite,aPOVM is defined by a family (M ) of positive operators with j |X(2qj −1)p˙j(t)|≤X|p˙j(t)|. jMj = 1 such that tr(Mjρ) is the probability for j j Pthe outcome j. Furthermore the connection between the POVM and the effect on the state is given by the Obviously we have p˙ (t)=0 since p (t)=1. We j j j j following consistency condition: There exist completely conclude P P positive maps G such that tr(G (ρ)) = tr(M ρ) and j j j that,giventheoutcomej,thepost-measurementstateis p˙(t) 2 q p˙ (t) , 1 j j k k ≥ |Xj | Gj(ρ)/tr(Mjρ). The unselected post-measurement state is hence given by hence ρ˜:= G (ρ). j t+∆t X p˙(t′) dt′ 1. j Z k k1 ≥ t Clearly, in this general setting it is even possible that Therefore we find that the average of p˙(t′) over the the measurement can decrease the entropy of the clock 1 interval [t,∆t] is at least 1/∆t. Since tkhis hoklds for ev- sincethemapsGj mayincludeprocessesthatarecooling ery t [0,T ∆t] the average of p˙(t) over the whole mechanisms for the clock and transport entropy to the 1 interv∈al [0,T−] is at least 1/∆t. Wke deknote this average environment. by However, we will show that each time measurement leadsunavoidablytoanentropyincreaseinthetotalsys- p˙(t) . tem consisting of the measurement apparatus and the 1 k k clock. Let be the Hilbert space of the measurement m For the average entropy generation we have apparatusaHndbeHˆ betheHamiltonianitsfreetimeevo- lution as long as no measurement interaction is active. 1 T 1 ∆S = S(ρ˜) S(ρ )dt ρ˜ ρ 2 We assume the state γ of the apparatus to be invariant T Z0 t − t ≥ 2k t− tk1 under its freeevolution. Otherwisethe measurementap- paratus would be a clock in its own since it contained by Lemma 2. Note that the average of the square is at some information about the time t. Then we switch on least the square of the average. We conclude the measurement interaction which leads to a unitary 1 2 transformation u on the space m (the definition ∆S ρ˜ ρ . H⊗H ≥ 2k t− tk1 of the apparatus includes its environment such that the total evolution is unitary). This process is often called a Due to Lemma 3 we have pre-measurement. The different outcomes j correspond 1 1 tomutualorthogonalsubspacesof m(“pointerstates”). ∆S , H ≥ 2(∆t∆E)2 Let (Qj) be the projections onto these subspaces. Now weassumethatdecoherenceonthemeasurementappara- aslongaswemeasurethe energyspreadinunits of~. In tus’ pointer takes place [8–10]. The process given by the physics, it is more commonto use SI-units which lead to unitary pre-measurement process u and the decoherence is described by 1 ~ 2 ∆S . ≥ 2(cid:16)(∆t∆E(cid:17) ρ γ (1 Qj)u(ρ γ)u†(1 Qj) ⊗ 7→ ⊗ ⊗ ⊗ X 2 j The entropy of this state is clearly the same as the en- tropy of III. ENTROPY GENERATED BY GENERALIZED u†(1 Q )u(ρ γ)u†(1 Q )u. j j MEASUREMENTS X ⊗ ⊗ ⊗ j BydefiningP :=u†(1 Q )uwehavereducedtheprob- The assumption that every measurement is described j ⊗ j lemtotheprobleminSectionIIwiththeonlygeneraliza- by projections (P ) and that the effect of the measure- j tionthat(P )isnotafamilyofprojectionsactingonthe ment is given by the projection postulate j Hilbert space of the clock but on the Hilbert space of H clock + measurement apparatus where the apparatus is ρ P ρP j j 7→ X atrest. Nowweshowthatthestationarityoftheappara- j tus state ensures that the bound in Theorem 1 holds for is notgeneralenough. Ingeneral,a measurementcanbe the considered kind of generalized measurement as well. describedbyapositiveoperatorvaluedmeasure(POVM) Formally, we claim: 4 Theorem 2 Let ρ γ be a density matrix on a Hilbert p˙(t) =tr(i[H,R(t)]ρ ) 2 H R(t) =∆E R(t) 1 t space ˆ. Let th⊗e time evolution of the composed sys- k k =∆E. ≤ k kk k k k tem beHg⊗ivHen by the Hamiltonian H := H 1+1 Hˆ c with [γ,Hˆ] = 0. Let (Pj) be a von-Neuma⊗nn meas⊗ure- NotealsotheconnectiontotheHeisenberguncertainty ment acting on the Hilbert