Exponential Sensitivity and its Cost in Quantum Physics

State selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system has to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schrödinger microscope) is possible; however, there is a strict bound on the number of copies needed.

Quantum technology progresses at a fast pace. Preparation, control and measurement of coherent quantum systems 1 became possible on an unprecedented level leading to a wealth of proposals of applications ranging from quantum information processing to high precision measurements and sensors. In these protocols, increasingly sophisticated sequences of coherent evolution, measurement and post-selection are applied in order to control the state of quantum systems. Dynamics achieved by state selective protocols was proven essential for a large number of quantum information protocols 2,3 and quantum communication 4 . Prominent examples of probabilistic protocols are the KLM scheme 2 for linear optical quantum gates or the entanglement purification protocols [5][6][7] employing measurement and selection in order to increase the entanglement between subsystems.
Manipulation by measurement and selection breaks the linearity of quantum mechanics, thereby broadening the possibilities for quantum evolution [8][9][10][11] . In contrast, in the well established field of quantum chaos 12 one studies the signatures of chaos in closed quantum systems with linear evolution. In such systems the linearity of time evolution prevents distance growing between two initial quantum states. However, the essential non-linearity of an iterated, state selective protocol can result in truly chaotic behaviour [13][14][15] , meaning that initially close quantum states can get separated rapidly. The existence of such sensitive quantum protocols has been shown in 13 , but sensitivity was proved only for a tiny fractal subset of initial states having zero measure. In this article we demonstrate that exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, we show that any complex rational polynomial map, including the example of the Mandelbrot maps 16 , can be directly realised using state selective protocols, bringing a whole new class of quantum protocols to life.
From a fundamental point of view, one can search for the most general evolution for a quantum system. A very general dynamics is sometimes imagined as the action of both unitary evolution and non-selective measurements on a system together with one or more ancillas. The evolution reduced for the system only is called a quantum channel. When talking about quantum states in practice, it is unavoidable to be able to repeat experiments on an ensemble of identically prepared initial states in order to uncover the underlying probabilistic laws. This ensemble view of quantum states allows for the following trick when designing the most general dynamics for a given initial state. Let us, for example, consider systems from the ensemble pairwise and let them interact with each other. After the interaction one can perform a measurement on one of the pairs and then discard the measured member of the pair. In case of selective measurement, one may also discard the unmeasured member of the pair, depending on the measurement result. The resulting ensemble will be reduced in size, but some of its properties may be changed in a beneficial way, e.g. entanglement between subsystems. The above procedure goes beyond the usual notion of quantum channels in the following sense. The initial step of the procedure, namely taking the systems pairwise, can be viewed as splitting the original ensemble into two parts and employing one part as an ancilla. In other words, the state of the ancilla will be dependent on the state of the system. The state dependent ancilla lies at the heart of the non-linearity of the process.

