33 research outputs found
Comparison of mixed quantum states
In this article, we study the problem of comparing mixed quantum states:
given 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
mixed quantum states arbitrarily chosen from a set of 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 and . 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 ancillas, the postselection
probability can be increased by a factor of while keeping the weak value
fixed (compared to uncorrelated attempts with one ancilla), which is the
optimal scaling with 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