space ˆ. Let ∆E be the principle H⊗H energy bandwidth of the left component of the composed ∆′E∆′t ~/2. system (given by the state ρ with Hamiltonian H). Let ≥ themeasurementhavethetimeresolution∆tinthesense where ∆′E is the energy spread of the system, i.e., the of Theorem 1. Then we have the following lower bound standarddeviationoftheenergyvaluesand∆′tthestan- on the average entropy increase ∆S of the composed sys- dard deviation of the time estimation [12] based on any tem caused by the measurement: measurement. However,using the symbols ∆′ instead of 1 ~ 2 ∆,weemphasizethatthesedefinitionsdonotagreewith ∆S . ours. Note that the energy spread ∆′E is at most the ≥ 2(cid:16)∆t∆E(cid:17) energybandwidth∆E. But∆′tcanexceedourtimeres- olution ∆t by an arbitrary large value. This shows the following example: assume the clock to be a two-level Proof: Clearly, the Hamiltonian Hˆ is irrelevant for the system with period T˜. Set ρt := ψt ψt with | ih | evolution of the state ρ γ due to ⊗ 1 ψ := (0 +exp(i2π/T˜)1 ). t exp( iH t)(ρ γ)exp(iH t)=exp( iHt)ρexp(iHt) γ. | i √2 | i | i c c − ⊗ − ⊗ Given the prior information that the time is in an in- Hencewecantreattheevolutionasifitwasimplemented by the Hamiltonian H 1, which can assumed to be terval [0,T] with T ≫ T˜ we can only estimate the time bounded as in Section II⊗. 2. up to multiples of T˜ and obtain an error in the order T. In contrast, the time resolution ∆t in our sense is T˜/2 Note that our arguments do not require that the time since we can certainly distinguish between the state at evolution u takes place on a smaller time scale than ∆t. the time t and at t+T˜/2. However, we conjecture that ThereasonisthatwehavearguedthatP :=u†(1 Q )u similarboundsasinTheorem1ontheentropygeneration j j ⊗ may formally be considered as projections of a measure- can be given for systems where the energy spectrum is ment performed before the interaction was switched on. essentiallysupportedbyaninterval[Emin,Emax]. There- Our results in Section II and III may be given an ad- fore our results suggest the following interpretation: ev- ditional interpretationas information-disturbancetrade- ery measurement that allows to estimate the time up to off relation. Lemma 1 shows that the entropy in- an error that is not far away from the Heisenberg limit crease caused by the measurement is the Kullback- producesanon-negligibleamountofentropy. Thefollow- Leibler distance between pre- and post-measurement ing example suggests that the necessary entropy genera- state. Information-disturbance trade-off relations are an tion may even be much above our lower bound if “time important partof quantuminformation theory [11]. Un- measurements” are used that are extremely close to the fortunately, Theorem 1 is restricted to von-Neumann Heisenberg limit. Consider the wave function of a free measurements. Theorem2extendsthestatementtogen- Schr¨odinger particle moving on the real line. A natural eral measurements as far as the disturbance of the total way to measure the time would be to measure its posi- state ρ γ of the “clock” and the ancilla system is con- tion. We may use, for instance, von-Neumann measure- t ⊗ sidered. However, this is interesting from the thermo- ments that correspondto a partitionof the realline into dynamical point of view taken in this article but not in intervalsoflength∆x. Assumeonewantstoimprovethe the setting of information-disturbance trade-off. In the time accuracy by decreasing ∆x arbitrarily. The advan- latter setting, the disturbance of the ancilla state that is tageforthetimeestimationissmallif∆xissmallerthan used to implement a non-von-Neumann measurement is the actual position uncertainty of the particle. However, not of interest. Therfore we admit, that our bound on if the state is pure it is easy to see that the generated the state disturbance does