Does Information Have Mass?

Transcription

1 P O I N T O F V I E W Does Inforation Have Mass? By LASZLO B. KISH Departent of Electrical and Coputer Engineering, Texas A&M University, College Station, TX USA CLAES G. GRANQVIST Departent of Engineering Sciences, The Ångströ Laboratory, Uppsala University, SE Uppsala, Sweden electron involved in signal transfer carries one bit of inforation [2], which akes the corresponding ass to be about 10 5 g. The difference between the two ass estiates is partially caused by the low energy efficiency of classical coputers [3] (note that the efficiency of quantu coputers is even worse [4]). We will address the fundaental question of iniu ass related to a bit of inforation fro the angles of quantu physics and special relativity. Our results indicate that there are different answers depending on the physical situation, and soeties the ass can even be negative. We tend to be skeptical about the earlier ass estiations, entioned above, because our results indicate that the electron s ass does not play a role in any one of the. Finally, in a seeingly related but actually different and independent context of easured inaccuracies of the gravitation constant, we refer to experients on the weight transient of eory devices after inforation writing and erasure. Does inforation have ass? This question has been asked any ties and there are any answers even on the Internet, including on Yahoo Answers. Usually the answer is no. Attepts have been ade to assess the physical ass of inforation by estiating the ass of electrons feeding the power-guzzling coputers and devices aking up the Internet [1], the result being around 50 g. Other efforts to calculate the ass of inforation have assued that each Digital Object Identifier: /JPROC I. MASS OF THE INFORMATION CARRIER DURING COMMUNICATION A. Single Electrons as Inforation Carriers Electrons have a finite rest ass of e ¼ kg, which is their ass at zero velocity. This is an Ó 2013 IEEE Vol. 101, No. 9, Septeber 2013 Proceedings of the IEEE 1895

2 extreely sall nuber, but it is infinitely larger than zero which is the sallest ass a photon theoretically ay have. B. Single Photons One cannot store inforation with photons, but it is possible to counicate inforation with the, either throughanopticalfiberorviafree space. In an exploration of the fundaental lower liits of the transferred ass related to inforation, it is, therefore, ore proper to use photons than electrons. In an ideal case, a single photon can carry one bit of inforation, for exaple, by using the two orthogonal polarization odes to represent the two bit values. According to quantu echanics, the energy E p of a photon is given by Planck s forula which states that E p ¼ hf p (1) where h ¼ Js is Planck s constant and f p is the frequency of the photon. The relativistic ass p of the photon is given by Einstein s faous equation E p ¼ p c 2 (2) where c ¼ /s is the speed of light. Equations (1) and (2) then yield that p ¼ h c 2 f p ¼ f p (kg): (3) So if a single optical photon with E p 1eV J carries one bit of inforation, the ass of the carrier of the counicated bit is p kg, which is about a illion ties less than the electron s rest ass. Equation (3) seeingly indicates that the ass of the inforation carrier can approach zero if one uses photons in the low-frequency liit where f p! 0. However, this is not so even if one were able to easure single photons of low frequency, and the reason is twofold: one ay neither disregard noise (i.e., the theral radiation background) nor the uncertainty principle. C. Photons With Theral Radiation Background Each photon ode has two therodynaic degrees of freedo. In accordance with Boltzann s energy equipartition theore, the ean theral energy E t in a photon ode is E t ¼ kt (4) where k is Boltzann s constant and T is the absolute teperature of the environent. Equation (4) is valid in the classical physical liit of therodynaics where hf p kt (5) and describes the theral radiation background, which represents the theral noise of the electroagnetic field. This noise energy is available for each ode, including the polarization odes. In order to have a sufficiently strong signal and high signal-to-noise ratio (SNR) to achieve a sall bit error probability (BEP), the energy representing the bit value ust be uch greater than the corresponding noise energy, which eans that kt ¼ E t E p ¼ hf p : (6) Equation (6) obviously contradicts (5), which is an indication that single photons cannot be used with high SNR and low BEP in the classical physical liit of therodynaics. On the other hand, the assuption f p! 0, which is needed for the photon ass p to approach zero, iplies the classical liit in (5) for any fixed