Phase Transition in Iterated Quantum Protocols for Noisy Inputs

Quantum information processing exploits all the features quantum mechanics offers. Among them there is the possibility to induce nonlinear maps on a quantum system by involving two or more identical copies of the given system in the same state. Such maps play a central role in distillation protocols used for quantum key distribution. We determine that such protocols may exhibit sensitive, quasi-chaotic evolution not only for pure initial states but also for mixed states, i.e. the complex dynamical behavior is not destroyed by small initial uncertainty. We show that the appearance of sensitive, complex dynamics associated with a fractal structure in the parameter space of the system has the character of a first order phase transition. The purity of the initial state plays the role of the control parameter and the dimension of the fractal structure is independent of the purity value after passing the phase transition point. The critical purity coincides with the purity of a repelling fixed point of the dynamics and we show that all the pre-images of states from the close neighborhood of pure chaotic initial states have purity larger than this. Initial states from this set can be considered as quasi-chaotic.

Introduction -The effect of noise is a threat to all information processing applications of quantum mechanics. Too much external noise or, with other words, randomness destroys quantum coherence and harms the successful operation of a quantum machine. In order to study noise in quantum computation on an elementary model, we assume that all essential ingredients of quantum information processing are available: initial state preparation, coherent manipulation and gaining information by measurement at the output. Any possible quantum computer capable of coherent manipulation of qubits is known to be reducible to an arrangement of identical two-qubit gates (e.g. a controlled NOT gate) and singlequbit gates. Perhaps the simplest arrangement of these elements is a CNOT gate followed by one single-qubit gate with an adjustable parameter and a von Neumann measurement on one of the output qubits. We will apply this arrangement on two qubits from an ensemble of identically prepared qubits and measure one of the output qubits [1] and then simply repeat the same procedure.
Iterative application of this scheme is known to exhibit surprisingly rich dynamics, including: complex chaos, true sensitivity to initial conditions and fractal bordered convergence regions to stable cycles of the dynamics in the space of initial states [2,3], which can be applied for useful purposes e.g. for quantum state discrimination [4][5][6]. None of these features would be possible by undisturbed unitary dynamics [7,8] or even in an open quantum system governed by a completely positive map, as they act linearly on superpositions of initial states. It is the postselection based on measurement results after each iterational step which gives rise to nonlinear behavior [9][10][11] and leads to the above sensitive phenomena.
But how sensitive are these effects to external noise? One may naïvely expect that for noisy initial states the fine details of the fractal structures characterizing the border between convergence regions in the space of initial states would be washed out [12,13].
For pure state inputs, measurement induced iterative dynamics of qubits can be effectively described by iterated maps with one complex variable, having the form of a quadratic polynomial or a quadratic rational function. It has been proven that by replacing the CNOT gate by another appropriate unitary gate, we can implement any quadratic rational function (including quadratic polynomials). Moreover, any rational function of arbitrary degree can be realized by higher dimensional unitary gates with more input qubits [14]. Thus it is not a surprise that the operation of the above scheme for pure initial states exhibits high resemblance to the well-known and extensively studied logistic map of chaos theory, being itself an iterated quadratic polynomial with one real variable [15][16][17]. Among the basic features of such maps we can list the appearance of period doubling, the appearance of limit cycles of high degree and the appearance of fractal structures specifying the boundary between different asymptotic stationary regimes of the system. For noisy (mixed state) inputs, however, the dynamics does not reduce to a map with one complex variable, therefore we can expect the occurrence of novel phenomena not included in previous studies which have been limited primarily to initial pure states which assumption considerably simplified the analysis.
In this Letter we show that the characteristic fractal structure of the borders between various convergence regions in a measurement-induced nonlinear protocol for qubits is surprisingly robust against randomness in the initial states. In particular, we prove that the fractal structure survives within initially noisy (mixed) states. We numerically estimate the fractal dimension of the appearing structure and prove that the dimension is independent of the initial purity above a specific threshold which is defined by a critical initial purity of the system. Below the critical purity, we find that the fractal dimension drops. This sudden change has the property of a first order phase transition [18] where the phase of the component is characterized by the presence of a fractal structure. Additionally, we study which noisy initial states have a close connection with the pure-state fractal and can thus be termed "quasi chaotic".