only hold for von-Neumann entropy goes to infinity for ∆x 0. → measurements. The following example shows that the entropy gen- eration in a time measurement can really go to zero when ∆E goes to infinity. Consider the Hilbert space IV. HOW TIGHT IS THE BOUND? := l2(Z) of square sumable functions over the set of iHntegers. Let j with j Z be the canonical basis vec- | i ∈ tors. Let the Hamiltonian be the diagonal operator Theentropyincreasepredictedbyourresultscannever be greater than 1/2. This can be seen by the following H j := j j . argument. Using the definitions of the proof of Lemma | i | i 3 we have Consider the state 5 1 k−1 two logical states 0 and 1 correspond to an orthogonal ψ := j . decomposition | i √k X| i j=0 = 0 1 By Fourier transformation is isomorphic to the set H H ⊕H L2(Γ) of square integrable fHunctions on the unit circle of subspaces of the Hilbert space of the composed sys- Γ and the dynamical evolution is the cyclic shift on Γ. tem. If and are non-isomorphic as Hilbert spaces 0 1 H H The period of the dynamics is T = 2π. In this picture, we extent the smaller space such that they are isomor- ψ is a wave package that has its maximum at the an- phic. Then we can assume without loss of generality g|lei φt = t with an uncertainty ∆φ in the order 1/k. to be of the form 0 C2 without assuming that thHe H ⊗ We assume to know that the true time is in the inter- bit is physically realized by a two-levelquantum system. val [0,2π) and construct a measurement that allows to The fact that the bit is classical will now be described decide for each t [0,π) whether the true time is t or by the fact that it is subjected to decoherence, i.e., that ∈ t+π with confidence that is increasing with k. We use all superpositions between 0 and 1 are destroyed and | i | i ameasurementwithprojectionsP1,P2,P3,P4 projecting changed to mixtures on a time scale that is not larger on the four sectors of the unit circle. If the angle uncer- than the switching time. Decoherence keeps the diago- taintyisconsiderablysmallerthanπ/2thismeasurement nalvaluesofthedensitymatrixwhereasitsnon-diagonal canclearlydistinguishbetweentandt+πwithhighcon- entries decay. If the process is a uniform time evolution, fidence. Hence, for k large enough, we get ∆t = π as i.e., given by a semi-groupdynamics, decoherence of the time resolution of the measurement. Note that we have two-levelsystemisgivenbyanexponentialdecayofboth only non-negligible entropy generation at that moments non-diagonalentries: where the main part of the wave packet crosses the bor- der between the sectors, i.e., when there are two sectors γt :=γ00 0 0 +exp( λt)γ01 0 1 | ih | − | ih | containinga non-negligiblepartofthe wavefunction. If, + exp( λt)γ 1 0 +γ 1 1 , 10 11 for instance, each sector contains about one half of the − | ih | | ih | probability,we generate the entropy one bit, i.e., the en- whereγ arethecoefficientsofthe densitymatrix. Note ij tropy ∆S =ln2 in natural units. thatitisassumedthattheeffectoftheenvironmentdoes The probabilitythatthe measurementis performedat notcauseanybit-flipsinthesystembutonlydestroysco- a time in which non-negligible entropy generation takes herence. The parameter λ 0 defines the decoherence ≥ place is about 2∆φ/π. The average entropy generation rate. Let P and P be the projections onto the states 0 1 decreases therefore for increasing k, i.e., for increasing 0 and 1 ,respectively. Thedecoherenceprocesscanbe ∆E = k 1. Let pk(t) for j = 1,...,4 be the probabil- |simi ulate|diby measurement processes that are performed − j ities for the 4 possible outcomes when the wave packet at randomly chosen time instants. This analogy is ex- with energyspreadk 1is measuredatthe time instant plicitly given as follows: Let − t. There are four moments where the maximum of the wave packet is exactly on the border of the sectors. The