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arXiv:quant-ph/0412155 · Physical Review A

State estimation on correlated copies

Rafał Demkowicz-Dobrzański

Published Jun 17, 2005• 11 citations
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Abstract

State estimation is usually analyzed in the situation when copies are in a product state, either mixed or pure. We investigate here the concept of state estimation on correlated copies. We analyze state estimation on correlated $N$-qubit states, which are permutationally invariant. Using a correlated state we try to estimate as good as possible the direction of the Bloch vector of a single-particle reduced density matrix. We derive the optimal fidelity for all permutation invariant states. We find the optimal state, which yields the highest estimation fidelity among the states with the same reduced density matrix. Interestingly, this state is not a product state. We also point out that states produced by optimal universal cloning machines are the worst, from the point of view of estimating the reduced density matrix.

DOI
10.1103/physreva.71.062321

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State estimation is usually analyzed in the situation when copies are in a product state, either mixed or pure. We investigate here the concept of state estimation on correlated copies. We analyze state estimation on correlated NNN-qubit states, which are permutationally invariant. Using a correlated state we try to estimate as good as possible the direction of the Bloch vector of a single-particle reduced density matrix. We derive the optimal fidelity for all permutation invariant states. We find the optimal state, which yields the highest estimation fidelity among the states with the same reduced density matrix. Interestingly, this state is not a product state. We also point out that states produced by optimal universal cloning machines are the worst, from the point of view of estimating the reduced density matrix.
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a d-dimensional unknown pure state with its N copies given as input, which was first obtained by Bruss and Macchiavello by establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we show optimal estimators with finite POVM exist but not generally universal, where universality means the fidelity is independent of input states. However any optimal estimator for M(>N) copies is universal if it is used for N copies as input.
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a ddd-dimensional unknown pure state with its NNN copies given as input, which was first obtained by Hayashi in terms of an infinite set of covariant positive operator valued measures (POVM's) and by Bru\ss{} and Macchiavello establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we construct optimal estimators with finite POVMs using exact quadratures on a hypersphere. These finite optimal estimators are not generally universal, where universality means the fidelity is independent of input states. However, any optimal estimator with finite POVM for M(>N)M(>N)M(>N) copies is universal if it is used for NNN copies as input.
We derive a tight upper bound for the fidelity of a universal $N\ensuremath{\rightarrow}M$ qubit cloner, valid for any $M\ensuremath{\ge}N$, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.
We investigate the asymptotic relationship between quantum cloning and quantum estimation from the global point of view where all the copies produced by the cloner are considered jointly. For an N-to-M cloner, we consider the overall fidelity between the state of the M output systems and the state of M ideal copies, and we ask whether the optimal fidelity is attained by a measure and-prepare protocol in the limit M -> \infty. In order to gain intuition into the general problem, we analyze two concrete examples: i) cloning qubit states on the equator of the Bloch sphere and ii) cloning two-qubit maximally entangled states. In the first case, we show that the optimal measure-and-prepare fidelity converges to the fidelity of the optimal cloner in the limit M -> \infty. In the second case, we restrict our attention to economical covariant cloners, and again, we exhibit a measure-and-prepare protocol that achieves asymptotically the optimal fidelity. Quite counterintuitively, in both cases the optimal states that have to be prepared in order to maximize the overall fidelity are not product states corresponding to M identical copies, but instead suitable M-partite entangled states.
Fidelity estimation is a necessary tool for evaluating noise in quantum measurement and quantum computation. The traditional fidelity estimation is to calculate the distance between two density matrices by employing direct fidelity estimation, which consumes too much copies of state. To reduce the number of copies of the state, we develop optimal fidelity estimation by proposing an optimal model. It calculates the minimum number of copies of state given a fixed value for the fidelity deviation. The result shows it saves a large number of copies of state compared with traditional approach (Direct Fidelity estimation) that is developed several years ago.The number of copies of the state employed increases slower than linear increase with increase of the dimension of density matrix when pauli measurement basis is applied. In addition, it consumes roughly a constant number of copies of the state with the increase of dimension of density matrix when the measurement bases are freely chosen.
The cloning of conjugate continuous quantum variables is analyzed based on the concept of Gaussian cloning machines, i.e., transformations that yield copies that are Gaussian mixtures centered on the state to be copied. The optimality of Gaussian cloning machines that transform N identical input states into M output states is investigated, and bounds on the fidelity of the process are derived via a connection with quantum estimation theory. In particular, the optimal N-to-MMM cloning fidelity for coherent states is found to be equal to $MN/(MN+M\ensuremath{-}N).$