The Transactional Nature of Quantum Information

The Transactional Nature of Quantum Information Subhash Kak Department of Computer Science Oklahoma State University Stillwater, OK 74078 ABSTRACT Information, in its communications se nse, is a transac tional property. If the receive d signals communicate choices made by the sender of the signals, then information has been transm itter by the sender to the receiver. Given this reality, the potential information in a n unknown pure quantum state should be non-zero. We exam ine transactional quantum information, which unlike von Neu mann entropy, depend s on the mutuality of the relationship b etween the sender and the receiver, associating inform ation with an unknown pu re state. The inform ation that can be obtained fr om a pure state in repeate d experi ments is pot entiall y infinite. INTRODUCTION The term “information” is used with a variety of meanings in different situation s. In the mathematical theory of communications [1], information is a m eas ure of the surprise associated with th e received signal. It is implied that the receiver has knowledge of the statis tics of the m essages produced, or to be produced, by the sender. A more likely signal carries less informati on as it comes with less surprise. Let the s ender and th e receiv er shar e a set of mes sage s from a specific alphabet and the statistics of the communications will allow us to d etermine the probability o f each letter of the alphabet. The information measure of the message x associated with probability p x is –log p x . Classical i nformational entropy is given by x x x p p X H log ) ( ∑ − = ( 1 ) where p x is the probability of the message x. The amount of information asso ciated with an object could be take n to mean the am ount necessary to com pletely describe it. Since this information will vary depending on the interactions the object has with other objects and fields, and t hus be vari able, it m ay be measured for the sit uation whe re the object is isolated. In the signal context in formation in a signal is sometimes seen from th e lens of complexity [ 2],[3],[4]. On frequency c onsiderations, t he two pat terns: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 1 (2) assumed to be generated by tosses of a fair coin must be taken to be equal ly random . But from t he perspective of complexity, the first pattern is less random than the second since it can be described more compactly. One may speak of inform ation associated with an object also from the perspective of uncertain ty, either in the limitations related to its employment in a computational sit uation, or in defining a d eeper symmetry with classical objects [5],[6]. Another view would be t o see how much inf ormation ca n be carried by the object. The physi cal nature of i nformation was emphasized i n differe nt ways by Wheeler and L andauer [ 7],[8],[ 9]. Let us consider now what is the info rmation associated with a qua ntum obj ect, say, a photon. Represen ted as a qubit , the photon will collaps e to or but since this collapse is random, it would not communicate any useful information to the receiver. Ma xi mum information will be comm unicated to the receiver ) 1 | 0 | ( 〉 + 〉 b a 〉 0 | 〉 1 | if the sender prepares the photon in on e of the two or thogonal states, say, or , which the receiver will be able to determine upon observation. It is assum ed th at the sender a nd the receiver have agreed upon the communicat ion protocol and the measurem ent bases. If the agreem ent on the nature o f the comm unication had not been made in adva nce, the receiver cannot , in general, obtain useful inform ation fr om the stream of photons (excepting in their prese nce or absence) since he does not know the basis states of the sender, or know if th ese states are fixed or variable. 〉 0 | 〉 1 | ) ( If he knew the basis states of the send er, he would obtain the maximum of one bit of inform ation from each qubit. Quantum informatio n is tradition ally measured by the von Neumann en tropy, ) log ( ρ ρ ρ tr S − = n , where ρ is the density operator associated with the state [10]. This entropy may be written also as x x x n S λ λ ρ log ) ( ∑ − = ( 3 ) where x λ are the eigenvalues of ρ . It follows that a pure state has zero von Neumann entro py (we will take “log” in the expressi on to bas e 2 im plying the units are bits ). The von Neum ann measure is independent of t he receiver. This is unsatisfactory from a communications pe rspective b ecause if the sender is send ing an unknown pure state to the receiver, it should, i