The Transactional Nature of Quantum Information

Transcription

1 The Transactional Nature of Quantu Inforation Subhash Kak Departent of Coputer Science Oklahoa State University Stillwater, OK 7478 ABSTRACT Inforation, in its counications sense, is a transactional property. If the received signals counicate choices ade by the sender of the signals, then oration has been transitter by the sender to the receiver. Given this reality, the potential oration in an unknown pure quantu state should be non-zero. We exaine transactional quantu oration, which unlike von Neuann entropy, depends on the utuality of the relationship between the sender and the receiver, associating oration with an unknown pure state. The oration that can be obtained fro a pure state in repeated experients is potentially inite. INTRODUCTION The ter oration is used with a variety of eanings in different situations. In the atheatical theory of counications [], oration is a easure of the surprise associated with the received signal. It is iplied that the receiver has knowledge of the statistics of the essages produced, or to be produced, by the sender. A ore likely signal carries less oration as it coes with less surprise. Let the sender and the receiver share a set of essages fro a specific alphabet and the statistics of the counications will allow us to deterine the probability of each letter of the alphabet. The oration easure of the essage x associated with probability p x is log p x. Classical orational entropy is given by H ( X ) = p x log px () where p x is the probability of the essage x. x The aount of oration associated with an object could be taken to ean the aount necessary to copletely describe it. Since this oration will vary depending on the interactions the object has with other objects and fields, and thus be variable, it ay be easured for the situation where the object is isolated. In the signal context oration in a signal is soeties seen fro the lens of coplexity [],[3],[4]. On frequency considerations, the two patterns: () assued to be generated by tosses of a fair coin ust be taken to be equally rando. But fro the perspective of coplexity, the first pattern is less rando than the second since it can be described ore copactly. One ay speak of oration associated with an object also fro the perspective of uncertainty, either in the liitations related to its eployent in a coputational situation, or in defining a deeper syetry with classical objects [5],[6]. Another view would be to see how uch oration can be carried by the object. The physical nature of oration was ephasized in different ways by Wheeler and Landauer [7],[8],[9]. Let us consider now what is the oration associated with a quantu object, say, a photon. Represented as a qubit ( a + b ), the photon will collapse to or but since this collapse is rando, it would not counicate any useful oration to the receiver. Maxiu oration will be counicated to the receiver

2 if the sender prepares the photon in one of the two orthogonal states, say, or, which the receiver will be able to deterine upon observation. It is assued that the sender and the receiver have agreed upon the counication protocol and the easureent bases. If the agreeent on the nature of the counication had not been ade in advance, the receiver cannot, in general, obtain useful oration fro the strea of photons (excepting in their presence or absence) since he does not know the basis states of the sender, or know if these states are fixed or variable. If he knew the basis states of the sender, he would obtain the axiu of one bit of oration fro each qubit. Quantu oration is traditionally easured by the von Neuann entropy, Sn ( ρ) = tr( ρ log ρ), where ρ is the density operator associated with the state []. This entropy ay be written also as Sn ( ρ) λx logλx = (3) x where λx are the eigenvalues of ρ. It follows that a pure state has zero von Neuann entropy (we will take log in the expression to base iplying the units are bits). The von Neuann easure is independent of the receiver. This is unsatisfactory fro a counications perspective because if the sender is sending an unknown pure state to the receiver, it should, in principle, counicate oration. Suppose that the sender and the receiver have agreed that the sender will choose, say, one of 6 different polarization states of the photon. The sender picks one of these and sends several photons of this specific polarization. Since all these photons are the sae pure state, the oration associated with the, according to the von Neuann easure, is zero. But, the oration in this choice, fro the counications point of view, is log 6 = 4 bits. Although the classical entropy and the von Neuann entropy easures as given by equations () and (3) look siilar atheatically, there is a difference between the two expressions, as the first one relates to probabilities and the second to the eigenvalues associated with a atrix. INFORMATION IN MIXED AND PURE STATES A given photon ay be in a pure or ixed quantu state, and these two cases are very different fro the point of view of easureent. A ixed state is a statistical ixture of coponent pure states, and its entropy coputed by the von Neuann easure is siilar to the entropy for classical states. The axiu oration provided by a single ixed state photon is one bit. 3 Exaple. Consider the ixed state Ψ = +. Its von Neuann entropy equals.8 bits. 4 4 This ixed state can be viewed to be generated fro a variety of ensebles of states. For exaple, it can be 3 3 Ψ = a a + b b, where a = + and b = viewed as the enseble In each of these ensebles the entropy would be the sae. Exaple. Consider an entangled pair of quantu objects represented by the pure state Ψ = ( + ) that corresponds to the density operator.5 ρ = (4)

