Photonic Communications and Quantum Information Storage Capacities

Transcription

1 Oti and Photoni Journal, 13, 3, doi:1436/oj133b3 Publihed Online June 13 (htt://wwwirorg/journal/oj) Photoni Communiation and Quantum Information Storage Caaitie William C Lindey Univerity of Southern California, Deartment of Eletrial Engineering, Lo Angele, CA, USA wlindey@gmailom Reeived 13 ABSTRACT Thi aer reent hotoni ommuniation and data torage aaitate for laial and quantum ommuniation over a quantum hannel Thee aaitie rereent a generalization of Shannon laial hannel aaity and oding theorem in two way Firt, it extend laial reult for bit ommuniation tranort to all frequenie in the eletromagneti etrum Seond, it extend the reult to quantum bit (qubit) tranort a well a a hybrid of laial and quantum ommuniation ature limit on the rate at whih laial and/or quantum information an be ent error-free over a quantum hannel uing laial and/or quantum error-orreting ode are reented a a funtion of the thermal bakground light level and Eintein zero-oint energy Grahial reult are given a well a numerial reult regarding ommuniation rate limit uing Plank natural frequeny and time-interval unit! Keyword: Quantum Communiation; Quantum Information Storage; Quantum Error-orreting Coding; ature Photoni Limit 1 Introdution Photoni modulation an be ued, reetively, to reliably tranort laial information bit a well a quantum information qubit, ee Figure 1 Uing the eondquantization of the eletromagneti field, quantum mehanial model for oherent hotoni tate and Shannon here-aking argument, the quantized analog of Shannon laial hannel aaity and oding theorem i derived when laial or quantum information bit are tranorted over a quantum hannel Uing thi reult, the unit information metri between a laial bit and a quantum bit, the qubit, i etablihed from whih the quantum hannel aaity and etral effiieny, quantum information torage denity and quantum information torage aaity are develoed It i hown that the quantized aaitie (ignal energy direte) redue to Shannon laial reult when the energy in the field i aumed to be ontinuou and the hannel enter frequeny Hz/ K i le than the artitioning frequeny Hz/ K, ie, 4 k / h Hz/ K, where k i Boltzman and h i Plank ontant ature limit on the rate at whih laial and quantum information an be ent error-free over a quantum hannel uing laial and/or quantum error-orreting ode are reented a funtion of the thermal bakground light level and Eintein zero-oint energy (ZPE) For ytem engineering deign, numerou grahial reult are lotted for both the quantized and quantum hannel aaitie, etral effiienie and the information torage aaitie The reult demontrate the feaibility of Terabit er e to Petabit er e data rate and Petabyte information torage aaitie of /ln bit/ hoton or one qubit er hoton In thi regard, it i hown that the qubit information unit equal two nat/qubit or /ln bit/qubit, ie, 1 qubit = nat = /ln bit! Finally, it i hown that error-free quantum ommuniation an be aymtotially aroahed in a wideband ritine environment uing a minimum of 345 hoton er bit or one hoton er qubit In a low temerature environment, it i hown that laial or quantum information, torage denity and ommuniation aaity do not deend uon energy but uon the ratio of two integer, viz, the ratio of the number of hoton er meage, to the number of dimenion er meage,, or equivalently, the interiale hoton time bandwidth rodut BT By etting a fundamental bandwidth limitation on the quantum hannel bandwidth B uing Plank natural frequeny and time interval unit at boundary BT 1, it i hown that Plank quantum ommuniation aaity aroximately equal 143 qubit/e or (9)143 bit/e It i further hown that there exit a quantum error orreting ode that ahieve zero MEP if and only if the ode rate R ln E Z ln where Z h / i Eintein zero-oint energy level From thi we obtain Coyright 13 SiRe