Nonlinear protocol -We consider the following nonlinear protocol. Given an ensemble of qubits prepared in the same quantum state ρ we take pairs and apply a controlled-NOT gate on them. Then we measure the target qubit in each pair, and we keep the control qubit only if the respective target was found in the |0 state. These kept qubits will form a new ensemble the state of which will be described by a density matrix where each element is squared in the computational basis [1,2,19]. If we subsequently act with a single-qubit gate, and then iterate this whole procedure, we can find a rich dynamical behavior: Attractive and repelling fixed cycles form nontrivial structures (fractals) on the Bloch sphere of pure initial states [14].
In order to analyze the dynamics of mixed initial states here we fix the single-qubit gate to be the Hadamard gate. One step of the above iterative dynamics reads where the symbol stands for elementwise (Hadamard) product in the computational basis. The density matrix can be parameterized with its Bloch-sphere coordinates u, v, w ∈ R as and the purity of the state is given by P = Tr(ρ 2 ) = (1 + u 2 + v 2 + w 2 )/2 ≤ 1, which leads to the condition u 2 + v 2 + w 2 ≤ 1. After one step of the protocol the parameters describing the state are transformed by the nonlinear map M : The analysis of M reveals that the (u, 0, w) plane is an invariant subset of the map, where all the four quadrants are mapped to the first quadrant within at most two steps in a counterclockwise manner (c.f. Fig 1a). Three of the fixed points of M (C 0 , C 1 , and C 2 ) and both of its length-2 cycles (C 3 and C 4 ) lie on this plane (see Table I).
C 0 and C 3 are attractive, while C 1 is repelling in every direction, this can be shown by examining the second iteration for a small neighborhood. Cycles C 2 and C 4 exhibit both attractive and repelling behavior depending on the direction. Furthermore, M has two other fixed points, C 5 and C 6 (see Table I), which do not lie on the (u, 0, w) plane. According to our numerical analysis, the map M does not have any globally attractive cycles beyond C 0 and C 3 . Pure-state dynamics -In the case of P =1 the evolution by the map M does not change the purity, the states remain pure [19]. For pure states one can introduce a complex variable z with |z| 2 = (1−w)/(1+w), and arg(z) = arctan(−v/u) to parameterize quantum states The iterative dynamics can then be described by the complex quadratic rational function f M (z) = (1 − z 2 )/(1 + z 2 ). Its behavior may be analyzed by the theory of complex rational maps [5,14,17,20,21]. It can be shown that the fixed points C 2 , C 5 and C 6 are repelling fixed points of f M , therefore they are part of its Julia set, while C 3 is its single attractive fixed cycle (c.f. Fig. 1b) corresponding to the states ψ Singlequbit pure states which converge to this attractive cycle form the so-called Fatou set of the map, all other pure states belong to the Julia set containing also the repelling cycles, which are everywhere dense in it. Perfectly pure initial states (states without any statistical noise) that belong to the Julia set remain in the Julia set under the iteration of f M . The Julia set is formed at the border between the different Fatou-components -in this case at the border between sets of points which converge to C after an even number of iterations. This is shown in Fig. 1b which is a stereographic projection of the P = 1 surface of the Bloch sphere. The Julia set is a fractal, its Hausdorff dimension can be calculated e.g. by the periodic point algorithm it is estimated to be 1.5655 [22]. Since the existence of chaos is due to the fact that the Julia set is nonvacuous [17], we can see that for pure initial states our protocol has chaotic dynamical regimes.
