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

• M. Owari1,2, M.B. Plenio1,2, E.S. Polzik3, A. Serafini4,

and M. M. Wolf3

1.

Institute of Mathematical Science, Imperial College, London

2.

QOLS, Blacket Laboratory, Imperial College, London

3.

Niels Bohr Institute, Copenhagen University,

4.

Department of Physics & Astronomy, University College London NJP, vol.10, 113014 (2008)

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.
• They known an unknown state is in .
• They may be also know the prior probability .
• An error is caused by an inevitable noise, and Bob gets

.

ω

ρ

Ω ∈ ω ω

ρ } {

Ω ∈ ω ω}

{p

) (

ω

ρ Γ Γ

Alice Bob

ω ω

ρ ρ ≠ Γ ) (

Ideal case:

is an identity channel

Γ

Impossible in a real experiment

Introduction

Quantum teleportation and Quantum memory.

Suppose an experiment is done, and we have data of and . However, looks far from the identity channel. Question: Is this process really “quantum”? At least, it should not be simulated by a “classical” scheme.

ω

ρ

) (

ω

ρ Γ Γ

Alice Bob

ω

ρ ) (

ω

ρ Γ Γ

Introduction

Quantum teleportation and Quantum memory.

Classical scheme (or Measure and Preparing scheme): (also called Entanglement breaking channel)

1.

Alice measure by POVM .

2.

Alice send a result of the measurement “i” to Bob.

3.

Bob choose a state depending on a classical information “i”.

ω

ρ

Alice Bob

ω

ρ

N i i

M

1

} {

=

i

σ

i

σ

i

M

“i” “i”

They want an average output state to be similar to

.

∑

=

⋅

N i i i

M Tr

1

) ( σ ρω

ω

ρ

Classical channel In prob.

) (

i

M Tr

ω

ρ

Introduction

Quantum teleportation and Quantum memory.

Classical scheme Entanglement breaking (EB) channel Suppose there exists another system For all , is separable.

AC

ρ

Alice Bob

C A AC

H H ⊗ ∈ ρ

iff

C A AC

H H ⊗ ∈ ρ

) (

AC

ρ Γ

) (

AC C

I ρ ⊗ Γ

C

I ⊗ Γ

Such a channel is useless: e.g. Repeater, Computation, etc

Introduction

Quantum teleportation and Quantum memory.

Our aim: By using experimental data (data of input and output states ), we want to show “a given channel can not simulated by classical scheme”.

ω

ρ

) (

ω

ρ Γ Γ

Alice Bob

ω

ρ

) (

ω

ρ Γ

Introduction

Quantum teleportation and Quantum memory.

Our aim: By using experimental data (data of input and output states ), we want to show “a given channel can not simulated by classical scheme”.

ω

ρ

Alice Bob

i

σ

i

M

“i” “i”

Classical channel

Quantum Benchmark

ω

ρ

) (

ω

ρ Γ

The optimal average fidelity

Most natural quantum benchmark is the optimal average fidelity between input and output states. For a given channel and an input ensemble , an average fidelity is given as: Then, the optimal average fidelity is derived as , : a set of all EB channels is a legitimate quantum benchmark:

1.

can be calculated by only experimental data of and .

2.

If , then, is not EB channel. This experiment can not simulated by a classical scheme.

Γ

Ω ∈ ω ω ω

ρ } , { p

ω ρ ρ

ω ω ω

d F F

∫

Ω ∈

Γ ≡ Γ )) ( | ( ) (

) (Γ F

) ( sup Γ ≡

Ε ∈ Γ

F F

b

) (Γ F

ω

ρ ) (

ω

ρ Γ

F

F F ≥ Γ) (

Γ

b

Ε

The optimal worst fidelity

Another popular quantum benchmark is the optimal worst fidelity between input and output states. For a given channel and an input ensemble , an worst fidelity is given as: Then, the optimal worst fidelity is defined as The optimal average fidelity is a legitimate quantum benchmark, too.

Γ

Ω ∈ ω ω

ρ } {

)) ( | ( inf ) (

ω ω ω

ρ ρ Γ ≡ Γ

Ω ∈ F

F

) (

0 Γ

F

) ( sup Γ ≡

Ε ∈ Γ

F F

b

F

is not depend on prior probability. Therefore, even in the case where we cannot define a reasonable prior probability, We can use .

F F

By definition, , and thus, .

