Optimal cloning of mixed Gaussian states

Guţǎ, Mădălin; Matsumoto, Keiji

doi:10.1103/physreva.74.032305

Cited by 10 publications

(17 citation statements)

References 35 publications

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“…In [14,13] it is shown that the quantum version of the local asymptotic normality with the Le Cam type convergence holds for identically prepared qubits with the limit experiment being a family of displaced thermal equilibrium states. In [15], the problem of optimal cloning of mixed quantum Gaussian states is solved along lines similar to the solution of the classical problem of finding the deficiency between two Gaussian shift experiments.…”

Section: Example 22mentioning

confidence: 99%

Local Asymptotic Normality in Quantum Statistics

Guţǎ

Jenčová

2007

Commun. Math. Phys.

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The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕ n θ 0 +u/ √ n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation relations. The convergence holds for all "local parameters" u ∈ R m such that θ = θ0 + u/ √ n parametrizes a neighborhood of a fixed point θ0 ∈ Θ ⊂ R m . In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For reader's convenience and completeness we review the relevant results of the classical as well as the quantum theory.

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Section: Example 22mentioning

confidence: 99%

Local Asymptotic Normality in Quantum Statistics

Guţǎ

Jenčová

2007

Commun. Math. Phys.

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“…In other words, the output state is the same as the state of the 'amplified' mode c when b and a are prepared in states τ and respectively Φ. From this point we follow closely the arguments used in [25] which were originally devised for finding the optimal amplification channel for displaced thermal states. The candidate channels are labelled by diagonal matrices τ and we denote by p τ the probability distribution consisting of the elements of the output state p…”

Section: H(dζ) := |ζ ζ|Dζ/2πmentioning

confidence: 86%

“…Now, since Φ is invariant under phase rotations, Φ = exp(iθN )Φ exp(−iθN ), we can apply the covariance argument again [25] to conclude that we can restrict to channels which are covariant under phase rotation, which amounts to taking τ to be diagonal in the Fock basis. Since for diagonal states f (ξ) = f (|ξ|) we obtain the following Schrödinger version of (7) Tr…”

Section: H(dζ) := |ζ ζ|Dζ/2πmentioning

confidence: 99%

“…We solve the same problem using the trace norm loss function, and again find this MAP scheme to be optimal. Interestingly, the optimality proof is based on the concept of stochastic ordering which was previously used for finding optimal cloning maps for thermal states [25]. This result allows us to exploit LAN and solve the benchmark problem for n i.i.d.…”

Section: Introductionmentioning

confidence: 99%

“…Stochastic ordering can be transformed into the desired optimality result by a standard argument [25] which has some interest in its own and is summarised in the following lemma. The key property needed here is that p Lemma IV.5.…”

Section: H(dζ) := |ζ ζ|Dζ/2πmentioning

confidence: 99%

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Quantum-teleportation benchmarks for independent and identically distributed spin states and displaced thermal states

2010

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A successful state transfer (or teleportation) experiment must perform better than the benchmark set by the 'best' measure and prepare procedure. We consider the benchmark problem for the following families of states: (i) displaced thermal equilibrium states of given temperature; (ii) independent identically prepared qubits with completely unknown state. For the first family we show that the optimal procedure is heterodyne measurement followed by the preparation of a coherent state. This procedure was known to be optimal for coherent states and for squeezed states with the 'overlap fidelity' as figure of merit. Here we prove its optimality with respect to the trace norm distance and supremum risk. For the second problem we consider n i.i.d. spin-1 2 systems in an arbitrary unknown state ρ and look for the measurement-preparation pair (Mn, Pn) for which the reconstructed state ωn := Pn • Mn(ρ ⊗n ) is as close as possible to the input state, i.e. ωn − ρ ⊗n 1 is small. The figure of merit is based on the trace norm distance between input and output states. We show that asymptotically with n the this problem is equivalent to the first one. The proof and construction of (Mn, Pn) uses the theory of local asymptotic normality developed for state estimation which shows that i.i.d. quantum models can be approximated in a strong sense by quantum Gaussian models. The measurement part is identical with 'optimal estimation', showing that 'benchmarking' and estimation are closely related problems in the asymptotic set-up.

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Quantum cloning machines and the applications

et al. 2014

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No-cloning theorem is fundamental for quantum mechanics and for quantum information science that states an unknown quantum state cannot be cloned perfectly. However, we can try to clone a quantum state approximately with the optimal fidelity, or instead, we can try to clone it perfectly with the largest probability. Thus various quantum cloning machines have been designed for different quantum information protocols. Specifically, quantum cloning machines can be designed to analyze the security of quantum key distribution protocols such as BB84 protocol, six-state protocol, B92 protocol and their generalizations. Some well-known quantum cloning machines include universal quantum cloning machine, phase-covariant cloning machine, the asymmetric quantum cloning machine and the probabilistic quantum cloning machine etc. In the past years, much progress has been made in studying quantum cloning machines and their applications and implementations, both theoretically and experimentally. In this review, we will give a complete description of those important developments about quantum cloning and some related topics. On the other hand, this review is self-consistent, and in particular, we try to present some detailed formulations so that further study can be taken based on those results.

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Optimal cloning of mixed Gaussian states

Cited by 10 publications

References 35 publications

Local Asymptotic Normality in Quantum Statistics

Local Asymptotic Normality in Quantum Statistics

Quantum-teleportation benchmarks for independent and identically distributed spin states and displaced thermal states

Quantum cloning machines and the applications

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