Phase transition for noisy inputs -Considering noisy (i.e., mixed) initial states, our nonlinear protocol can be analyzed by examining the convergence of each initial state within the Bloch sphere under the iterative application of the map M. There are three characteristically different behaviors: an initial state can converge to either one of the two globally attractive cycles C 0 and C 3 or not converge stably to any of these. Fig. 1(a) shows the different convergence regions for initial states on the (u, 0, w) plane. Since C 0 corresponds to the completely mixed state ρ 0 = (|0 0|+|1 1|)/2, while C 3 corresponds to the pure-state cycle ψ , the question naturally arises whether a general state with purity 1/2 < P < 1 will become pure or completely mixed by the iteration of the protocol. It turns out that initial states with the same purity and close to each other may converge to the same or to different attractive cycles according to some complicated pattern. In order to visualize this behavior, in Fig. 1(c)-(d) we present stereographic projections of surfaces corresponding to initial states with a given P (with the respective value represented by the dotted circles on Fig. 1(a)), colored according to which cycle the states converge to.
As the purity of initial states is decreased below 1, in addition to the pure-state cycle ψ , one can find states which converge to the maximally mixed state ρ 0 . This means that the presence of noise in the initial quantum state can lead to a different output of the protocol than one would get for an ideal pure-state input. However, the border between states converging to the different attractive cycles preserves its fractalness. Fig. 1(c) shows the case when the fractal is still visible with the naked eye at the same magnification and resolution as the pure-state fractal. For lower values of the purity, the fractal further shrinks and can only be seen under magnification, our numerical simulations indicate that it is present only above a certain threshold which we found to be P c ≈ 0.76929 (which is equal to the purity P 1 of the fixed point C 1 up to this numerical precision). Furthermore, we have calculated the dimension of the fractals using the box counting method [23] for different P values, as shown in Fig. 2. The fractal dimension approximately remains the same value as for P = 1 until the purity is decreased to P c , where it suddenly drops to the value ≈ 1, which means that the convergence regions become regular (i.e., non-fractal) below this value. Note, that the applied numerical method systematically underestimates the box counting dimension D bc due to the finite resolution of the numerically determined fractal.
The fact that the fractal dimension remains the same for noisy initial states is a nontrivial result, since convergence to the completely mixed state (color red) is present for all purities less than one and some parts of the original fractal structure visibly disappear. Calculation of the box counting dimension confirms that although the fractal looks different from the pure-state case, its fractal  nature and even its calculated dimension does not change when noisy inputs are used as long as the initial purity is above a certain critical value P c . It changes only when more noisy (i.e., less pure) states are used as inputs, in which case the fractal dimension suddenly drops due to the fact that the border between the different convergence regions loses its fractalness. This effect resembles a first-order phase transition where the purity plays the role of the control parameter and the sudden appearance of a fractal at the critical purity with a jump in the fractal dimension marks the transition point [18].
Quasi-chaotic noisy initial states -In the case of pure inputs corresponding to the Julia set, iterations are numerically unstable and lead out of the set due to numerical errors. However, under the inverse map f −1 M (which is, in fact, a pair of maps since each point has two preimages) the Julia set is attractive and iterations are numerically stable. Moreover, the set of pre-images is everywhere dense in the Julia set. We assume that some points with purity P < 1 will tend to the Julia set in a similarly unstable manner as it happens for the points of the Julia set itself. We can find such points numerically by applying backwards iteration to initial states close to the Julia set, but with purity P < 1. With backwards iteration we have two pre-images in each step (M −1 + and M −1 − ) thus we can choose which branch we consider [12]. There are two special branches: M −1 + • n and M −1 − • n . In order to find points which tend to the Julia set we have iterated randomly chosen branches of the inverse map starting from randomly chosen mixed initial states from the vicinity of the Julia set (see Fig. 3). We found that none of the resulting points have a purity smaller than P 1 (our numerical results approach P 1 from above up to a numerical deviation of 8.6 × 10 −6 ). The special branch M −1 − • n does not decrease the purity asymptotically, as it maps points towards its two fixed points C 5 and C 6 , both corresponding to pure states. The other special branch M −1 + • n on the other hand, tends to P 1 for any such initial state (this property is due to the fact that this branch has C 1 as a fixed point). Branches with a combination of the two types of pre-images have not been found to show any other tendency. The fact that the fractal of the pure case thus does not have any preimages with purity smaller than P 1 together with our results regarding the drop of the fractal dimension and the disappearance of the fractal-like structures at approximately P 1 indicates that the critical purity P c coincides with P 1 . Another question is, whether we can find a relationship among the border points separating the convergence regions in the case of different initial purities. The iterative behavior of these points is -similarly to the case of the Julia set -numerically unstable. However, using the inverse map M −1 , iterations are stable, and we can look for pre-images of such points on surfaces of different purity.
