Computation in Sofic Quantum Dynamical Systems

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( s ) s , U A} ∈ n a and state | Q | - , 2 ics. They are the only way to observe a quantum pro- In [11] the authors established a hierarchy of process cess’s current internal state. languages and the corresponding quantum and classi- A quantum generator operates as follows. Uij gives cal computation-theoretic models that can recognize and the transition amplitude from internal state qi to internal generate them. state qj . Starting in state vector hψ0| the generator updates its state by applying the unitary matrix U. Then the state vector is projected using P (s) and renormal- B. Alternative quantum finite-state machines ized. Finally, symbol s ∈ A is emitted. In other words, starting with state vector hψ0|, a single time-step yields Said most prosaically, we view quantum generators as hψ(s)| = hψ0| U · P (s), with the observer receiving mea- representations of the word distributions of quantum pro- surement outcome s. cess languages. Despite similarities, this is a rather different emphasis than that used before. The first men- tion of quantum automata as an empirical description of A. Process languages physical properties was made by Albert in 1983 [13]. Al- bert’s results were subsequently criticized by Peres for The only physically consistent way to describe a quan- using an inadequate notion of measurement [14]. In a tum system under iterated observation is in terms of computation-theoretic context, quantum finite automata ↔ were introduced by several authors and in varying ways, the observed sequence S ≡ ...S 2S 1S0S1 ... of dis- − − crete random variables St. We consider the family of but all as devices for recognizing word membership in a word distributions, {Pr(st+1,...,st+L): st ∈ A}, where language. For the most widely discussed quantum au- Pr(st) denotes the probability that at time t the ran- tomata, see Refs. [15, 16, 17]. Ref. [18] summarizes the dom variable St takes on the particular value st ∈ A different classes of languages which they can recognize. and Pr(st+1,...,st+L) denotes the joint probability over Quantum transducers were introduced by Freivalds and sequences of L consecutive measurement outcomes. We Winter [19]. Their definition, however, lacks a physical assume that the distribution is stationary: notion of measurement. We, then, introduced quantum finite-state machines, as a type of transducer, as the gen- Pr(St+1,...,St+L) = Pr(S1,...,SL) . (1) eral object that can be reduced to the special cases of quantum recognizers and quantum generators of process L We denote a block of L consecutive variables by S ≡ languages [11]. L S1 ...SL and the lowercase s = s1s2 ··· sL denotes a particular measurement sequence of length L. We use the term quantum process to refer to the joint distribution III. INFORMATION PROCESSING IN A SPIN-1 Pr(↔S) over the infinite chain of random variables. A DYNAMICAL SYSTEM quantum process, defined in this way, is the quantum analog of what Shannon referred to as an information We will now investigate concrete examples of quantum source [12]. processes. Consider a spin-1 particle subject to a mag- Such a quantum process can be described as a stochas- netic field which rotates the spin. The state evolution tic language L, which is a formal language with a prob- can be described by the following unitary matrix: ability assigned to each word. A stochastic language’s 1 1 word distribution is normalized at each word length: √2 √2 0 U =  0 0 −1  . (5) L 1 1 Pr(s )=1 ,L =1, 2, 3,... (2)  − √2 √2 0  XL s { ∈L} Since all entries are real, U defines a rotation in R3 L π with 0 ≤ Pr(s ) ≤ 1 and the consistency condition around the y-axis by angle 4 followed by a rotation L L π Pr(s ) ≤ Pr(s s). around the x-axis by an angle 2 . A process language is a stochastic language that is sub- Using a suitable representation of the spin operators word closed: all subwords of a word are in the language. Ji [20, p. 199]: We can now determine word probabilities produced by a QFG. Starting the generator in hψ |, the probability 0 0 0 0 0 i 0     of output symbol s is given by the state vector without Jx = 0 0 i , Jy = 0 00 , renormalization:  0 −i 0   −i 0 0  0 i 0 Pr(s)= hψ(s)|ψ(s)i . (3)   Jz = −i 0 0 , (6) While the probability of outcome sL from a measurement  0 00  sequence is 2 the relation Pi =1 − Ji defines a one-to-one correspon- L L L Pr(s )= hψ(s )|ψ(s )i . (4) dence between the projector Pi and the square of the spin 3

