Squeezing the limit: Quantum benchmarks for the teleportation and - PowerPoint PPT Presentation

Squeezing the limit: Quantum benchmarks for the teleportation and storage of squeezed states NJP, vol.10, 113014 (2008) M. Owari 1,2 , M.B. Plenio 1,2 , E.S. Polzik 3 , A. Serafini 4 , and M. M. Wolf 3 Institute of Mathematical Science, Imperial College, London 1. QOLS, Blacket Laboratory, Imperial College, London 2. Niels Bohr Institute, Copenhagen University, 3. Department of Physics & Astronomy, University College London 4.

Introduction Quantum teleportation and Quantum memory. Both processing can be written down the following processing: Alice and Bob are spatially or temporally separated. � Alice wants to send an unknown quantum state to Bob. � ρ } { They known an unknown state is in . ω ω ∈ Ω � { p ω } They may be also know the prior probability . � ω ∈ Ω Γ ρ ≠ ρ ( ) An error is caused by an inevitable noise, and Bob gets . � ω ω ρ Γ Γ ρ ( ) ω ω Bob Alice Γ Ideal case: Impossible in a real experiment is an identity channel

Introduction Quantum teleportation and Quantum memory. ρ Γ ρ ( ) Suppose an experiment is done, and we have data of and . ω ω Γ However, looks far from the identity channel. Question: Is this process really “quantum”? At least, it should not be simulated by a “classical” scheme. ρ Γ Γ ρ ( ) ω ω Bob Alice

Introduction Quantum teleportation and Quantum memory. Classical scheme (or Measure and Preparing scheme): (also called Entanglement breaking channel) ρ N { } M Alice measure by POVM . ω 1. = 1 i i Alice send a result of the measurement “i” to Bob. 2. σ Bob choose a state depending on a classical information “i”. 3. i ρ ω σ i M i Classical channel “i” “i” Bob Alice ρ ( ) In prob. Tr M ω i N ρ ∑ ρ ω ⋅ σ ( ) They want an average output state to be similar to Tr M . ω i i = i 1

Introduction Quantum teleportation and Quantum memory. Classical scheme Entanglement breaking (EB) channel iff ρ ∈ H ⊗ H Suppose there exists another system AC A C ρ ∈ H ⊗ Γ ⊗ ρ H ( ) I For all , is separable. AC A C C AC Γ ρ ρ ( ) AC AC Γ ⊗ I C Bob Alice Such a channel is useless: e.g. Repeater, Computation, etc

Introduction Quantum teleportation and Quantum memory. Our aim: ρ Γ ρ ( ) By using experimental data (data of input and output states ), ω ω we want to show “a given channel can not simulated by classical scheme”. ρ Γ Γ ρ ( ) ω ω Bob Alice

Introduction Quantum teleportation and Quantum memory. Our aim: ρ Γ ρ ( ) By using experimental data (data of input and output states ), ω ω we want to show “a given channel can not simulated by classical scheme”. Quantum Benchmark ρ ω σ i M i Classical channel “i” “i” Bob Alice

The optimal average fidelity Most natural quantum benchmark is the optimal average fidelity between input and output states. ρ Γ { , } p For a given channel and an input ensemble , ω ω ω ∈ Ω ( Γ an average fidelity ) is given as: F ∫ Γ ≡ ρ Γ ρ ω ( ) ( | ( )) F F d ω ω ω ∈ Ω Then, the optimal average fidelity is derived as ≡ Γ Ε sup ( ) F F , : a set of all EB channels b Γ ∈ Ε b is a legitimate quantum benchmark: F ρ Γ ρ ( Γ ( ) can be calculated by only experimental data of and . ) ω F 1. ω Γ Γ ) ≥ If , then, is not EB channel. ( F F 2. This experiment can not simulated by a classical scheme.

The optimal worst fidelity Another popular quantum benchmark is the optimal worst fidelity between input and output states. ρ } Γ { For a given channel and an input ensemble , ω ω ∈ Ω 0 Γ ( ) F an worst fidelity is given as: Γ ≡ ρ Γ ρ ( ) inf ( | ( )) F ∈ F ω ω 0 ω Ω Then, the optimal worst fidelity is defined as ≡ Γ sup ( ) F F 0 0 Γ ∈ Ε b F The optimal average fidelity is a legitimate quantum benchmark, too. 0 F is not depend on prior probability. 0 Therefore, even in the case where we cannot define a reasonable prior probability, We can use . F 0 Γ ≥ Γ ≥ ( ) ( ) F F F F By definition, , and thus, . 0 0

Known results: finite dimension ρ } { F Driving or is equal to solving a normal estimation problem of . F ω ω ∈ Ω 0 Many results have been derived as the state estimation problem. (example) ψ { , } U dU For an ensemble of pure states distributed ∈ ( ) U SU D according to Haar measure in a D-dimensional system. dU 2 (Werner 98, Horodecki × 3 99) = = F F 0 + 1 D In this talk, I concentrate on a infinite dimensional system.