teperature T. To avoid this contradiction in the classical liit, one ust stay away frousingsinglephotons. Looking now at a nuber N p of photons, the requireent of having uch greater signal energy than noise energy yields kt N p E p ¼ N p hf p : (7) If this is translated to the ass w of a bit-carrying classical electroagnetic wave, one gets a frequencyindependent result according to w kt c 2 (8) which corresponds to w kg at roo teperature and for zero-decibel SNR. D. The Uncertainty Principle Suppose that the bits ust be sent with extraordinarily high speed (which is not typical in today s optical counication owing to liitations in light sources and detectors). Then, the energy-tie uncertainty principle poses a new liitation which reads DEDt h 2 (9) where the energy uncertainty DE is the root ean square (RMS) error in the energy easureent, the bit duration Dt is the width of the tie window for generating the photon or photon package with the required bit value, and h ¼ h=2. Expression (9) can be written DE h=ð2dtþ, andthe ass uncertainty of the counicated bit is D b h 2c 2 Dt : (10) Expression (10) becoes significant only when the ass uncertainty is larger than the ass related to the 1896 Proceedings of the IEEE Vol. 101, No. 9, Septeber 2013

3 counicated bit, because zero is the low liit of the resultant ass. A bit duration of one picosecondv corresponding to a clock frequency of 1000 GHz in a serial photonic one-bit counication portvyields D b kg: (11) The latter result indicates that, with a theral background corresponding to roo teperature, the effect of the uncertainty principle would begin to increase the ass of the bit only at bit frequencies exceeding 10 5 GHz for the case of a serial one-bit counication line. Finally, one should note that it is possible to counicate without eitting any signal energy into the counication channel, naely, by odulating the statistical properties of its theral noise or quantu noise [5]. But even in this case the noise ediu carrying the inforation has a ass. II. MASS OF STORED INFORMATION Mass related to inforation storage is different fro the proble studied above, because this ediu has ass even without storing any inforation. The question to address is this: How large is the ass change per bit of the storage ediu if one stores inforation bits there? The sae question could have been asked about the photonics-based counication discussed before, but then the ass is independentofthebitvalueandthe inforation content unless the odulationissofastthatitintroducesa signal bandwidth coparable to or greater than the frequency of the photons. We will see next that the answer to the question concerning inforation storage can even be negative ass, depending on the definition of the erased state. We note in passing that two irrelevant ters soeties pop up on Internet discussions sites: entropy Fig. 1. (a) Syetric and (b) asyetric double-well-potential eories. The two bit values correspond to (a) identical and (b) different energies, respectively. and ass of electrons that charge up a solid-state (flash) eory. A calculation of the therodynaic entropy does not give any clue to the correct answer, however, because the increase of entropy is relevant for the energy dissipation (which is then released to the environent) rather than for the change of the energy and ass of the syste. Siilarly, the electron ass does not play any role in these eories because they operate with charge neutrality (i.e., their total charge is always zero, iplying that the total nuber of electrons is constant at any inforation content). Siilarly to any capacitor or etal oxide seiconductor (MOS) transistor in a circuit, the charges at the surface of the gate capacitance are always copensated by an equal aount of opposite charges at the other electrode of this capacitor, which is the channel between the drain and the source. Otherwise, a pen drive containing certain inforation would act like an insulating body charged up to a voltage of V; it would attract dust and sall pieces of papers and ake one s hair stand up when being in its vicinity. To gain a physical forulation of inforation storage, we first consider the erased state of the eory. In the case of noral erasure, oneonlyalters the address identifying the free eory, and the old inforation is not deleted but siply readdressed so that it belongs to the free eory, where it will eventually be written over by new inforation. Therefore, we discuss secure erasure and suppose that 1) all bits have zero value in the (secure-) erased eory; furtherore, we assue that the bits do not interact and ask the question: 2) what is the energy difference between the stored values (0 or 1) of a physical bit? Fig. 1(a) shows a diagra of the potential energy for a single-bit eory based on a syetric double-well potential. A agnetic eory cell, for exaple, can approxiate this situation. The height E 1 of the potential barrier characterizes the energy that ust be invested (dissipated) to change the bit value. Flips corresponding to 0! 