Results
A linear optical experimental scheme implementing a family of non-linear maps. We propose here a simple experimental setup that implements a non-linear process exhibiting exponential sensitivity to the initial state. Our scheme is inspired by an experimentally tractable entanglement purification protocol 17,18 and uses only linear optical elements. At the beginning of each iterative step we form pairs of photonic qubits from the ensemble of identically prepared photons and apply a post-selective transformation on the pairs. After measuring the polarization of one photon we either keep or discard the other photon depending on the measurement result. The post-selection induces a non-linear, deterministic transformation on the remaining photons; therefore, the kept photons remain identically prepared.
The key element of our scheme is the polarizing beam splitter (PBS). When two photons arrive at the same time but from different spatial input modes, this linear optical element introduces entanglement between the spatial modes and the polarization degrees of freedom, see Fig. 1. We apply post-selection and accept the output of the PBS only if there is a photon in both spatial output modes.
Let H and V denote the horizontal and vertical polarization states for our photonic qubits. Consider the effect of the PBS acting on a product state of two incoming photons: After post-selection the remaining quantum state is we may apply a Pauli-Z gate whenever we measure − ( Fig. 1(f)) or simply neglect such cases, introducing another level of post-selection ( Fig. 1(g)). Either way the protocol implements a non-linear transfor- which maps the identical qubit states of an ensemble to another qubit state of a smaller identical ensemble. If we iterate this process S amended with an additional unitary step U, the iterates (US) n exhibit increasingly rich dynamics.
It has been shown 13 that by iteratively applying US on an ensemble of identically prepared qubits, the one qubit state of the ensemble after n iterations ψ α β = + H V n n n may evolve sensitively. By sensitivity we mean that for some initially similar quantum states ψ ψ , ′ 0 0 the evolving states ψ ψ , ′ n n can get very different during iteration, i.e. using some quantum information distance d (e.g. the Bures distance) we can get 0 . More precisely we call the process sensitive at some initial state ψ 0 if arbitrarily close quantum states can get separated from it to a constant distance C, i.e. ε ψ . We call this exponential sensitivity if it also holds requiring ~ε ( / ) n log 1 . Sensitivity has been shown for initial states lying on a fractal subset of the Bloch sphere called the Julia set 16,19 , see Fig. 2. The specific fractals that were examined regarding the protocol US 13 all had zero measure. However certain choices of unitaries from the family θ θ U e e cos s in sin c os i i seem to produce increasingly saturated Julia set images (see Fig. 2) suggesting that it may reach a point where the whole Bloch sphere is covered by sensitive initial states. A candidate for such a transformation is Φ = π π Exponential sensitivity for all initial states. In order to handle the arising non-linear maps better we project the surface of the Bloch sphere to the complex plane using stereographic projection. Thus a (photonic) qubit ψ α β = + H V may be described using a single complex parameter including infinity: This representation yields () a new description of our protocol in terms of rational functions 16,19 : : Using this formalism, it turns out that f is one of a few special so-called Lattès maps 20 and as such has gained a lot of attention in the theory of complex dynamical systems 19 . We can better understand the special properties of our Lattès map by analysing its relationship to the corresponding linear transformation of the 2 dimensional torus. We will represent the torus C Z m od 1 , which rotates and folds the torus 2 times over itself. The correspondence between the torus and the sphere is established via the so called Weierstrass elliptic function 21 Relating the two surfaces gives rise to the identity showing that iterating f on  has essentially the same effect as repeatedly applying multiplication ⋅( − ) Fig. 3 and Methods. Viewing Φ through these glasses it becomes clear that it shows chaotic behaviour on the whole Bloch sphere. The map representing Φ on the torus uniformly stretches the surface of the torus by a factor of 2 and folds over itself two times. It is intuitively clear that the iterative application of such a transformation shows exponential sensitivity to the initial position on the torus and has a positive Lyapunov exponent. Even more strikingly, it exhibits exponential mixing, yielding that during the iteration of Φ even a tiny uncertainty about the initial state evolves exponentially fast to a complete uncertainty, meaning that the iterated state may be any point of the Bloch sphere, as depicted on Fig. 3. For a rigorous derivation of the exponential mixing see Methods.  A notable consequence is that we found a direct quantum physical realisation of the Mandelbrot maps → + z z c 2 and can devise a quantum circuit for it. Figure 4 shows a possible quantum circuit implementation for this family of maps. The scheme demonstrates how the corresponding family of 2-qubit unitaries may be constructed using only controlled 1-qubit gates, which are considered experimentally more feasible in general.
For the sake of completeness we note that recently another strong connection between rational functions and quantum computing with post-selection was discovered 22 from an algorithmic perspective. The cost of non-linearity and the Schrödinger microscope. We have just shown that using post-selection one can implement a wide range of non-linear maps, which may be useful for various tasks. A highly non-linear map, like Φ , provides a sort of "Schrödinger microscope" 23 enabling one to exponentially magnify tiny differences between quantum states. Consider for example the behavior of Φ around the fixed state + = Φ + . Using our representation in terms of rational functions this fixed point becomes 1 = f (1).
translates to that Φ doubles infinitesimal distances around + , analogously to a microscope. Having such a tool we are tempted to develop powerful quantum algorithms utilising it. It is well known that introducing post-selection to Quantum Computing makes it extremely powerful -the corresponding complexity class PostBQP 24 includes NP and even PP. The question of efficiency and resource needs naturally arises. We address it using a black box argument considering results about state discrimination [25][26][27] .
Suppose we have a quantum device implementing n iterations of Φ processing a qubit ensemble of size N. The size M of the successfully processed output ensemble may be probabilistic. We would like to determine its success rate, i.e. the average ratio M/N. To derive a bound on the success rate consider applying this quantum device to the qubit ensemble having state either ψ 0 or ψ ψ ≠ j n for j = 0, 1 then, after n iterations of our process Φ , the distance of the two states increases by a factor of roughly 2 n since Φ doubles infinitesimal distances around the fixed point + . If the device outputs M copies of the transformed states then we can distinguish the ensembles with error probability . For large N the success rate is roughly constant because of the law of large numbers; thus we can treat the value M/N fixed. But ⋅ / M N 2 n cannot exceed 1 as the error probability of discrimination cannot decrease and so the success rate is upper bounded by 4 −n . This holds for states lying close to + , in better cases the rate may be higher. For our implementation scheme each iteration has a success rate at least 1/4 (up to a negligible term − 1/N due to parity) implying that this scheme provides the best possible worst case success rate.
In this way we have shown that exponentially many copies are needed for n iterations of the process. Similar upper bound can be devised to any non-linear map that have a region where the separation of close states can be described by a multiplicative factor λ > 1. If we follow the above argument it turns out that the worst case success rate of such protocol is bounded by 1/λ 2 . Note that the particular choice of metric by which we measured separation does not limit the scope of the argument too much -we could use any other metric that agrees infinitesimally, e.g. the Bures metric. Thus it turns out that the implementation of any kind of Schrödinger microscope needs exponentially many copies of the states in terms of magnification steps (more precisely, quadratically many in terms of the total magnification).