Let us first focus on the asymptotics for initial states on the (u, 0, w) invariant plane. Here we find that points from the vicinity of C 2 (which is itself a border point on the P = 1 surface) converge to C 1 under the special branch M −1 + • n and the other branches produce the pre-images of these points in the other quadrants (see Fig. 1(a)). Points from the vicinity of C 4 also converge to C 1 under the special branch M −1 + • n , while the other branches result in the pre-images in the other quadrants. Reversing this behavior, we see that on this plane the points on the border of the light blue and dark blue regions converge (unstably) to the point C 2 , which belongs to the Julia set, while the points on the border between the red and blue regions converege (unstably) to the fixed cycle C 4 .
We used the same method for two sets of initial points which lie outside the invariant plane: (i) points in the vicinity (P < 1) of randomly chosen points of the Julia set and (ii) points from the vicinity of the red-blue border on the (u, 0, w) plane. We found a similar behavior as on the invariant plane. Namely, with the different branches of the backwards iterated map, the border points of the two blue regions remained on such type of borders, and the red-blue border points also stayed on the border between the red and blue convergence regions. This is illustrated in Fig. 4, which shows points which are pre-images of the two sets (i) and (ii) on the P = 0.9 surface.
Reversing the results of the backwards iterational method reveals that the border points of the light and dark blue regions converge (unstably) to the points of the Julia set. Since the points of the Julia set are related to the chaotic dynamical behavior of our protocol, these mixed initial states may be considered quasi-chaotic. Let us note that the states which correspond to the red-blue border points cannot be directly connected to the pure chaotic states with this method.
Conclusions-We studied the iterated dynamics of a particular measurement-induced nonlinear quantum protocol. We demonstrated that the dynamics is sensitive to initial conditions, and a fractal structure describes the asymptotics of initially pure input states. We found that the fractalness of the asymptotics is robust towards noise in the initial states in the sense that the border between the different convergence regions remains a fractal with the same dimension as in the pure case. We have shown that there is a sudden transition from fractal border to non-fractal border between convergence regions of states as a function of the initial purity of input states. The transition takes place at a single critical initial purity which resembles a first order phase transition known from thermodynamics, where the purity plays the role of the control parameter. We found similar results by arguing that the pre-images of the pure-case fractal cannot be found for purities below the critical purity. In fact, they can be traced back to points lying at the border between the regions which converge to the pure-state cycle for pu-rities P 1 ≤ P < 1. As these points converge to chaotic points, they can be considered quasi chaotic. The other types of fractal-border points cannot be shown to have a connection with the chaotic regime of the dynamics.
Our model quite naturally resembles the well-known logistic map. We studied other properties than those usually analyzed for quantum chaos [24]. Due to their inherent complex character, our results underline much more the high degree of complexity which iterated rational maps can exhibit. We demonstrate that the application of such protocols requires a careful check of the parameters with which they are applied as the resulting chaotic dynamics can appear in a rather wide range of (physically relevant) parameters. A challenge is then to study protocols for higher dimensions relevant, e.g., for entanglement distillation [21,25,26]. When iterated functions of more complex variables are involved, it is a formidable task to derive analytic results [27]. Another one of the open questions remaining to be analysed is what happens for nonlinear protocols with different rotations chosen in place of the Hadamard gate, or for higher dimensional inputs.