G˜(γ):=P0γP0+P1γP1 probability to meet these times is zero. For all the other times t there is one j such that 1 pk(t) tends to zero be the effect of the measurement that distinguishes the − j two logical states. Then we have with O(1/k2) by standard Fourier analysis arguments. Note thatameasurementattime t producesthe entropy γ = lim ((1 λ/n)id+λG˜/n)n(γ)=exp(λt(G˜ id))(γ). t n→∞ − − ∆S = pk(t)lnpk(t). − j j Thesecondexpressionprovidesthefollowingintuitiveap- X j proximationoftheprocess: Ineachsmalltimeintervalof length 1/na measurementis performedwith probability We conclude by elementary analysis that ∆S tends to zerowith O((lnk)/k2) for alltimes t except from4 irrel- λ/n. Let G := 1 G˜ be the extension of G˜ to the total ⊗ system. We assume that the dynamical evolution of the evant values t and therefore the average entropy genera- tion tends to zero with O((lnk)/k2). Note that the de- total system is generated by creaseofthelowerboundofTheorem1isasymptotically F :=i[.,H]+(G id), (2) alittlebitfastersinceitisO(1/k2)(dueto∆E =k 1). − − i.e., the decoherence of the bit is the only contact of the system to its environment. Define the switching time as V. CONTROLLING A CLASSICAL BIT SWITCH thelengthofthetimeinterval[0,∆t]wheretheprobabil- BY A MICROSCOPIC CLOCK ity of one of the logical states changes from 1/4 to 3/4. Thereasonthatwedonotassumeittoswitchbetween0 Nowweconsiderthesystemconsistingoftheclock,its and1 is thatthis is impossible within afinite time inter- environmentandtheclassicalbit. Forthemomentweig- val with a Hamiltonian that has limited spectrum. This nore the fact that the bit is quantum and claim that the can be seen by the fact that all expectation values with 6 Hamiltonians of lower (!) bounded spectrum are analyt- p˙ (t)=tr(P F(ρ ρ˜))=tr(P i[ρ ρ˜,H]) j j t t j t t − − ical functions [13]. Consider the case that the decoher- ence time is small compared to the switching time ∆t, due to tr(P (G id)γ) = 0 for every state γ. It should j i.e., λ∆t 1. Then the probability to have more than be emphasized t−hat this argument would fail if the two- ≪ one measurement during the switching process is small level system was not only subjected to decoherence but (of second order in λ) and the probability that one mea- also to relaxation. If L is the generator of a relaxation surement occurs is about λ∆t. Given that it occurred, process that causes directly transitions from the state theprobabilitiesofitstimesofoccurrenceisequallydis- 1 to the state 0 the equation tr(P Lγ) = 0 does not j tributed in [0,∆t]. Since entropy is convex the entropy |hoild, i.e., L wou|ldinot be irrelevant for our proof. This generated by the switching process is at least λ∆t times is consistent with the observation in Section I that the the entropy that is generated if a measurement has oc- “relaxationclock”canbereadoutwithoutgeneratingen- curred during the switching process. Also by convexity tropy by the measurement. We conclude that the bound arguments, we conclude that the entropy generated by (3)holdsforarbitrarilyhighdecoherencerate. Formally, the process “performa measurementat a randomlycho- the bound predicts infinite entropyproduction if ∆t and sen(unknown)timeinstantin[0,∆t]”isatleasttheaver- ∆E are constant and λ . However, this asymp- ageentropygenerationifthe time instantis known. The totics ismeaninglessdue t→o th∞e quantumzeno effect(see latter situation meets exactly the assumptions of Theo- e.g. [14]). Infinite decoherence would stop the switch- rem 1 with the following parameters: Set T := ∆t, i.e., ing process completely. Consider a short time t after a assumethatthepriorinformationisthatthetruetimeis measurement has been performed on the state. Then in [0,∆t). Furthermore the switching time ∆t coincides the derivative of the probabilities p corresponding to a j with the time resolution ∆t