( ) ( ) F F Γ ≥ Γ F F ≥

Known results: finite dimension

For an ensemble of pure states distributed according to Haar measure in a D-dimensional system.

) (

} , {

D SU U

dU U

∈

ψ dU

1 2 + = = D F F

(Werner 98, Horodecki×3 99)

In this talk, I concentrate on a infinite dimensional system.

Driving or is equal to solving a normal estimation problem of . Many results have been derived as the state estimation problem.

F F

Ω ∈ ω ω

ρ } {

(example)

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

• For an ensemble of coherent states ,

where : Especially, in the limit of flat distribution ,

{ , } pα

α

α

∈

) | | exp( ) (

2

α λ π λ α − = p

λ λ + + = 2 1 F

(Braunstein et al. 2000, Hammerer et al. 2005)

∞ → λ

1 2 F =

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 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.

for a noisy channel .

• The ensemble is rotationally invariant.

Λ

Ω ∈ Ω ∈

Ν =

ω ω ω ω ω ω ω

ψ ψ ρ } ), ( { } , { p p

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ρ σ ρ σ ρ Tr F ) || (

The optimal average fidelity between an ideal input pure state and a output state:

Discussion about the first restriction

(The first restriction) : States became mixed by a fixed rotationally covariant noisy channel. for a noisy channel s.t. . This is a natural assumption for experiment (e.g. attenuation channel). Under this restriction, we can redefine a quantum benchmark as follows:

Ν

( )

T

T

F F

b

Ε ∈

≡ sup

( ) ω

ρ ψ ψ ω ρ ψ ψ

ω ω ω ω ω ω ω ω

d Tr d F F

∫ ∫

Ω ∈ Ω ∈

Γ ⋅ = Γ ≡ ) ( )) ( || ( ) (T

Ω ∈ Ω ∈

Ν =

ω ω ω ω ω ω ω

ψ ψ ρ } ), ( { } , { p p

is still a legitimate quantum benchmark.

F

We succeeded to remove non-linearity from the definition of benchmark!

( )

( )

* * θ θ θ θ

ρ ρ U U U U Ν = Ν

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

Suppose is invariant under the action of a symmetric group . That is, and unitary representation s.t. . Then, we can choose an group covariant optimal strategy. is covariant w.r.t. (Proof for a compact group) Suppose is a optimal classical strategy. Define a covariant by . Then, We can do the same discussion for .

Ω ∈ ω ω ω

ρ } , { p G

) (

,

ω ω g

p p G g = ∈ ∀

∃

* ) ( g g g

U U

ω ω

ρ ρ = ) ( ) ( ,

* * g g g g

U U U U ρ ρ ρ Γ = Γ ∀

G

define

Γ

∫

Γ = Γ

g g g g g

U U U U dg ) ( ) (

* *

ρ ρ

) ( )) ( || ( )) ( || ( )) ( || ( ) (

) ( ) ( ) (

1

Γ = Γ = Γ ≥ Γ = Γ

∫∫ ∫∫ ∫

−

F F p dg d F p dg d F p d F

g g g ω ω ω ω ω ω ω ω ω

ρ ρ ω ρ ρ ω ρ ρ ω

F

Even for a “non-compact” group this statement is valid!

Γ Γ

Main results

Under the two restriction,

• States became mixed by a fixed rotationally covariant noisy channel.
• The ensemble is rotationally invariant.

We derive the following results for an ensemble of squeezed states:

• 1. For input states with uniform rotations and displacement,

we derive an analytical formula of .

• 2. For input states with uniform rotations and general displacement,

we derive an upper-bound of described as a finite dimensional SDP. So, we can efficiently calculate it by a numerical calculation.

F F

Notations

• An ensemble of squeezed states ,

, is characterized by the covariant matrix (CM) and the displacement vector : In other words, is derived from a squeezed vacuum as , where is a displacement (Weyl) operator, and is a phase-rotation operator.

