On the central limit theorem for unsharp quantum random variables

We study the weak-convergence properties of random variables generated by unsharp quantum measurements. More precisely, for a sequence of random variables generated by repeated unsharp quantum measurements, we study the limit distribution of relative frequency. We provide a representation theorem for all separable states, showing that the distribution can be well approximated by a mixture of normal distributions. Furthermore, we investigate the convergence rates and show that the relative frequency can stabilize to some constant at best at the rate of order 1 / N for all separable inputs. On the other hand, we provide an example of a strictly unsharp quantum measurement where the better rates are achieved by using entangled inputs. This means that in certain cases the noise generated by the measurement process can be suppressed by using entanglement. We deliver our result in the form of quantum information task where the player achieves the goal with certainty in the limiting case by using entangled inputs or fails with certainty by using separable inputs.


Introduction
Quantum theory predicts probability distribution of measurement outcomes. In practice, we identify probabilities with the relative frequencies of measurement outcomes in the limit of large number of experimental runs. The identification is justified by the so-called i.i.d. assumption, which demands that a certain physical process (e.g. use of quantum channel or state) is repeated arbitrarily many times identically and independently of other processes. The convergence to probability is guaranteed by the weak law of large numbers and the errors are quantified by the central limit theorem (CLT). However, one can naturally ask what happens if the i.i.d. assumption no longer applies? Clearly, such a framework is much less structured and it opens-up new possibilities and imposes new limitations both for quantum foundations and quantum information processing [1][2][3].
In classical probability theory the weak-convergence properties are fairly well understood for independent variables. The pioneering works by Kolmogorov, Chebyshev, Lindeberg and Lyaponov provided a good set of conditions for CLT and asymptotic normality to hold (see for example [4,5]). On the other hand, dependent variables are much more difficult to tackle. Generally, one has to impose certain restrictions, otherwise there is not much to say in the most general case. For example, the set of sufficient conditions for CLT to hold can be provided for the weaklydependent variables [6]. Furthermore, for random variables under the symmetry constraints such as exchangeability, one can derive the exact necessary and sufficient conditions [7]. In quantum setting, apart from the i.i.d. scenario [8][9][10][11], in recent years we have seen a plethora of CLT-type results mainly in the context of quantum many-body dynamics [12][13][14][15][16]. Yet, the main focus in these studies are the properties of quantum states and observables without counting the measurement effects. However, these effects are unavoidable in general. Our main concern here is to investigate how the noise generated by the measurement process affects the limiting distribution of measured quantities, i.e. the distribution in the asymptotic limit of many repeated measurement runs. For this matter, we investigate the limiting distribution for correlated inputs subjected to unsharp (POVM) quantum measurements. We have found two very different behaviors: (a) for separable (classically correlated) inputs asymptotic normality is Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. recovered, and (b) entangled inputs do not follow asymptotic normality in general. We derive the representation theorem for separable inputs, where the output distribution is well approximated by a mixture of normal distributions. It is very similar to the de-Finetti-type representation theorems (see [17][18][19] for classical and [20][21][22][23][24][25][26] for quantum scenario). Interestingly, unlike these cases, our result does not require any symmetry constraint, such as the permutational invariance (exchangeability), as long as the inputs are subjected to strictly unsharp measurements. Furthermore, we investigate the convergence rates and show that the relative frequency can stabilize to some constant value at best at the rate of order N 1 for all separable inputs. On the other hand, we demonstrate that entangled inputs behave in a very different way (as compared to separable ones). We provide a simple example where the entangled inputs can significantly increase the convergence rate. We deliver our result in the form of a quantum game [27] where the player is able to accomplish the task with certainty in the asymptotic limit by using entangled inputs or fails with certainty by using any separable inputs. M Tr r [ ]ˆ, respectively. We see that the uncertainty operator produces additional noise that comes solely due to measurement (note that V 0  Dˆin general). For all projective (von Neumann) measurements V 0 D = , and so this term vanishes. We focus on strictly unsharp measurements, i.e. we consider σ − σσ + , with σ − >0 being strictly positive for all states ρ. Furthermore, we assume that the third moment r X X T For a sequence of random variables X 1 , K, X N , where X i ä{x 1 , x 2 , ...}, generated by repeated measurements, we set X ( N) =X 1 +...+X N and R X N N N 1 = ( ) to be the relative frequency. Furthermore, we define the standardly normalized sum S X X The distribution of the relative frequency R N is the central object of our investigation. More precisely, we ask: what is the probability that R N takes some value in the limit of large number of experimental runs?