1 and 1! 0 require the sae lower liit of energy E 1. Fig. 1(b) illustrates a potential energy diagra of a single-bit eory based on an asyetric doublewell potential. A floating-gate etal oxide seiconductor field-effect transistor (MOSFET), which is the flash-eory eleent, is an exaple of this situation (although the energy levels for values 0 and 1 ay be inverted). At one of the bit values, the floating gatevwhich is ebedded in an insulating oxide or nitridevust be charged up by a quantu-tunneling current via the insulator in order to keep the channel of the floating-fieldeffect transistor in the required state. Vol. 101, No. 9, Septeber 2013 Proceedings of the IEEE 1897

4 In Fig. 1(b), the charged position is represented by the high-energy state 1 and the discharged position is represented by the low-energy state 0. The height E 1 of the potential barrier characterizes the energy that ust be invested to trigger the bit-value change 0! 1, and E 2 is needed for the 0! 1 flip. These potential barriers represent theral activation in the two directions. (Note that energies that are uch larger than E 1 or E 2 are used in practice to facilitate highspeed operation via quantu tunneling, and thus the actual energy dissipation is greater than indicated in our siplified analysis.) What is the iniu energy barrier E 1 needed in these devices in order for the not to forget their inforation? Supposing that nothing but theral activation leakage takes place, one gets [6], [7] E 1 > kt ln 1 t " (12) where t is the lifetie of the eory, " 1 is the probability of bit error during this lifetie, and is the correlation tie of the theral fluctuations driving the theral excitation. Assue that the eory contains N bits and that the goal is that the probability of even a single bit error should be negligible during the eory s entire lifetie. Then, the relation " 1=N ust hold, which leads to E 1 > kt ln N t : (13) For a one-terabyte solid-state drive with ten years lifetie and ¼ s, one obtains E Jatroo teperature, which corresponds to 75 kt. Though this energy is dissipated by the writing operation during changing the state, the energy and the ass of the syste after the operation reain the sae as their original values. On the other hand, the energy change of the asyetric syste in Fig. 1(b) is characterized by DE ¼ E 2 E 1,whichistheenergyneeded to keep the MOSFET channel closed (or open, depending on the type of MOSFET). The BEP of that MOSFET channel is controlled by an expression that is analogous to the one for E 1 in (13) [3], and it follows that the energy change DE of the syste ust satisfy DE > kt ln N t : (14) In conclusion, the iniu ass change of a one-bit flash eory, when flipped into bit value 1, is f kt c 2 ln N t (15) where Einstein s relation in (2) has been used. This yields f kg at roo teperature. Coincidentally, the absolute value of the ass in (15) is very siilar to that of a counicated quantu bit at the chosen conditions. However, the ass in (15) is negative if one defines the erased eory eleent to be a MOSFET with charged gate. III. EXPERIMENTS: NEGATIVE WEIGHT TRANSIENTS OF MEMORIES Experients devised to copare the weight of inforation storage edia before and after recording/erasure have been carried out to 10 8 kg accuracy by use of a precision balance, and significant differencesvthe order of 10 6 kgvhave been found [8], [9], as discussed below. The original goal of these experients was not to copare weights, because the considerations in the section above ake it evident that the expected differences are too sall to be easurable. Instead the weight transients were discovered during tests [8] inspired by explorations of a hypothetical fifth force that conceivably copleents electrostatic, strong, weak, and gravitational interactions [10]. The reason behind the quest for a fifth force is the extraordinary inaccuracy of the Newtonian gravitational constant G, and that specific aterials and environents see to influence easureents that theoretically should offer excellent accuracy. The fifth force hypothesis was forwarded by Fischbach et al. in the 1980s but was found untenable. Nevertheless, it stiulated several laboratories to carry out easureents during the 1990s with ephasis on enhancing the accuracy of G [11]. These efforts backfired, however, and the new and supposedly iproved easureents yielded even greater inaccuracies than before for the