dorsal/arxiv
View SchemaOptimal estimation of qubit states with continuous time measurements
| Authors | Madalin Guta, Bas Janssens, Jonas Kahn |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0608074 |
| URL | https://arxiv.org/abs/quant-ph/0608074 |
| DOI | 10.1007/s00220-007-0357-5 |
| Journal | Commun. Math. Phys. 277, 127-160 (2008) |
Abstract
We propose an adaptive, two steps strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given $n$ identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size $n^{-1/2}$ of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large $n$, the statistical model described by $n$ identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term `local' refers to a shrinking neighborhood around a fixed state $\rho_{0}$. An essential result is that the neighborhood radius can be chosen arbitrarily close to $n^{-1/4}$. This allows us to use a two steps procedure by which we first localize the state within a smaller neighborhood of radius $n^{-1/2+\epsilon}$, and then use LAN to perform optimal estimation.
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"abstract": "We propose an adaptive, two steps strategy, for the estimation of mixed qubit\nstates. We show that the strategy is optimal in a local minimax sense for the\ntrace norm distance as well as other locally quadratic figures of merit. Local\nminimax optimality means that given $n$ identical qubits, there exists no\nestimator which can perform better than the proposed estimator on a\nneighborhood of size $n^{-1/2}$ of an arbitrary state. In particular, it is\nasymptotically Bayesian optimal for a large class of prior distributions.\n We present a physical implementation of the optimal estimation strategy based\non continuous time measurements in a field that couples with the qubits.\n The crucial ingredient of the result is the concept of local asymptotic\nnormality (or LAN) for qubits. This means that, for large $n$, the statistical\nmodel described by $n$ identically prepared qubits is locally equivalent to a\nmodel with only a classical Gaussian distribution and a Gaussian state of a\nquantum harmonic oscillator.\n The term `local\u0027 refers to a shrinking neighborhood around a fixed state\n$\\rho_{0}$. An essential result is that the neighborhood radius can be chosen\narbitrarily close to $n^{-1/4}$. This allows us to use a two steps procedure by\nwhich we first localize the state within a smaller neighborhood of radius\n$n^{-1/2+\\epsilon}$, and then use LAN to perform optimal estimation.",
"arxiv_id": "quant-ph/0608074",
"authors": [
"Madalin Guta",
"Bas Janssens",
"Jonas Kahn"
],
"categories": [
"quant-ph"
],
"doi": "10.1007/s00220-007-0357-5",
"journal_ref": "Commun. Math. Phys. 277, 127-160 (2008)",
"title": "Optimal estimation of qubit states with continuous time measurements",
"url": "https://arxiv.org/abs/quant-ph/0608074"
},
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