We compare different strategies aimed to prepare an ensemble with a given density matrix ρ. Preparing the ensemble of eigenstates of ρ with appropriate probabilities can be treated as 'generous' strategy: it provides maximal accessible information about the state. Another extremity is the so-called 'Scrooge' ensemble, which is mostly stingy in sharing the information. We introduce 'lazy' ensembles which require minimal effort to prepare the density matrix by selecting pure states with respect to completely random choice. We consider two parties, Alice and Bob, playing a kind of game. Bob wishes to guess which pure state is prepared by Alice. His null hypothesis, based on the lack of any information about Alice's intention, is that Alice prepares any pure state with equal probability. Then, the average quantum state measured by Bob turns out to be ρ, and he has to make a new hypothesis about Alice's intention solely based on the information that the observed density matrix is ρ. The arising 'lazy' ensemble is shown to be the alternative hypothesis which minimizes type I error.
We compare different strategies aimed to prepare an ensemble with a given density matrix ρ. Preparing the ensemble of eigenstates of ρ with appropriate probabilities can be treated as 'generous' strategy: it provides maximal accessible information about the state. Another extremity is the so-called 'Scrooge' ensemble, which is mostly stingy in sharing the information. We introduce 'lazy' ensembles which require minimal effort to prepare the density matrix by selecting pure states with respect to completely random choice. We consider two parties, Alice and Bob, playing a kind of game. Bob wishes to guess which pure state is prepared by Alice. His null hypothesis, based on the lack of any information about Alice's intention, is that Alice prepares any pure state with equal probability. Then, the average quantum state measured by Bob turns out to be ρ, and he has to make a new hypothesis about Alice's intention solely based on the information that the observed density matrix is ρ. The arising 'lazy' ensemble is shown to be the alternative hypothesis which minimizes type I error.
The Scrooge ensemble associated with a density matrix ρ is the maximally random ensemble whose average density matrix is ρ. It has attracted growing interest for its applications to quantum many-body dynamics and quantum information. The intrinsic statistical properties of the Scrooge ensembles are encoded in their higher-order moments, yet they are nontrivial to analyze and prepare for they feature a non-polynomial Haar integrand, blocking direct use of Haar integration machinery. Existing physical mechanisms to sample multiple copies of states from the Scrooge ensembles, meanwhile, typically require costly postselection, rendering them inefficient for large system size. In this work, we develop efficient and accurate methods for analyzing and implementing Scrooge moments. We first introduce a polynomial approximation to the Scrooge k-th moments whose error is tunable down to a threshold that improves from inverse polynomial to exponential in 1/∣∣ρ∣∣∞ over previous results. Building on our approximation, we then give the first efficient quantum algorithm that estimates observable expectation values using O(k3/ε5) copies of ρ for any target error ε above this threshold. We further provide a construction of a block encoding of the Scrooge k-th moment when the preparation circuit of ρ is known. Of independent interest could be a key subroutine in our algorithms, which efficiently implements projection onto the k-fold symmetric subspace for ρ⊗k using O(k3/ε2) samples of ρ, avoiding the naive k! postselection cost of such a projection. These results provide new tools for both the theoretical analysis and practical prediction of properties of Scrooge ensembles, with potential applications to quantum many-body dynamics and quantum information processing.
Alice prepares two large qubit-ensembles E1 and E2 in the following states: She individually prepares each qubit of E1 in |0> or |1>, the eigenstates of Pauli-z operator Z, depending on the outcome of an unbiased coin toss. Similarly, she individually prepares each qubit of E2 in |+> or |-> the eigenstates of Pauli-x operator X. Bob, who is aware of the above states preparation procedures, but knows neither which of the two is E1 nor Alice's outcomes of coin tosses, needs to discriminate between the two maximally mixed ensembles. Here we argue that Bob can partially purify the mixed states (E1, E2), using the information supplied by central limit theorem. We will show that, subsequently Bob can discriminate between ensembles E1 and E2 by individually rotating each qubit state about the x-axis on Bloch sphere by a random angle, and then projectively measuring Z. By these operations, the variance of sample mean of Z measurement outcomes corresponding to the ensemble E1 gets reduced. On the other hand, qubit states in E2 are invariant under the x-rotations and therefore the variance remains unaltered. Thus Bob can discriminate between the two maximally mixed ensembles. We analyse the above problem both analytically as well as numerically, and show that the latter supports the former.
We compare different strategies aimed to prepare an ensemble with a given density matrix rho. Preparing the ensemble of eigenstates of rho with appropriate probabilities can be treated as `generous' strategy: it provides maximal accessible information about the state. Another extremity is the so-called `Scrooge' ensemble, which is mostly stingy to share the information. We introduce `lazy' ensembles which require minimal efforts to prepare the density matrix by selecting pure states with respect to completely random choice. We consider two parties, Alice and Bob, playing a kind of game. Bob wishes to guess which pure state is prepared by Alice. His null hypothesis, based on the lack of any information about Alice's intention, is that Alice prepares any pure state with equal probability. Then, the average quantum state measured by Bob turns out to be rho, and he has to make a new hypothesis about Alice's intention solely based on the information that the observed density matrix is rho. The arising `lazy' ensemble is shown to be the alternative hypothesis which minimizes the Type I error.