short-range value of G [11]. In 1999, the Coittee on Data for Science and Technology (CODATA), under the International Council for Science, decided [11] to increase the uncertainty of G fro 128 to 1500 pp (0.15%). The original 0.01% accuracy was reclaied the next year, but even this uncertainty is orders of agnitude beyond that of the acceptable level for other physical constants. The first author (L. B. Kish) hypothesized that the fifth force ight be due to soe short-range interaction between structures with siilar inforation patterns, shared aong the test bodies, and their environent [8], [9]. Such an interaction goes beyond today s physics. Tests were nevertheless executed but no easurable interaction force, at a resolution of 10 7 N, was found between two identical 4-GB flash drives filled with identical noise sequences [8], [9]. However, a serendipitous result eerged fro these efforts to explore the fifth force, naely, negative weight transients [8], [9] of the order of 10 5 N (corresponding to a weight of about 1 g) during the recording or secure-erasing inforation (noises and periodic signals) in the gigabyte rangeonflash,cd,anddvdedia. The relaxation of the weight transient was nearly exponential, with tie constants ranging fro a few inutes to 15 in [8], [9] Proceedings of the IEEE Vol. 101, No. 9, Septeber 2013

5 An elevated teperature of the storage ediu after recording/erasure could cause a lifting force due to heat convection flow and the ensuing Bernoulli force [8], but the observations were inconsistent with theral relaxation tie constants, which are uch shorter than those observed. For exaple, the weight transients of the DVD disks typically have order-ofagnitude longer tie constants than expected for theral relaxation. Another conceivable explanation for the weight changes is the loss of stored oisture in hygroscopic parts of the edia during recording, and a corresponding ass relaxation when oisture is regained. However, the observed dependence of the tie constant on the huidity of abient air does not support this assuption, and neither are other siple explanations satisfactory, such as those involving a changed Archiedes force due to theral expansion (which is a negligible effect). We ephasize that the experients were done carefully to avoid artifacts of shocks to the precision balance during placing/reoving eories as well as direct contact of the balance plate with the war flash drives. Tests against artifacts included repeated placing of the sae flash drive without inforation recording to ascertain that the values were reproducible and tie independent in the absence of inforation being written into the edia. In conclusion, weight transients observed after writing or secureerasing inforation into a eory have not been satisfactorily explained. Experients with better control of the abient conditions, for exaple, by use of vacuu, would be desirable. Finally, we note that the observed effects offer a security application, and weight transients conceivably could serve as a soking gun to evidence that a storage ediu has been in recent use. h REFERENCES [1] R. Seitz, Weighing the web, ADAMANT, Apr. 9, [Online]. Available: adaant.typepad.co/seitz/2007/04/ weighing_the_we.htl. [2] S.Cass, Howuchdoestheinternet weigh? Discover Mag., Jun.2007.[Online]. Available: /jun/how-uch-does-the-internetweigh. [3] L. B. Kish, Moore s law and the energy requireent of coputing versus perforance, Proc. Inst. Electr. Eng.VCircuits Devices Syst., vol.151, pp , [4] J. Gea-Banacloche and L. B. Kish, Future directions in electronic coputing and inforation processing, Proc. IEEE, vol. 93, no. 10, pp , Oct [5] L. B. Kish, Stealth counication: Zero-power classical counication, zero-quantu quantu counication and environental-noise counication, Appl. Phys. Lett., vol. 87, 2005, Article [6] L.B.KishandC.G.Granqvist, Energy requireent of control., Europhys. Lett., vol. 98, 2012, Article [7] L. B. Kish and C. G. Granqvist, Electrical Maxwell deon and Szilard engine utilizing Johnson noise: Measureent, logic and control, PLoS ONE, vol.7,2012.[online]. Available: info:doi/ /journal.pone [8] L.B.Kish, Gravitationalass of inforation? Fluct. Noise Lett., vol.7, pp. C51 C68, [9] L. B. Kish, Weight fluctuations of inforation storage edia, presented at the 5th Int. Conf. Unsolved Probl. Noise, Lyon, France, Jun. 2, [Online]. Available: ens-lyon.fr/en/downloads_files/kish.pdf. [10] E. Fischbach and C. Taladge, 6 years of the 5th force, Nature, vol.356, pp ,1992. [11] F.Nolting,J.Schurr,S.Schlainger,and W. Kundig, Deterination of the gravitational constant G by eans of a bea balance, Europhys. News, vol.31,pp.25 27, Vol. 101, No. 9, Septeber 2013 Proceedings of the IEEE 1899