FIG. 1: The Golden Mean quantum generator. component along the i-axis. The resulting measurement answers the yes-no question, Is the square of the spin component along the i-axis zero? FIG. 2: Golden Mean process language: Word {00} has zero 2 Consider the observable Jy . Then the following pro- probability; all others have nonzero probability. The loga- jection operators together with U in Eq. (5) define the rithm base 2 of the word probabilities is plotted versus the bi- L L quantum finite-state generator: nary string s , represented as base-2 real number “0.s ”. To allow word probabilities to be compared at different lengths, P (0) = |010i h010| the distribution is normalized on [0, 1]—that is, the probabilities are calculated as densities. and P (1) = |100i h100| + |001i h001| . (7)

A graphical representation of the automaton is shown in Fig. 1. The process language generated by this QFG is the so-called Golden-Mean Process language [1]. The word distribution is shown in Fig. 2. It is characterized by the set of irreducible forbidden words F = {00}: no consecutive zeros occur. In other words, for the spin-1 particle the spin component along the y-axis never vanishes twice in a row. This restriction—the dominant structure in the process—is a short-range correlation since the measurement outcome at time t only depends on the immediately preceding one at time t − 1. If the outcome is 0, the next outcome will be 1 with certainty. If the outcome is 1, the next measurement is maximally uncertain: outcomes 0 and 1 occur with equal probability. FIG. 3: The Even Process quantum generator. Consider the same Hamiltonian, but now use instead 2 the observable Jx . The corresponding projection operators deﬁne the QFG:

P (0) = |100i h100| system tracks the evenness or oddness of number of consecutive measurements of “spin component equals 0 along and P (1) = |010i h010| + |001i h001| . (8) the x-axis”.

The QFG defined by U and these projection operators Note that changing the measurement, specifically is shown in Fig. 3. The process language generated by 2 choosing Jz as the observable, yields a QFG that gen- this QFG is the so-called Even Process language [1, 21]. erates Golden Mean process language again. The word distribution is shown in Fig. 4. It is defined by the infinite set of irreducible forbidden words The two processes produced by these quantum dynam- 2k 1 F = {01 − 0}, k = 1, 2, 3, .... That is, if the spin com- ical systems are well known in the context of symbolic dy- ponent equals 0 along the x-axis it will be zero an even namics [5]—a connection we will return to shortly. Let number of consecutive measurements before being ob- us first, though, turn to another important property of served to be nonzero. This is a type of infinite corre- finite-state machines and explore its role in computa- lation: For a possibly infinite number of time steps the tional capacity and dynamics. 4

dynamics. Once they are known and one is synchro- nized, the process becomes optimally predictable. A ﬁ- nal, related reason why determinism is desirable is that closed-form expressions can be given for various information processing measures, as we will discuss in Sec. VI.