Known results: infinite dimension Of course, quantum benchmark in an infinite dimensional system is also really important as an technological application. Difference between infinite and finite dimensional systems: A set of pure states is non-compact. � It is impossible to make all pure states in an experiment. � We are interested in a particular set of states.

Quantum benchmark for a set of coherent states α { , } p α For an ensemble of coherent states , � α ∈ � λ α = − λ α 2 where : ( ) exp( | | ) p π + λ 1 = 2 (Braunstein et al. 2000, Hammerer et al. 2005) F + λ 1 λ → ∞ F = Especially, in the limit of flat distribution , 2 However, a coherent state is a “classical” state. People are interested in a quantum teleportation and quantum memory for more quantum states So, we want to derive a quantum benchmark for squeezed states.

Quantum benchmark for squeezed states (Difficulty) Hammerer et al.’s trick does not work for squeezed states. � In experiment, a pure squeezed states rapidly becomes mixed, � because of attenuation of light fields. ⎡ ⎤ Therefore, we should treat mixed states ρ σ = ρ σ ρ ( || ) F Tr ⎢ ⎥ ⎣ ⎦ However, the fidelity for mixed states is non-linear! Under two restrictions, we will give a way to calculate a benchmark! (Restriction) States became mixed by a fixed rotationally covariant noisy channel. � Λ ρ = Ν ψ ψ for a noisy channel . { , } { ( ), } p p ω ω ω ∈ Ω ω ω ω ω ∈ Ω The ensemble is rotationally invariant. �

Discussion about the first restriction (The first restriction) : States became mixed by a fixed rotationally covariant noisy channel. ρ = Ν ψ ψ { , } { ( ), } p p ω ω ω ∈ Ω ω ω ω ω ∈ Ω ( ) ( ) Ν Ν ρ = Ν ρ for a noisy channel s.t. . * * U U U U θ θ θ θ This is a natural assumption for experiment (e.g. attenuation channel). Under this restriction, we can redefine a quantum benchmark as follows: The optimal average fidelity between an ideal input pure state and a output state: ( ) ω ∫ ∫ ≡ ψ ψ Γ ρ ω = ψ ψ ⋅ Γ ρ ( T ) ( || ( )) ( ) F F d Tr d ω ω ω ω ω ω ω ∈ Ω ω ∈ Ω ( ) ≡ sup F F T ∈ Ε T b is still a legitimate quantum benchmark. F We succeeded to remove non-linearity from the definition of benchmark!

Discussion about the rotational invariance (The second restriction) The ensemble is rotationally invariant. =We should rotate a input state randomly in the phase space. But, this is easily done in an experiment. We do not need to do anything, but just wait for a short time! (Rotation in the phase space is just a natural time evolution.) However, the rotational invariance makes the problem much simpler! Group invariance of an ensemble Group covariance of the optimal strategy

Group invariance and Group covariance ρ { , } p G Suppose is invariant under the action of a symmetric group . ω ω ω ∈ Ω ∀ ∈ = , That is, g G p p ω ω ( ) g ∃ ρ = ρ * and unitary representation s.t. . U U ω ω ( ) g g g Then, we can choose an group covariant optimal strategy. Γ ∀ ρ Γ ρ = Γ ρ * * , ( ) ( ) U U U U is covariant w.r.t. G define g g g g (Proof for a compact group) Γ Suppose is a optimal classical strategy. ∫ Γ Γ ρ = Γ ρ * * ( ) ( ) dg U U U U Define a covariant by . g g g g g ∫ Γ = ω ρ Γ ρ Then, ( ) ( || ( )) F d p F ω ω ω ∫∫ ≥ ω ρ Γ ρ ( || ( )) d dg p F ω ω ω ( ) ( ) g g ∫∫ = ω ρ Γ ρ = Γ ( || ( )) ( ) d dg p F F ω ω − 1 ω ( ) g F We can do the same discussion for . 0 Even for a “non-compact” group this statement is valid!