3 Its eigenvalues are,,, and, and, therefore, its von Neuann entropy is zero. But consider the two objects separately; their density operators are.5 ρ =.5 each, which eans that their entropy is bit each. We are now confronted with the situation that the two coponents of an entangled quantu syste have non-zero entropy although the syste taken as a whole has entropy of zero! Intuitively, this is not reasonable because one expects the oration of a syste to be function of the su of its parts. The zero entropy of the entangled state is unreasonable also fro the perspective that it could have been in one of any different fors and therefore having oration potential. For exaple, with the assuption that the probability aplitudes be real and equal, the state function could have been either ( + ) or ( ), counicating oration equal to one bit. Inforation in classical theory easures the reduction in uncertainty regarding the essage based on the received counication. It is predicated on the utuality underlying the sender and the receiver. Therefore the receipt of an unknown pure state should counicate oration to the recipient of the state. In our exaination of the question of oration in an unknown pure state Ψ = a + b, we take it that ultiple copies of the state are available. The received copies can be used by the recipient estiate the values of a and b. If the objective is the estiation of the density atrix ρ, we can do so by finding average values of the observables ρ, Xρ, Yρ, Zρ, where X, Y, Z are the Pauli operators, by eans of the expansion: ρ ( tr( ρ) I + tr( Xρ) X + tr( Yρ) Y + tr( Zρ) Z) = (5) The operators tr(xρ) etc are average values and therefore the probabilistic convergence of ρ to its correct value would depend on the nuber of observations ade. Considering a ore constrained setting, assue that the oration that user A, the preparer of the states, is trying to counicate to the user B, is the ratio a/b, expanded as a nuerical sequence,. For further siplicity it is assued that A and B have agreed that a and b are real, then a + = and b = ±. One ay + use either a pure state φ = a + b or a ixed state consisting of an equal ixture of the states φ = a + b and φ = a b. Alternatively, one ay counicate the ratio a /b =. In this case, a = and b = ±. + + Sinceb =, one needs to erely deterine the sequence corresponding to the reciprocal of + for the + probability of the coponent state. In yet another arrangeent, the sender and the receiver ay agree in advance to code the contents of the essage in the probability aplitudes in soe other anner. For exaple, they ay agree on the following table relating binary sequence and the probability aplitude a: 3

4 /8 /8 3/8 4/8 5/8 6/8 7/8 5 3 Thus the binary sequence will ap to the qubit. 8 8 The preparer of the quantu state ay choose out of inity of possibilities, and depending on the utual relationship between the preparer and the receiver the pure state s oration will vary fro one receiver to another. In case the pure state chosen by the sender is aligned to the easureent basis of the receiver, the oration transitted will be zero. In general, the oration generated by the source equals the probability of choosing the specific state out of the possibilities available (this is the states a priori probability). If the set of choices is inite, then the oration generated by the source is unbounded. On the other hand, due to the probabilistic nature of the reception process, not all the oration at the source is obtained at the receiver by the easureent. The ore the nuber of copies of the photon the receiver is sent, the ore oration will be extracted. A single photon will of course counicate at best only one bit of oration. PARALLEL BETWEEM CLASSICAL AND QUANTUM PROTOCOLS We have spoken of sending oration using a single pure quantu state using any of its copies. This ay be copared, in the classical case, to the representation of the contents of an entire book by a single signal, V, obtained by converting the binary file of the book into a nuber by putting a dot before it to ake it less than. If a large nuber of copies of this signal are sent, it is possible that the receiver could, in principle, deterine its value even in the presence of noise, ε, taken to be syetrically distributed about zero. This would be accoplished by suing a large nuber of received signals W, so that E(W) = E(V) + E(ε) = V + E(ε) ay be calculated accurately. As the nuber of copies available to be sued increases, the value E(ε) goes to zero and E(W) V Although such a ethod is not practical for an entire file, ore than one binary sybol of a essage are coded into a single aplitude in non-binary systes. If one wished to use a pure state to transit oration in several binary bits in a practical ipleentation, the transission should contain several copies of the pure state to enable the receiver to extract the relevant oration fro it. If the pure state is used to transit a single bit, then we have the case of the oration being sent by choice between two alternatives. TRANSACTIONAL INFORMATION IN A QUANTUM STATE In 7, I proposed a easure of entropy that covers both pure and ixed states []. In this easure, which I call quantu orational entropy, S (ρ), the entropy is related only the diagonal ters of the density atrix: S ( ρ) = ρ log ρ (6) i ii ii Let the density atrix be n n, that is it is associated with n qubits. Then, S (ρ) n (7) 4