of Theorem 1 since reading second measurement is only of the order ∆Et. The es- out the logical state can distinguish between the times 0 sential consequence is that in particular for small ∆E and ∆t with error probability at most 1/4. Taking into when our bound would predict large entropy generation accountthatameasurementoccursonlywithprobability fast switching processes are impossible. λ∆t we multiply the bound of Theorem 1 by this factor In order to show how to apply our bound to realistic and obtain: situations consider the following example: A light signal λ∆t ~ 2 λ~2 (guidedbyanopticalfiber,forinstance)issenttoanap- ∆S = . (3) paratus that contains a two-level system. The incoming ≥ 2 (cid:16)∆t∆E(cid:17) 2∆t(∆E)2 signal triggers the transition 0 1 from the lower to | i7→| i the upper level (see Fig.1). Note thatwedo notspeak aboutaverage entropygen- eration since we consider (in contrastto Theorem1) the entropy of a single density matrix which is already an averageoverasetofdensitymatriceswhichareobtained light pulse when the measurement occurs at different times. Consider now the case that the decoherence time is so small that more than one measurement during the two-level system with switching processis likely. Thenwe haveto worryabout optical fiber decoherence thepost-measurementstateandwhetheritsenergyspec- trum is still bounded. If not, we cannot apply our argu- FIG.1. Classicalbitswitchcontrolledbyalightpulse. For ments. If it is, we can use the bound (3) nevertheless. time t→−∞ the time evolution is approximatively the free This is less obvious than it may seem at first sight. The evolution of the light field without any interaction with the totaldynamicsisassumedtobethesemi-groupevolution apparatus. ρt :=exp(Ft)(ρ) The apparatus may be a huge physicalsystem and we are only interested in the fact that the transition time with F as in eq. (2). Now we investigate the amount is given by the incoming light signal, the energy may be of entropy generated by a measurement at time t. Note providedbytheapparatusitself. Assumethelightsignal that we do not have a lower bound by Theorem 1 since consists of photons with frequency bandwidth ∆ω. The the dynamical evolution leading to the state ρt was not signal is assumed to contain at most k photons. Than Hamiltonian. However,the bound the energy bandwidth of the signal is at most ∆ωk. For t thetimeevolutionofthetotalsystemisevolving p˙(t) 2 →−∞ ∆S k k1 approximatively as ≥(cid:16) ∆E (cid:17) holds nevertheless. This is seen by the observation that ρt γ ⊗ the generator G is irrelevant if the proof of Lemma 3 should be converted to the situation here. With ρ˜ := whereγ isthe(stationary)stateoftheapparatus. When t P ρ P we have thesignalmeetstheapparatusthetimeevolutionisgiven j j t j P 7 byanunknownHamiltonianH ofthetotalsystemcom- [p,p+∆p]. Let ρ be the particle’s density matrix and c bined with the decoherence of the two-levelsystem. The ρ the state translated by x R. Then we conclude x ∈ energy bandwidth of the total system according to H that every measurement that is suitable to distinguish c is unknown. The “measurement” on the two-level sys- between ρ and ρ in the sense of Theorem 1 pro- x x+∆x tem takes definitely place at a moment where the to- duces at least the entropy (~/(∆x∆p))2/2. Note that tal evolution of the system is generated by the unknown the measurement is not necessarily a position measure- Hamiltonian H . Nevertheless we can apply the bound ment, it has noteven to be compatible with the position c of Theorem 2. In Section III we have argued that The- operator. Inthissense,ourresultsmaybeinterpretedas orem 2 does not assume that the measurement interac- a kind of “generalized uncertainty relation”. tion was only switched on during a time interval that is Another very natural application is to consider the small compared