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Computation in Sofic Quantum Dynamical Systems Computation in Sofic Quantum Dynamical Systems 1, 1, Karoline Wiesner ∗ and James P. Crutchfield † 1Center for Computational Science & Engineering and Physics Department, University of California Davis, One Shields Avenue, Davis, CA 95616 We analyze how measured quantum dynamical systems store and process information, introduc- ing sofic quantum dynamical systems. Using recently introduced information-theoretic measures for quantum processes, we quantify their information storage and processing in terms of entropy rate and excess entropy, giving closed-form expressions where possible. To illustrate the impact of measurement on information storage in quantum processes, we analyze two spin-1 sofic quantum systems that differ only in how they are measured. I. INTRODUCTION shifts in sequence space. Definitions of entropy followed from this. However, no connection to quantum finite- Extending concepts from symbolic dynamics to the state automata was established there. quantum setting, we forge a link between quantum dy- In the following we develop an approach that differs namical systems and quantum computation, in general, from this and other previous attempts since it explic- and with quantum automata, in particular. itly accounts for observed sequences, in contrast to se- quences of (unobservable) quantum objects, such as spin Symbolic dynamics originated as a method to study chains. The recently introduced computational model general dynamical systems when, nearly 100 years ago, class of quantum finite-state generators provides the re- Hadamard used infinite sequences of symbols to analyze quired link between quantum dynamical systems and the the structure of geodesics on manifolds of negative cur- theory of automata and formal languages [11]. It gives vature; see Ref. [1] and references therein. In the 1930’s access to an analysis of quantum dynamical systems in and 40’s Hedlund and Morse coined the term symbolic terms of symbolic dynamics. Here, we strengthen that dynamics [2, 3] to describe the study of dynamics over link by studying explicit examples of quantum dynam- the space of symbol sequences in their own right. In the ical systems. We construct their quantum finite-state 1940’s Shannon used sequence spaces to describe infor- generators and establish their sofic nature. In addition mation channels [4]. Subsequently, the techniques and we review tools that give an information-theoretic analy- ideas have found significant applications beyond dynam- sis for quantifying the information storage and processing ical systems, in data storage and transmission, as well as of these systems. It turns out that both the sofic nature in linear algebra [5]. and information processing capacity depend on the way On the flip side of the same coin, computation theory a quantum system is measured. codes symbol sequences using finite-state automata. The class of sequences that can be coded this way define the regular languages [6]. It turns out that many dynamical systems can also be coded with finite-state automata us- II. QUANTUM FINITE-STATE GENERATORS ing the tools of symbolic dynamics [5]. In fact, Sofic sys- tems are the particular class of dynamical systems that To start, we recall the quantum finite-state generators are the analogs of regular languages in automata theory. (QFGs) defined in Ref. [11]. They consist of a finite set The study of quantum behavior in classically chaotic of internal states Q = {qi : i = 1,..., |Q|}. The state systems is yet another active thread in dynamical sys- vector is an element of a |Q|-dimensional Hilbert space: arXiv:0704.3075v1 [quant-ph] 23 Apr 2007 tems [7, 8] which has most recently come to address the hψ| ∈ H. At each time step a quantum generator outputs role of measurement. Measurement interaction leads to a symbol s ∈ A and updates its state vector. genuinely chaotic behavior in quantum systems, even far The temporal dynamics is governed by a set of |Q|- from the semi-classical limit [9]. Classical dynamical sys- dimensional transition matrices {T (s)= U ·P (s),s ∈ A}, tems can be embedded in quantum dynamical systems whose components are elements of the complex unit disk as the special class of commutative dynamical systems, and where each is a product of a unitary matrix U and a which allow for unambiguous assignment of joint proba- projection operator P (s). U is a |Q|-dimensional unitary bilities to two observations [10]. evolution operator that governs the evolution of the state P An attempt to construct symbolic dynamics for quan- vector hψ|. = {P (s): s ∈ A} is a set of projection oper- tum dynamical systems was made by Alicki and Fannes ators—|Q|-dimensional Hermitian matrices—that deter- [10], who defined shifts on spin chains as the analog of mines how the state vector is measured. The operators span the Hilbert space: s P (s)= ½. Each output symbol sPis identified with the measure- ment outcome and labels one of the system’s eigenvalues. ∗Electronic address: [email protected] The projection operators determine how output symbols †Electronic address: [email protected] are generated from the internal, hidden unitary dynam- 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 up- dates 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 dif- ferent 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.
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