n prin ciple, communicate inform ation. Suppose that the sender a nd the receiver have agreed t h at the sender will choose, say, one of 16 different polarization states of the photon. T he sender picks on e of t hese and sends several photons of this speci fic polarization. Since all these photons are the sam e pure st ate, the information associat ed with them , according to the von Neumann m easure, is zero. But, t he information in this choice, from the communications point of view, is log 2 16 = 4 bits. Although the cl assical entropy and t he von Neum ann entro py measures as give n by equat ions (1) and (3) look similar mathematically, there is a diffe rence between the two expressions, as the first one relates to probabilities and the second to the eigenvalue s associated with a matrix. INFORMATION IN MIXED AND PU RE STATES A given phot on may be in a pure or m ixed quantum state, and these two cases are very diffe rent from the poi nt of view of measurem ent. A mixed stat e is a statisti cal mixture of com ponent pure st ates, and its entro py computed by the von Neum ann measure is similar t o the entropy for classical states. Th e maxim um inform ation pro vided by a single mi xed state photon i s one bit. Example 1. C onsider the mixed state | 1 1 | 4 1 | 0 0 | 4 3 | 〉〈 + 〉〈 = 〉 Ψ . Its von Neum ann entropy equal s 0.81 bits. This mixed st ate can be viewed to be generated from a variety of ensembles of states. For example, it can be viewed as the ensem ble | | 2 1 | | 2 1 | b b a a 〉〈 + 〉〈 = 〉 Ψ , where 〉 1 | + 〉 0 = 〉 4 1 | 4 3 a | and 〉 − 〉 = 〉 1 | 4 1 0 | 4 3 | b . In each of these ensembles the entropy would be the same. Example 2. C onsider an enta ngled pair of quantum objects represe nted by the pure state ) 11 | 00 (| 2 1 | 〉 〉+ = 〉 Ψ that corresponds to the density o perator ( 4 ) ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 5 . 0 0 5 . 0 0 0 0 0 0 0 0 5 . 0 0 5 . ρ 2 Its eigenvalues are 0, 0, 0, and 1, and, therefore, its von Neumann entr opy is zero. But consider the two obj ects separately; their density operators are each, which means that th eir ent ropy is 1 bit each. ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 5 . 0 0 5 . ρ We are now c onfronted with the si tuation t hat the t wo com ponents of an entangled quantum system have non-zero entropy although the system taken as a whole has en tropy of zero! Intuitively, this is not reasonable because one expects the inf ormation of a system to be function of the sum of its parts. The zero entropy of the entangled state is unreasona ble also from the perspective that i t could have be en in one of many diffe rent form s and theref ore havin g inform ation po tential. For example, with the assumption that the probability amplitu des be real and equal, the state fu nction could hav e been either ) 11 | 00 (| 2 1 〉 〉+ or ) 11 | 00 (| 2 1 〉 〉− , communicating information equal to one bit. Information in classical theory measur es the reduction in uncertainty regard ing the me ssage based on the received communication. It is predicat ed on th e mutuality underlying the sender and t h e receiver. Therefore the receipt of an unknown pure state should co mmunicate in formation to the recipient of the state. In our exam ination of the question of inform ation in an unknown pu re state 〉 + 〉 = 〉 Ψ 1 | 0 | | b a , we take it that multiple copies of the state a re available. The received copi es can be used by the recipient estim ate the values of a and b . If the objectiv e is the estimation of the den sity matrix ρ , we can do so by findi ng average values of t he observables ρ , X ρ , Y ρ , Z ρ , where X , Y, Z are t he Pauli ope rators, by m eans of the e xpansion: ) ) ( ) ( ) ( ) ( ( 2 1 Z Z tr Y Y tr X X tr I tr ρ ρ ρ ρ ρ + + + = (5) The operators tr(X ρ ) etc are average values and therefor e the probabilistic co nvergence of ρ to its correct value would depe nd on t he number of observati ons m ade. Considering a more constrained setting, assume that the in formation that user A, the preparer of the states, is trying to communicate to th e user B, is the ratio a/b , expanded as a numerical seque nce, m. For further simplicity it is assumed that A and B have agreed that a and b are real, then 2 1 m m a + = and 2 1 1 m b + ± = . One may use either a pure state 〉 + 〉 = 〉 1 | 0 | | b a φ or a mixed state consisting of an equal mixture of the states 〉 + 〉 = 〉 1 | 0 | | 1 b a φ and 〉 − 〉 = 〉 1 | 0 | | 2 b a φ . Alternatively, one may comm unicate the ratio a 2 /b 2 =m. In this case, m m a + = 1 and m b + ± = 1 1 . Since m b + = 1 1 2 , one needs to m erely determine the seque nce correspon ding to the reciprocal of 1+m for the probability of the component state . 