to the time resolution ∆t. We observed group of rotations around a specific axis. Consider a that the measurement can formally be considered as a spin-k/2 particle and the group of rotations measurement that had been performed before the inter- actionhadbeenswitchedon. Analogously,wearguethat (exp(iLzα/~))α∈R the unknown interaction between light field and appara- tus is irrelevant: Let σt be the state of the total system onits HilbertspaceCk+1,whereLz isthe operatorofits at the time t. Let the switching process happen during angular momentum in z-direction. We have ∆L = k~ z the interval [0,∆t]. Let (ut)t∈R be the time evolution of and conclude the following: Eachmeasurement that dis- the total system implemented by a possibly unbounded tinguishes between ρ and ρ with error probability α α+∆α Hamiltonian. Theentropygeneratedby“measurements” at most 1/4 produces at least the entropy 1/(2(k∆α)2). P performed on the state σ during this period due to j t the decoherence is the same as produced by u†P u per- s j s formed on the state u σ u†. For s the family of states (u σ u†) s tevsolve appro→xim−a∞tively like the VII. CONCLUSIONS s t s t∈[0,∆t] free evolutionρ γ of the light field. We conclude that t ⊗ the bound in eq. (3) can be applied and the relevant en- Our bound on the entropy that is generated when in- ergy bandwidth is the energy bandwith of the free light formationabout the actual time is extracted from quan- field. The entropy generation is at least tumsystemholdsonlyforHamiltoniantimeevolution. A simple counterexample in Section I has shown that time λ . readout without state disturbance is possible for some 2∆t(n∆ω)2 dissipative semi-group dynamics. This leads to an in- Statementsofthiskindmayberelevantinfuturecom- teresting question: In physical systems, each dissipative putertechnologywhenminiaturizationreducessignalen- semi-group dynamics that is induced by weak coupling ergies on the one hand and requires on the other hand to a reservoir in thermal equilibrium is unavoidably ac- reduction of power consumption. Our results show that companied by some loss of free energy. This shows that reductionofsignalenergydowntothelimitoftheenergy- the loss of free energy caused by the time measurement timeuncertaintyprincipleleadsunavoidablytoheatgen- can only be avoided by systems that loose energy dur- eration as long as the signal control classical bits. The ing its autonomous evolution. It would be desirable to entropy generation ∆S leads to an energy loss of ∆SkT know whether there is a general lower bound (including (where k is Boltzmann’s constant and T is the absolute dissipative dynamics)on the total amountof free energy temperature) due to the second law of thermodynamics. that is lostas soonas timing informationis convertedto classical information. VI. GENERALIZATION TO OTHER ONE-PARAMETER GROUPS ACKNOWLEDGMENTS Obviously, our results generalize to other one- parameter groups since the proofs do not rely on the Thanks to Thomas Decker for helpful comments. interpretationoftheunitarygroupexp( iHt)asthesys- This work has been supported by grants of the DFG − tem’s time evolution. project“Komplexit¨atund Energie”of the “Schwerpunk- Consider for instance the case that the momentum of tprogramm verlustmarme Informationsverarbeitung”Be a Schr¨odinger wave package is restricted to the interval 887/12. 8 [1] M.KoashiandN.Imoto.Whatispossiblewithoutdisturbingpartiallyknownquantumstates? quant-ph/0101144v2, 2002. [2] D. Janzing and Th. Beth. Are there quantum bounds on the recyclability of clock signals in low power computers? In Proceedings of the DFG-Kolloquium VIVA,Chemnitz, 2002. quant-ph/0202059. [3] D.JanzingandT.Beth.Quasi-orderofclocksandtheirsynchronismandquantumboundsforcopyingtiminginformation. to appear in IEEE Trans. Inform. 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