An extension to computational mechanics complexity measure is proposed in order to tackle quantum states complexity quantification. The method is applicable to any $n-$partite state of qudits through some simple modifications. A Werner state was considered to test this approach. The results show that it undergoes a phase transition between entangled and separable versions of itself. Also, results suggest interplay between quantum state complexity robustness rise and entanglement. Finally, only via symbolic dynamics statistical analysis, the proposed method was able to distinguish separable and entangled dynamical structural differences.
Over the last two decades, diverse attempts to quantify complexity have been proposed, a significant amount of them make use of information-theoretic or computational tools to address this issue [1,2]. Their use in various systems analysis justifies these efforts of complexity quantification in order to better understand complex systems, unraveling underlying structures and sometimes bridging together very distinct systems. To assess systems quantum systems there are already proposed quantum informational complexity measures [3][4][5]; however, they do not have desired features from common complexity measures proposed to classical systems because all of them are a quantum extension of Kolmogorov's algorithmic complexity [6].
Hence, they all will present the latter attribute: monotonically increasing function of disorder [7] That is an undesirable feature for a modern complexity measure to have since its objective is to measure the degree of organization between the periodic and the random, and not how disordered a system is. To this goal, there are a number of entropies to be used [8].
Therefore, a modern quantum complexity measure is a still new and unknown path to follow, by the use of quantum information theory together with modern complexity measures concepts. Thus, in this work a well-established complexity measure is extended to accommodate quantum informational framework in order to properly tackle the quantum states complexity issue.
In this Letter, we briefly introduce computational mechanics complexity measure framework [9]. Next, an iterative successive measurement procedure is defined for quantum states to adequate them to the method. Then, the full approach is applied to a bipartite mixed state of qubits, the Werner state. Finally, results are discussed and future work is addressed.
Originally proposed by Crutchfield and Young [9] as a new way to quantify dynamical systems complexity, and then extended to a whole research field, computational mechanics is mainly based on a probabilistic automaton construction to imitate the analyzed system symbolic dynamics. Through this automaton, it is possible to quantify the intrinsic computation a dynamical system performs, hence its complexity, as well.
The first step to reconstruct the automaton is a symbolic dynamics extraction through defining a state space generating partition M ǫ , constructed with cells of size ǫ which is sampled every τ time unities.
Then, using this defined measuring instrument {M ǫ , τ }, a sequence of states {x} is mapped to a string of symbols {s, s ∈ A}, where
is the number of partitions, and m bed is the data set embedded dimension.
In fact, there are a number of methods to reconstruct this automaton from the original proposition. Other approaches were developed to reconstruct an automaton only from the observed string outputted by {M ǫ , τ }. One of these is the Causal-State Splitting Reconstruction (CSSR) algorithm [10].
After the automaton reconstruction, it is described with a transition matrices set {T (γ) :
γ ∈ A}, one for each symbol in the alphabet, defined by
where p α , ij is the probability of being in state i to go to state j through outputting symbol γ. Then the probabilistic connection matrix is defined
The largest eigenvalue λ of T is positive [9], and its associated eigenvector p = {p s : s ∈ S} has non-negative elements representing the asymptotic state probabilities, where S is the states set.
Finally, the α-order automaton complexity is defined as p Rènyi entropy
Thus, it is clear that C α is generally an intensive quantity, i.e., scale invariant; hence only automaton information fluctuation is considered. However, if α → 1, C 1 is a extensive quantity; thus, besides automaton information fluctuation, also the number of states will influence the C 1 value.
Therefore, higher values of C 1 are a product of larger and more probabilistic equally distributed automaton.
The use of computational mechanics complexity with quantum states is not straightforward. It is necessary to define an iterative successive measurement process to characterize the underlying quantum state probability distributions and this will be called Quantum State Sampling (QSS).
QSS is described in Figure 1. The analyzed quantum state is produced by a perfect Source. The source generates identical quantum states and for each one of them a sequential eigenstate projective measurement procedure occurs in its sub-systems, which constitute the original quantum state. After the last sub-system measurement takes place, the original state is completely destroyed and another one is generated by the source to undergo the same routine. This is done iteratively, and the outcomes are generated sequentially constructing a string (s 0 s 1 s 2 s 3 • • •). Finally, this string is the input for the CSSR algorithm and therefore to computational mechanics complexity.
This approach to adapt computational mechanics complexity measure to quantum
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