2 13 W C LIDSEY the energy er bit to noie ondition E / Z ln, or equivalently, bit rate R b S / Z ln for all [ B, ] Thi omare with the laial reult of Shannon where E / Z ln and R b S / ln for Finally, information quanta are identified and related to Plank Eintein energy quanta Sytem Funtional Arhiteture In thi etion, funtional arhiteture for quantizedlaial and quantum ommuniation ytem are reented along with ytem arameter and erformane metri Sytem arameter inlude: time interval, hannel bandwidth, bit and qubit ignal energie and aoiated thermal bakground light level Performane metri inlude: hannel aaitie, information torage denitie and information torage aabilitie Relationhi onneting thee erformane metri are etablihed together with thoe that relate quantum aet to their laial ounterart 1 Claial-Quantum Communiation Sytem Model For tranmiion, laial bit are enoded into odeddigit uing a laial [, K] error-orreting ode; ee Figure 1 More eifially, aume Mequirobable meage ontaining K log M bit er meage, ee Figure 1 Eah meage i aumed to lat for T (log M) T b e er meage; Tb i the time er bit Eah K-tule i enoded into oded-digit uing a laial [, K] errororreting ode of ode rate R K (log M) bit er dimenion Eah oded digit lat for T e uh that T T eond er meage are ued to modulate the olarization of a oherent hoton oure of 1 rate T hoton er eond and enter frequeny B, Hz at ode rate R log M bit er hoton; here i the average number of ignal hoton in eah meage Thu every T e an -dimenional hotoni ignal ontaining energy E E h i tranmitted with energy er qubit E h Here h i Plank ontant Conider now the extended laial-quantum ytem hown in Figure 1 Here a laial error-orreting ode [, K] i onatenated with a quantum error-orreting K, ode ontaining K qubit er meage with quantum ode rate Q h K dimenion er qubit The notation K, imlie an -dimenional quantum ode roteting K -qubit Sine the ytem in Figure 1 ontain both laial and quantum ubytem, we refer to thi ytem a a hybrid ytem In thi regard, we an write the time interval-oding equation for Figure 1, viz, T KT (log log M) T T K T T (1) b b where T i the time er qubit and K K Dividng both ide of (1) by log M T, the bit rate q an be related to the quantum ode rate R / K dimenion er qubit, the quantum-modulator rate h QM K qubit er hoton The ombined tranmit ode rate R R q h RQM i related to the hannel hoton rate and bit rate, ie, h h R R Q Q bit / e () b M Figure 1 Quantum ommuniation ytem funtional arhiteture with onatenated laial and quantum error orreting ode Coyright 13 SiRe

3 W C LIDSEY 133 Furthermore, (1) and () allow u to onnet all rate to the number of laial meage R b R T K T R M in the hybrid ytem tranmit alhabet The variou energy akage are related to the energy er meage E to the energy er hotoni qubit E h through E (log M) E b E E h (4) where E b i the energy er bit and E i the energy er dimenion The orreonding ower-energy relationhi i log b (3) E ST M ST ST K S ST (5) where S watt i the average ower er meage From (5), we note that the number of hoton er bit (qubit/bit) 1 i given by the ratio Pb Eb E R whih may be ued a a meaure of the energy effiieny of a quantum ommuniation ytem to tranort laial information The time interval-oding equation, ode rate, energy-ower relationhi and the alhabet ize will be ueful when the erformane metri of the quantizedlaial ytem are omared to their quantum ounterart of Figure 1 In artiular, the quantized hannel aaity C bit er e and C bit er dimenion (etral effiieny in (bit/e)/hz) and the quantized information torage aaity C bit er hoton are related to the average information tored in bandwidth W Hz through I CT C C bit / meage (6) The quantum ounterart are the quantum hannel aaity of Q qubit er eond and Q qubit er dimenion and the quantum information torage denity Q qubit er e are related to the average quantum information I Q torage in bandwidth W Hz through IQ QT Q Q qubit / meage (7) With all arameter of our ytem model defined and onneting relationhi introdued, we are in a oition to reent the quantized laial aaitie, the quantum aaitie, their onnetion and nature limit regarding error-free tranmiion Before doing o, we reent the quantum hannel model 3 Quantum Communiation Channel Model From elementary quantum mehani, the vibrational tate of an atomi harmoni oillator have energie that deend on frequeny Enn 1/h, n,1, and the robability of finding the oillator in vibrational tate n i P n ex ex n [1 ex( )] (8) where kt / h i the natural frequeny of the quantum hannel, k i Boltzmann ontant and T i the bakground temerature in degree Kelvin Thu uing thi notation one an how that [3] E oth Z oth Z where kt ejoule haraterize the energy level of thermal noie defined in laial ytem and Z h / In limit a ν/ν aroahe zero, E while limit a ν/ν aroahe infinity, E Z ; Z i Eintein zero-oint energy (ZPE) found in quantum mehani where all thermal energy in the bakground light vanihe We will ue thi ondition to artition the eletromagneti etrum into a laial region and a quantum region, ee Figure In the laial region / / 4 and tanh x x In thi region, E and energy may be treated a a ontinuou variable (hoton energy level are mall and infiniteimally loe together) while for / / 4 we may onider thi to be the quantum region For / 4, we will how that all quantized aaity reult redue to Shannon laial reult [1] Figure 3 deit the notion of our quantum hannel of bandwidth W B Hz, note 4 At room temerature, T3K and 5 THz Further, a T aroahe zero, aroahe zero and all thermal energy vanihe By letting h aroah (9) Figure Partitioning the laial and quantum region Coyright 13 SiRe

