Comparison of mixed quantum states

In this article, we study the problem of comparing mixed quantum states: given $n$ unknown mixed quantum states, can one determine whether they are identical or not with an unambiguous quantum measurement? We first study universal comparison of mixed quantum states, and prove that this task is generally impossible to accomplish. Then, we focus on unambiguous comparison of $n$ mixed quantum states arbitrarily chosen from a set of $k$ mixed quantum states. The condition for the existence of an unambiguous measurement operator which can produce a conclusive result when the unknown states are actually the same and the condition for the existence of an unambiguous measurement operator when the unknown states are actually different are studied independently. We derive a necessary and sufficient condition for the existence of the first measurement operator, and a necessary condition and two sufficient conditions for the second. Furthermore, we find that the sufficiency of the necessary condition for the second measurement operator has simple and interesting dependence on $n$ and $k$. At the end, a unified condition is obtained for the simultaneous existence of these two unambiguous measurement operators.Comment: 9 page

Entanglement-assisted weak value amplification

Large weak values have been used to amplify the sensitivity of a linear response signal for detecting changes in a small parameter, which has also enabled a simple method for precise parameter estimation. However, producing a large weak value requires a low postselection probability for an ancilla degree of freedom, which limits the utility of the technique. We propose an improvement to this method that uses entanglement to increase the efficiency. We show that by entangling and postselecting $n$ ancillas, the postselection probability can be increased by a factor of $n$ while keeping the weak value fixed (compared to $n$ uncorrelated attempts with one ancilla), which is the optimal scaling with $n$ that is expected from quantum metrology. Furthermore, we show the surprising result that the quantum Fisher information about the detected parameter can be almost entirely preserved in the postselected state, which allows the sensitive estimation to approximately saturate the optimal quantum Cram\'{e}r-Rao bound. To illustrate this protocol we provide simple quantum circuits that can be implemented using current experimental realizations of three entangled qubits.Comment: 5 pages + 6 pages supplement, 5 figure

Optimization of probabilistic quantum search algorithm with a priori information

A quantum computer encodes information in quantum states and runs quantum algorithms to surpass the classical counterparts by exploiting quantum superposition and quantum correlation. Grover's quantum search algorithm is a typical quantum algorithm that proves the superiority of quantum computing over classical computing. It has a quadratic reduction in the query complexity of database search, and is known to be optimal when no a priori information about the elements of the database is provided. In this work, we consider a probabilistic Grover search algorithm allowing nonzero probability of failure for a database with a general a priori probability distribution of the elements, and minimize the number of oracle calls by optimizing the initial state of the quantum system and the reflection axis of the diffusion operator. The initial state and the reflection axis are allowed to not coincide, and thus the quantum search algorithm rotates the quantum system in a three-dimensional subspace spanned by the initial state, the reflection axis and the search target state in general. The number of oracle calls is minimized by a variational method, and formal results are obtained with the assumption of low failure probability. The results show that for a nonuniform a priori distribution of the database elements, the number of oracle calls can be significantly reduced given a small decrease in the success probability of the quantum search algorithm, leading to a lower average query complexity to find the solution of the search problem. The results are applied to a simple but nontrivial database model with two-value a priori probabilities to show the power of the optimized quantum search algorithm. The paper concludes with a discussion about the generalization to higher-order results that allows for a larger failure probability for the quantum search algorithm.Comment: v2: Main text expanded to include analysis of the first-order optimization result. Close to the published versio

Multi-particle quantum walks in one-dimensional lattice

Quantum walk is a counterpart of classical random walk in the quantum regime that exhibits non-classical behaviors and outperforms classical random walk in various aspects. It has been known that the spatial probability distribution of a single-particle quantum walk can expand quadratically in time while a single-particle classical random walk can do only linearly. In this paper, we analytically study the discrete-time quantum walk of non-interacting multiple particles in a one-dimensional infinite lattice, and investigate the role of entanglement and exchange symmetry in the position distribution of the particles during the quantum walk. To analyze the position distribution of multi-particle quantum walk, we consider the relative distance between particles, and study how it changes with the number of walk steps. We compute the relative distance asymptotically for a large number of walk steps and find that the distance increases quadratically with the number of walk steps. We also study the extremal relative distances between the particles, and show the role of the exchange symmetry of the initial state in the distribution of the particles. Our study further shows the dependence of two-particle correlations, two-particle position distributions on the exchange symmetry, and find exponential decrement of the entanglement of the extremal state with the number of particles