Separable inputs
The answer to the previous question heavily depends on the type of input state. For example, if one supplies in each run the same state ρ, the overall input state is described by an i.i.d. state ρ ( N) =ρ ⊗ N , where N is the number of experimental runs. The weak law guarantees the convergence of the relative frequency converges to the mean value X á ñ r and the CLT states that the distribution of S N converges to the normal distribution. A slightly more delicate example is the one of independent inputs, i.e. ρ ( N) =ρ 1 ⊗...⊗ρ N , where ρ i s are different in general.
Here we can define the mean variance and σ − Σ N σ + as each individual variance is bounded. We can apply the Lindeberg's condition for CLT [4], i.e. when N  +¥, therefore the normalized sum converges to the standard normal distribution. To quantify the deviation for finite N, we can use the Berry-Esseen theorem [29,30].
and C 0 is an absolute constant. We see that any product input state is subjected to CLT because the measurements are strictly unsharp (the variance is strictly bounded from bellow by σ − ). From here, we are ready to establish the representation theorem for separable states. For a given separable input ]of the relative frequency satisfies the following bound: The last inequality follows from (5). , Note that the bound (6) does not depend on any structure/symmetry of the underlying input state. This is in contrast to the de-Finetti-type representation theorems that heavily rely on symmetry.

Convergence rates and quantum game
In this section we will show that entangled states can behave very differently in certain cases compared to separable states with the respect to the distribution of the relative frequency. To illustrate our findings we will define the problem as an information-theoretic game between two players, Alice and Bob.
Suppose that Alice performs some POVM and generates a random variable Xä{x 1 , x 2 , ...} which is strictly unsharp, i.e. x [ ] for all ρ. As previously, we assume that third moments are bounded by T>0. She asks Bob to supply her with inputs, and his goal is to make the relative frequency R X X ...
) as close as possible to some pre-defined value X c . More precisely, he will try to maximize the probability with ò, α>0 being fixed parameters. The parameter α quantifies the convergence rate of the relative frequency to the constant X c . Our goal here is to show that the probability P N is negligible whenever α>1/2 for all separable states. And indeed, the bound (6) states that the distribution of R N is a mixture of Gaussians, therefore the error (as quantified by the convergence rate) cannot scale better than N 1 . We fix α>1/2.
for all separable inputs.

Proof. For a separable input
. Therefore it is sufficient to prove (8) for a product state. We set ρ ( N) =ρ 1 ⊗...⊗ρ N and, as previously N N i The last inequality follows from the Berry-Esseen bound (5). For a>0 the function Φ(x+a)−Φ(x−a) reaches its absolute maximum for x=0, hence x a x a a a a . Finally, we have The bound (8) states that the winning probability vanishes asymptotically P 0 N  with N  +¥, for all α>1/2. Therefore, Bob will fail to win the game with certainty by using separable inputs. Now we will provide a simple example where entanglement is able to beat the bound given by (8).

Entanglement counterexample
Consider a qubit three-outcome POVM with the elements on 'equilateral triangle' E m i i , , x y z s s s s  = { }is the vector of three Pauli matrices. We define the corresponding random variable with three possible values X ä{−1, 0, 1} and we set X c =0. It is convenient to introduce two operators A x , where x and z are components of the Bloch vector of the state ρ.
Clearly x 2 +z 2 1. A simple calculation shows that X Var , thus the third moment is bounded and we have T=1. The bound (8) applies to all separable inputs and α>1/2.
On the other hand, let Bob use the following input state , we can lower-bound the winning probability by using the Chebyshev's inequality where s X Var for L 0  . The second inequality follows from concavity of x 1for 0x1. Now we can derive the bound for variance b-. Since β<1/2 the last term is negligible. Furthermore, we see that the best rate is achieved for β=1/3.

Concluding remarks
In this letter we investigated the limit distribution of relative frequency for correlated inputs subjected to strictly unsharp quantum measurements. More precisely, for N quantum systems prepared in a generic state ρ, individually measured with POVMs that are strictly unsharp (i.e. the variance for individual outcomes is strictly positive), we found two different types of behavior for separable and entangled input state ρ. We found the standard behavior for all separable states (Theorem 1), meaning that errors and convergence rates of the relative frequency are at best at the order of N 1 . This scaling factor comes exclusively due to the measurement uncertainty. On the other hand, we have provided a concrete counterexample of an entangled input state ρ for which better convergence rates are achievable. This implies that entanglement can reduce the amount of noise generated by the measurement process. Therefore, our result shows a potential application for quantum metrology [32]. In addition, our results can be used for probabilistic entanglement detection [33][34][35]. Namely, we have shown that the winning probability (7) asymptotically reaches 1 for certain entangled states, whereas it asymptotically vanishes for all separable states, thus one can verify the presence of entanglement (in target state) with a very high probability even by using a single-copy of a quantum system (provided that N is sufficiently large). This is in agreement with our recent result presented in [33].