V. SOFIC SYSTEMS

In symbolic dynamics, sofic systems are used as tractable representations with which to analyze continuous-state dynamical systems [1, 5]. Let the al- phabet A together with an n × n adjacency matrix (with entries 0 or 1) define a directed graph G = (V, E) with V FIG. 4: Even Process language: Words {012k−10},k = the set of vertices and E the set of edges. Let X be the 1, 2, 3, ... have zero probability; all others have nonzero prob- set of all infinite admissible sequences of edges, where ad- ability. missible means that the sequence corresponds to a path through the graph. Let T be the shift operator on this sequence; it plays the role of the time-evolution operator IV. DETERMINISM of the dynamical system. A sofic system is then defined as the pair (X,T ) [23]. The Golden Mean and the Even The label determinism is used in a variety of senses, process are standard examples of sofic systems. The Even some of which are seemingly contradictory. Here, system, in particular, was introduced by Hirsch et al in we adopt the notion, familiar from automata theory the 1970s [21]. [6], which differs from that in physics, say, of non- Whenever the rule set for admissible sequences is finite stochasticity. One calls a finite-state machine (classical one speaks of a subshift of finite type. The Golden Mean or quantum) deterministic whenever the transition from process is a subshift of finite type. Words in the language one state to the next is uniquely determined by the out- are defined by the finite (single) rule of not containing the put symbol, or input symbol for recognizers. It is impor- subword 00. The Even Process, on the other hand, is not tant to realize that a deterministic finite-state machine of finite type, since the number of rules is infinite: The can still behave stochastically—stochasticity here refer- forbidden words {012k+10} cannot be reduced to a finite ring to the positive probability of generating symbols. set. As we noted, the rule set, which determines allowable Once the symbol is determined, though, the transition words, implies the process has a kind of infinite memory. taken by the machine to the next state is unique. Thus, One refers, in this case, to a strictly sofic system. what is called a stochastic process in dynamical systems theory can be described by a deterministic finite-state The spin-1 example above appears to be the first time generator without contradiction. a strictly sofic system has been identified in quantum dy- We can easily check the two quantum finite-state ma- namics. This ties quantum dynamics to languages and chines in Figs. 1 and 3 for determinism by inspecting each quantum automata theory in a way similar to that found state and its outgoing transitions. One quickly sees that in classical dynamical systems theory. In the latter set- both generators are deterministic. In contrast, the third ting, words in the sequences generated by sofic systems 2 QFG mentioned above, defined by U in Eq. (5) and Jz , correspond to regular languages—languages recognized is nondeterministic. by some finite-state machine. We now have a similar Determinism is a desirable property for various rea- construction for quantum dynamics. For any (finite- sons. One is the simplicity of the mapping between ob- dimensional) quantum dynamical system under observa- served symbols and internal states. Once the observer tion we can construct a QFG, using a unitary operator synchronizes to the internal state dynamics, the output and a set of projection operators. The language it gen- symbols map one-to-one onto the internal states. In generates can then be analyzed in terms of the rule set of eral, though, the observed symbol sequences do not track admissible sequences. One interesting open problem be- the internal state dynamics (orbit) directly. (This brings comes the question whether the words produced by sofic one to the topic of hidden Markov chains [22].) quantum dynamical systems correspond to the regular For optimal prediction, however, access to the internal languages. An indication that this is not so is given by state dynamics is key. Thus, when one has a determin- the fact that finite-state quantum recognizers can accept istic model, the observed sequences reveal the internal nonregular process languages [11]. 5