4 134 W C LIDSEY Figure 3 Quantum hannel onet zero (or aroahe infinity), all quantum mehanial effet are eliminated and the hannel model redue to the laial white noie model 4 Quantized Shannon Communiation Channel and Information Storage Caaitie We are now in a oition to develo the quantized verion of Shannon laial hannel aaity for all frequenie in the eletromagneti etrum We will how that the quantized reult redue to the ontinuou energy ae of Shannon in the frequeny region where Baed uon the quantum mehanial reult derived in [] and the ue of Shannon here aking argument [1], there exit a laial [, K] ode for the ytem, Figure 1, and a onatenated laial ode [, K] with a hybrid quantum error-orreting ode K, for the ytem of Figure 1, uh that the meage error robability (MEP) an be made arbitrarily mall when the number of bit in M equally meage, are le than the average information torage I, ie, log log u 1 M I I dbit / meage W (1) l On the other hand, for log M I, then the MEP aroahe one for all ode [3] I ( / )log [1 4Dtanh( v/ )] (11) where D / / W and the limit l and u define the quantum hannel band edge, ee Figure 3 5 Grahial Reult A we have een, the arameter D / in (11) lay a key role in etablihing value for all quantum aaitie Sine WT 1, the arameter D atifie 1 D / / W WT T / T Thu D an be viewed a one of hoton denity er dimenion or a the invere of the hoton-time bandwidth rodut The ondition 4 erve to artition the eletromagneti etrum into two dijoint region The region 4 hold for laial ommuniation (quantized or unquantized) in that quantum effet do not manifet themelve and Plank ontant i abent from all erformane reult In addition, in thi frequeny region the hotoni energy in the ommuniation ignal may be aumed ontinuou For all, quantum effet in the bakground light begin to manifet themelve Figure 4 and 5 demontrate quantum ommuniation aaity-bandwidth tradeoff veru E Figure 4 lot quantum ommuniation torage aaity / R Q in qubit/hoton veru energy er qubit to noie ratio for variou hoton time duration-bandwidth rodut BT Figure 5 lot quantum ommuniation aaity R Q in qubit er dimenion veru qubit energy-to-noie ratio for variou value of BT Figure 4 Quantum ommuniation torage aaitybandwidth (qubit/hoton) tradeoff veru energy-erqubit to thermal noie ratio Coyright 13 SiRe

5 W C LIDSEY 135 Figure 5 Quantum ommuniation aaity in qubit er dimenion veru for variou hoton-time bandwidth rodut, From thee urve we ee that erformane i, for all ratial uroe, inenitivity to the normalized bandwidth arameter B / q Figure 6 lot Pb 1/ Q whih i the minimum number of hoton er qubit to ahieve quantum ommuniation aaity Q From Figure 4 and 6 we oberve the limit of one hoton er qubit i theoretially ahievable 6 Aknowledgement The author wihe to thank Profeor Debbie Van Alhen of Cal State orthridge for her diligent and untiring Figure 6 Minimum number of hoton er qubit required to ahieve aaity ; ( hoton/qubit) uort in roviding the grahial reult and omment on the manurit REFERECES [1] C E Shannon, A Mathematial Theory of Communiation, Bell Sytem Tehnial Journal, Vol 7, 1948, [] W C Lindey, On Quantum Information Storage Caaity, ubmitted for ubliation IEEE Tranation, 1 [3] W C Lindey, Photoni Communiation and Quantum Information, Storage Caaitie, ubmitted for ubliation IEEE Tranation, 13 Coyright 13 SiRe