Discussion
While exploring the possible dynamical properties of state selective protocols, we found that any complex rational map can be implemented using state selection. Such a general and natural correspondence between a physical system and the theory of complex dynamical systems is unique to our knowledge. We could also devise a realistic optical experimental scheme, which implements particularly interesting quadratic rational dynamics.
We showed that a specific setting of the proposed optical scheme implements an exponentially mixing map. At several regions of the Bloch sphere this protocol magnifies initial differences between quantum states almost uniformly, thus we may call it a Schrödinger microscope. The term Schrödinger microscope was introduced by Lloyd and Slotine 23 in connection with a non-linear quantum protocol emerging from collective weak measurements and coherent feedback. Even though 23 introduces Schrödinger microscope at a conceptual level, there was no explicit example shown unlike in this article. Although the collective weak measurement approach seems very different from our post-selective scheme, they are connected at a deep level: the effective non-linearity comes from the underlying ensemble of identical quantum states in both cases.
During our study of the emergence of exponential sensitivity, we were led to the analysis of implementation cost, which turned out to be exponential. We found a general bound on the number of copies needed for the successful implementation of any expanding non-linear map. We proved that a protocol capable of magnifying differences between close quantum states by a factor λ > 1 necessarily has a worst case rate of loss at least 1 − 1/λ 2 in the number of copies of the unknown input quantum states. This "Quantum magnification bound" is basically another reformulation of the fact that one cannot bootstrap quantum information without an external source, somewhat resembling the quantum no-cloning theorem. We used the "Quantum magnification bound" principle to show the optimality of our implementation of a Schrödinger microscope. In general, this principle helps to understand the advantages and limitations of any kind of Schrödinger microscope, regardless of the actual implementation method. Thus it also overcomes the difficulties coming from approximative arguments required to describe complex systems, like the protocol 23 involving collective weak measurements and coherent feedback. This fundamental bound may be applied to other quantum information protocols, providing a general tool for bounding the success rate of particular probabilistic protocols.
Looking at general processes, with inspiration coming from this principle, may also provide a new insight to the relation of classical and quantum chaos 28 , suggesting that classical deterministic chaos may be just an approximation with a characteristic time scale. Classical deterministic chaotic systems explode quickly, and observing the deterministic evolution of the system, even at a macroscopic level, enables the determination of the initial conditions increasingly precisely 29 . But there is a level of precision that is prohibited by quantum uncertainty relations. This is an apparent philosophical contradiction, provided we believe classical physics is based on quantum mechanics; a possible dissolution is saying that on long time scales one cannot treat a classical process deterministically chaotic, just chaotic in some statistical sense. is a self inverse Möbius transformation 19 . Since conjugation by Möbius transformation does not change the iterative features, it implies that f exhibits the same dynamics as  f . To give a physical meaning to this Möbius transformation, we mention that it corresponds to a rotation of the Bloch sphere, i.e. f and  f essentially describe the same process, just written in a different qubits basis.

Methods
As we already indicated in this article, f is conjugate to the map . Now we use the well known property ℘′ = ℘ − ℘ − g g 4 2 3 2 3 . Since It is easy to show that after multiplying each point of  Initially the state of the n-tuples is the following product state: As before we use the parametrisation z = α/β for a qubit α β ( + ) 0 1 . Then the parameter of the unmeasured, post-selected qubit can be described as follows: 2 is non-zero. Then this success probability is also greater than some p probability for a fixed map, regardless the state φ due to compactness of the state space. (However, depending on the map, this lower bound may be arbitrarily low.) Thus using the above defined V unitary we can implement the rational function ∑ where z = α/β and N is a norming factor.