5 The proof that the least value of S (ρ) is zero is obvious when the transitted state is pure and aligned to the coputational basis of the receiver. The axiu value of S (ρ) is n when the diagonal ters are equal. Many properties of S (ρ) are siilar to those of S ). For exaple, for the joint state ρ AB of quantu systes A and B, n(ρ S A, B) S ( A) + S ( B) (8) ( An exaple of this is the entangled state (Exaple ), which has orational entropy of bit and the orational entropy of each of the coponents is also bit. The entropy function satisfies the inequality: S piρi ) pis ( ρi) (9) ( i i The intuition behind this is that the knowledge of the enseble that has led to the generation of the quantu state reduces the uncertainty and, therefore, the entropy of the right hand side is less. Due to the sae reason, we can write that S (ρ) S n(ρ) () If the receiver has knowledge that the enseble consists of a specific probabilistic cobination of pure and ixed coponents, then the partial entropy, Sp(ρ) = p i S i (ρ i ), is lower copared to when he has no such knowledge..7.5 Exaple 3. Consider ρ =..5.9 The orational entropy S (ρ) is S (ρ) = -.7 log.7.9 log.9 =.868 bits. The eigenvalues of ρ are.4 and.758 and, therefore, the von Neuann entropy S ) for this case is equal to S n(ρ) = -.4 log log.758 = =.798 bits. The orational entropy exceeds the von Neuann entropy by.7 bits. Partial Inforation Case. Assue that partial oration is available for Exaple 3, and we know that the enseble consists of a pure and a ixed coponent in the following anner: n(ρ.5 ρ = Then the partial entropy associated with it will be the su of the pure and ixed entropies and this turns out to be equal to =.85 bits. 5

6 Partial Inforation Case. On the other hand, if we are told that the enseble consists of pure and ixed coponent in the following anner: ρ = The partial entropy will now be equal to: = =.865 bits As in the previous case, in this partial easure, entropy has two coponents: one orational (related to the pure coponents of the quantu state), and the other that is therodynaic (which is receiver independent). For a two-coponent eleentary ixed state, the ost oration in each easureent is one bit, and each further easureent of identically prepared states will also at ost be one bit. For an unknown pure state, the oration in it represents the choice the source has ade out of the inity of choices related to the values of the probability aplitudes with respect to the basis coponents of the receiver s easureent apparatus. Each easureent of a two-coponent pure state will provide at ost one bit of oration, and if the source has ade available an unliited nuber of identically prepared states the receiver can obtain additional oration fro each easureent until the probability aplitudes have been correctly estiated. Once that has occurred, unlike the case of a ixed state, no further oration will be obtained fro testing additional copies of this pure state. QUANTUM CRYPTOGRAPHY The BB84 [] and the three-stage quantu cryptography protocols [] ay be seen as eerging naturally fro the different perspectives of coputing signal operations on two sets of bases (ixed state scenario) and a single set (for each user). DISCUSSION The approach of this paper is consistent with the positivist view that one cannot speak of oration associated with a syste excepting in relation to an experiental arrangeent together with the protocol for easureent. The experiental arrangeent is thus integral to the aount of oration that can be obtained and it eans that oration in a unknown pure state is non-zero. The receiver can ake his estiate by adjusting the basis vectors so that he gets closer to the unknown pure state. The oration that can be obtained fro such a state in repeated experients is potentially inite in the ost general case. A distinction ay be ade in the situation for discrete (and finite) and continuous geoetries. Since the easure of oration in a pure state is a consequence of its distance fro the observer s coputational basis, oration in a continuous geoetry would for alost each observer (excepting for the one who starts out exactly aligned, the probability of which is zero) reain inite. Conversely, for a finite discrete geoetry, a saller distance will translate into lesser oration, aking soe observers ore privileged than others. 6

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