VI. INFORMATION-THEORETIC ANALYSIS Quantum Spin-1 Dynamical System Particle 2 2 Observable J J The process languages generated by the spin-1 particle y x under a particular observation scheme can be analyzed hµ [bits/measurement] 0.666 0.666 E [bits] 0.252 0.902 using well known information-theoretic quantities such as Shannon block entropy and entropy rate [12] and others TABLE I: Information storage and generation for example introduced in Ref. [24]. Here, we will limit ourselves to quantum processes: entropy rate hµ and excess entropy E. the excess entropy. The applicability of this analysis to quantum dynamical systems has been shown in Ref. [25], where closed-form expressions are given for some of these where H(R) is a sum over |A|R terms. Given that the quantities when the generator is known. quantum generator is deterministic we can simply employ We can use the observed behavior, as reflected in the the above formula for hµ and compute the block entropy word distribution, to come to a number of conclusions at length R to obtain the excess entropy for the order-R about how a quantum process generates randomness and quantum process. stores and transforms historical information. The Shan- Ref. [25] computes these entropy measures for various non entropy of length-L sequences is defined example systems, including the spin-1 particle. The results are summarized in Table I. The value for the excess H(L) ≡− Pr(sL)log Pr(sL) . (9) X 2 entropy of the Golden Mean process obtained by using sL L ∈A Eq. (13) agrees with the value obtained from simulation It measures the average surprise in observing the “event” data, shown in Table I. The entropy hµ = 2/3 bits per sL. Ref. [24] showed that a stochastic process’s informa- measurement for both processes, and thus they have the tional properties can be derived systematically by taking same amount of irreducible randomness. The excess en- derivatives and then integrals of H(L), as a function of tropy, though, differs markedly. The Golden Mean pro- 2 E L. For example, the source entropy rate hµ is the rate of cess (Jy measured) stores, on average, ≈ 0.25 bits at 2 increase with respect to L of the Shannon entropy in the any given time step. The Even Process (Jx measured) large-L limit: stores, on average, E ≈ 0.90 bits, which reflects its longer memory of previous measurements. hµ ≡ lim [H(L) − H(L − 1)] , (10) L →∞ where the units are bits/measurement [12]. VII. CONCLUSION Ref. [25] showed that the entropy rate of a quantum process can be calculated directly from its QFG, when the latter is deterministic. A closed-form expression for We have shown that quantum dynamical systems store the entropy rate in this case is given by: information in their dynamics. The information is ac- cessed via measurement. Closer inspection would suggest Q 1 Q 1 | |− | |− even that information is created through measurement. 1 2 2 hµ = −|Q|− |Uij | log |Uij | , (11) In any case, the key conclusion is that, since both pro- X X 2 i=0 j=0 cesses are represented by a 3-state QFG constructed from the same internal quantum dynamics, it is the means The entropy rate hµ quantifies the irreducible random- of observation alone that affects the amount of memory. ness in processes: the randomness that remains after the This was illustrated with the particular examples of the correlations and structures in longer and longer sequences spin-1 particle in a magnetic field. Depending on the are taken into account. choice of observable the spin-1 particle generates differ- The latter, in turn, is measured by a complementary ent process languages. We showed that these could be quantity. The amount I(←S; →S) of mutual information [12] analyzed in terms of the block entropy—a measure of shared between a process’s past ←S and its future →S is uncertainty, the entropy rate—a measure of irreducible given by the excess entropy E [24]. It is the subextensive randomness, and the excess entropy—a measure of struc- part of H(L): ture. Knowing the (deterministic) QFG representation, these quantities can be calculated in closed form. E = lim [H(L) − hµL] . (12) We established a connection between quantum au- L →∞ tomata theory and quantum dynamics, similar to the Note that the units here are bits. way symbolic dynamics connects classical dynamics and Ref. [24] gives a closed-form expression for E for order- automata. By considering the output sequence of a re- R Markov processes—those in which the measurement peatedly measured quantum system as a shift system we symbol probabilities depend only on the previous R − 1 found quantum processes that are sofic systems. Taking symbols. In this case, Eq. (12) reduces to: one quantum system and observing it in one way yields a subshift of finite type. Observing it in a different way E = H(R) − R · hµ , (13) yields a (strictly sofic) subshift of infinite type. Conse- 6 quently, not only the amount of memory but also the soficity of a quantum process depend on the means of observation. The preceding attempted to forge a link between quan- This can be compared to the fact that, classically tum dynamical systems and quantum computation by the Golden Mean and the Even sofic systems can be extending concepts from symbolic dynamics to the quan- transformed into each other by a two-block map. The tum setting. We believe the results suggest further study adjacency matrix of the graphs is the same. A similar of the properties of quantum finite-state generators and situation arises here. The unitary matrix, which is the the processes they generate is necessary and will shed corresponding adjacency matrix of the quantum graph, light on a number of questions in quantum information is the same for both processes. The processes can be processing. One open technical question is whether sofic transformed into each other by expressing one set of quantum systems are the closure of quantum subshifts of projection operators in the eigenbasis of the other. This finite-type, as they are for classical systems [23]. There transformation always exists since the operators simply are indications that this is not so. For example, as represent different orthonormal basis sets spanning the we noted, quantum finite-state recognizers can recognize Hilbert space. nonregular process languages [11].

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