〉 1 | In yet another arrangem ent, the sender and the receive r may agree in advance to code th e contents of t he message in the probability amplitud es in some other manner. For example, they may ag ree on the following table relating binary sequence and th e probability amplitud e a : 3 000 0 001 1/8 010 2/8 011 3/8 100 4/8 101 5/8 110 6/8 111 7/8 Thus the binary sequen ce 101 will map to the qubit 〉 − 〉 1 | 8 3 0 | 8 5 . The preparer of the quantum state may choose out of infinity of po ssibilities, and depending on the mutual relationship between the pre parer and the receiver the pu re state’s inform ation will vary from one receiver to another. In case the pure state chosen by the sender is aligned to the measurem ent basis of the receiver, the information transmitted will be zero. In general, the information generated by the source equ als the probability o f choosing the specific state o ut of the possibilities available (this is the states a priori probability). If the set of choices is infinite, th en the information generated by the source is unbounde d. On the other hand, due to the probabilistic nature of the reception process, not all the information at the source is obta ined at the receiver by the m easurement. The more th e numb er of copies of the photon th e receiver is sent, the more information will be extracted. A single photon will of cou rse communicat e at best only one bit of inform ation. PARALLEL BETWEEM CLASSICA L AND QUANTUM PROTOC OLS We have spok en of sending i nformati on using a single p ure quantum state using m any of its co pies. This m ay be compared, in the classical case, to the representation of the contents of an entir e book by a single signal, V , obtained by converting th e binary file of the book in to a number by putting a “dot” before it to make it less than 1. If a large number of copies of this sign al are sent, it is possible that the recei ver could, in principle, de termine its value even in the presence of noise, ε , taken to be symmetrical ly distributed a bout zero. T his would be accomplished by summing a large number of receive d signals W , so that E(W) = E(V) + E( ε ) = V + E( ε ) may be calculated accurately. As the num ber of copies available t o be summed increases, the value E( ε ) goes to zero and E(W) → V Although suc h a method is n ot practical for an ent ire file, m ore than one binary sy mbol of a m essage are coded into a single amplitude in non -binary systems. If one wished to use a pure state to transmit information in sev eral binary bits in a practical implementation, the transmission should contain several copie s of the pure state to enable the recei ver to extract the releva nt information from it. If the pure state is used to transmit a single bit, then we have the case of the informatio n being sent by choice between two al ternatives. TRANSACTIONAL INF ORMATION IN A QUA NTUM STATE In 2007, I pr oposed a measure of entropy tha t covers both p ure and mixed st ates [11]. In this m easure, which I call quantum informational en tropy , S inf ( ρ ) , the entropy is related only the diagon al terms of the density matrix: ii i ii S ρ ρ ρ log ) ( inf ∑ − = ( 6 ) Let the density matrix be 2 n ¯ 2 n , that is it is associated with n qubits. Then, 0 ≤ S inf ( ρ ) ≤ n (7) 4 The proof that the least value of S inf ( ρ ) is zero is obvi ous when t he transmitt ed state is pure and al igned to t he computati onal basis of t he receiver. The maxim um value of S inf ( ρ ) is n when the diagonal terms are equa l. Many prope rties of S inf ( ρ ) are similar to thos e of ) ( ρ n S . For example, for the joint state ρ AB of quantum systems A and B, ) ( ) ( ) , ( inf inf inf B S A S B A S + ≤ (8) An example of this is the entang led state (Example 2), which has informational entrop y of 1 bit and the informational entropy of each of the c omponents is also 1 bit. The entropy functio n satisfies the inequality: ) ( ) ( inf inf i i i i i i S p p S ρ ρ ∑ ∑ ≥ ( 9 ) The intuition behind th is is that the knowledge of the en se mble that has l ed to the gener ation of the quantum state reduces the uncertainty an d, therefore, the entropy of the right ha nd si de is less. Due t o the same reason, we can write that S inf ( ρ ) ≥ ) ( ρ n S (10) If the receiver has knowledge that the en semble consists of a specific probab ilistic combination of pure and m ixed component s, then the partia l entropy, Sp( ρ ) = ∑ p i S i ( ρ i ) , is lower compared to when he has no su ch knowledge. Example 3. C onsider . ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 29 . 0 15 . 0 15 . 0 71 . 0 ρ The informational entropy S inf ( ρ ) is S inf ( ρ ) = -0.71 log 2 .71 – 0.29 log 2 .29 = 0.868 bits. The eigenvalues of ρ are 0.242 and 0.758 and , therefore , the von Neumann entr opy ) ( ρ n S for this case is equal to ) ( ρ n S = -0.242 log 2 .242 – 0 .758 log 2 .758 = .242 ¯ 2.047 + .758 ¯ .400=0 .798 bits. The informational entropy e xceeds the von Neum ann entropy by 0.07 bits. Partial Information Ca se 1. Assume that partial information is availab le for Example 3, and we know that the ensemble consi sts of a pure a nd a mixed c omponent in t he followi ng manner: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ × + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ × = 2 . 0 0 8 . 7 . 0 5 . 5 . 5 . 5 . 3 . 0 ρ Then the partial en tropy associated with it wi ll be the sum of the pure and mixed entr opies and this t urns out to be equal to 0. 3 + 0.7 ¯ 0.7 22= 0. 805 bits. 5 Partial Informatio n Case 2. On the other hand, if we are told that the ensemble consists of pure an d mixed component in the fol lowing m anner: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ × + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ × = 269 . 0 0 0 731 . 0 684 . 0 3 1 3 2 3 2 3 2 316 . 0 ρ The partial entropy will now be equal to: 0.316 ¯ 0.918+0.684 ¯ 0.84=0.2 90+0.5 75=0.865 bi ts As in the p revious case, in this partial m easure, entr opy has two components: one info rmational ( related to t he pure compone nts of the qua ntum state) , and the othe r that is therm odynami c (whic h is receiver indepe ndent). For a two-com ponent elementary mixed state, the m ost information in each m easurement is one bit, and each further measurem ent of identically prepar e d states will also at most be one bit. For an unknown pure state, the information in it represen ts the choice the sou rce has made out of the infinity of ch oices related to the v alues of the probability amplitudes with res pect to the basis compone nts of the receiver’s measurem ent apparatus. Each measurement of a two-com ponent pure state will provide at m ost one bit of information, and if the source has made available an un limited number of identically p repared states the receiv er can obtain addition al information from each measurement until the probability am plitudes have been correctly estimated. Once that has occurred, unlike the case of a mixed st ate, no further informati on will be obt ained from testing a dditional co pies of this pu re state. QUANTUM CRYPTOGRAPHY The BB84 [10] and the three- stage quantum cryptography protocols [ 12] may be seen as em erging natu rally from the different perspectives of co mputing signal operations on tw o sets of base s (m ixed state scenario) and a single set (for each user). DISCUSSION The approach of this paper is con sistent with the positiv ist view that one canno t speak of information associated with a system excepting in rela tion to an experim ental arrangem ent toget her with the prot ocol for measurem ent. The experim ental arrangem ent is thus inte gral to the am ount of inform ation that can be obtained a nd it means t hat information in a unknown pure state is non-zero. The receiver can m ake his estimate by adju sting the basis vectors so that he gets closer to the unknown pure state. The information that can be obtained from such a state in repeated exp eriments is potentially infinite in the most general case. A distinction may be made in th e situation for discrete (and finite) and continuo us geometries. Since the measure of information in a pure stat e is a consequence of its “distance” from the observe r’s c omputational basis, information in a continuous geom etry would for alm ost each observer (e xcepting for the on e who starts out exactly aligned, the probability of which is zero) remain infinite. Conversely, for a finite discrete geometry, a smaller “distance” will translate into lesser information, making som e observers m ore privileged than ot hers. 6 7 REFERENCES [1] Shanno n, C.E., “A mathem atical theory of communi cation.” Bel l System Technical Journal 27: 37 9-423, 623- 656 (1948). [2] Solom onoff, R., “ A form al theory of i nductive in ference.” Inf ormation a nd Contr ol 7: 1- 22, 224-2 54 (1964) . [3] Kak, S., “Classification of random binary sequen ces using Walsh-Fourier ana lysis.” IEEE Trans. On Electromagnetic Compatibility EMC-13, 74-77 (1970). 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