Ω ∈ ω ω ω

ρ } , { p

) , ( θ ξ ω = ] 2 , [ π × = Ω C

ω

ρ

ω

ρ

γ

ω

ρ

d

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = θ θ θ θ θ θ θ θ γ

ω

ρ

cos sin sin cos / 1 cos sin sin cos s s ξ

ω

ρ =

d

ω

ρ

s

ρ

* * ξ θ θ ξ ω

ρ ρ W U U W

s

=

ξ

W

θ

U

(Fixed squeezing)

Main theorem: Uniform displacement

Main theorem (pure states): Main theorem (mixed states): For a noisy channel s.t. and ,

[ ]

s s Tr s F

b

+ = Γ =

Ω ∈ Ε ∈ Γ

1 ) ( inf sup ) (

ω ω ω

ρ ρ

Ν

I η γ γ

ρ ρ

+ =

Ν ) (

( ) ( ) [ ]

2 / 1

2 1 1 2 1 inf sup ) (

− Ω ∈ Ε ∈ Γ

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = Ν Γ = s s Tr s F

b

η η ρ ρ

ω ω ω

ρ ρ

d dN =

) (

For pure squeezing states, for any .

2 / 1 ) ( < s F 1 ≠ s

Without the rotation, for any .

2 / 1 ) ( = s F s

Proof of the main theorem

(Proof for a pure ensemble) From the phase space invariance of the ensemble, We just need to optimize over squeezed vacuums. Then, we use the following lemma. Lemma (Holevo 96) For a phase space covariant channel , is completely positive, iff there exist a state s.t. it has the form , where is the time reversal operator defined by .

( )

[ ]

( )

[ ]

* * ] 2 , [ * * * * ] 2 , [ ) , (

inf sup inf sup ) (

θ θ θ θ π θ ξ θ θ ξ ξ θ θ ξ π θ ξ

ρ ρ ρ ρ U U U U Tr W U U W W U U W Tr s F

s s s s C

b b

Γ = Γ =

∈ Ε ∈ Γ × ∈ Ε ∈ Γ

Γ ϑ

• Γ

( )

( )

ξ ξ ξ

τ W W Tr W

2 *

= Γ

τ

( )

ξ ξ

ϑ

Z

W W = ϑ

All EB channel satisfies the complete positivity of .

ϑ

• Γ

( is a set of all covariant EBs)

b

E Γ

Proof of the main theorem

By using the lemma and the Parseval relation: Thus, Since consists of just diagonal elements of , we can conclude .

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

τ ρ τ ρ ξ π τ ρ ξ π ρ ρ ξ π ρ ρ

ξ ξ ξ ξ ξ ξ

Tr W Tr W Tr d W Tr W Tr d W Tr W Tr d Tr 2 1 2 1 2 1 ) ( 2 1

2 2 2 2 2 2 2

= = = Γ = Γ

∫ ∫ ∫

• p

s s

d U U d U U Tr F ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ≤

∫ ∫

π θ θ π θ θ τ

θ ρ π θ ρ π

2 * 2 *

4 1 4 1 sup

∫

π θ θ

θ ρ π

2 *

2 1 d U U

s

s

ρ s s F

s

+ = ≤ 1 2 1 ρ

(Optimal strategy) Bob Prepares after Alice’s heterodyne measurement .

ξ

W

{ }

π

ξ ξ

2 / W W The same optimal strategy w.r.t. the coherent states case. Moreover, the following strategy can achieve this upperbound!

Finite displacement

So far, we have derived an analytical formula of the benchmark in the case of uniform rotation and uniform displacement in phase space. However, uniform displacement is impossible in an experiment! We need to find a way to calculate a value of benchmark for an ensemble with finite (or exponentially dumping) displacement. Here, we give an upper-bound of which is in the form of a finite dimensional SDP. (Therefore, efficiently calculable)

F

Main theorem: finite displacement

For an ensemble of squeezed states , where , we derived the following theorem: Main theorem For any probability density and rotationally covariant noise channel , we have , where , , and .

] 2 , [ ) , ( * ,

)} ( , {

π θ ξ θ ξ θ

ξ ρ

× ∈C s

q U U

* , ξ ξ ξ

ρ ρ W W

s s

= ) (ξ q

Ν

( )

( )

( )

{ }

( )

* * , , [0, 2 ] ( )

sup ( ) 2 sup | 0, 0, 1

b c

def s s C c c B A c c B supp R

d F q Tr U U U U d Tr P P Tr I Tr P P

θ ξ θ θ ξ θ θ π ξ

θ ξ ρ ρ ξ π η η

∈Ε ∈ ∈ Γ Ω∈

⎡ ⎤ = Ν ⋅ ⎣ ⎦ ≤ Ω Ω ≥ Ω ≥ Ω = + −

∫ ∫

T

T

def

= η

( )

ξ π θ ρ ρ ξ η

π θ ξ θ ξ θ θ ξ θ

d d U U U U q

C s s def

2 ) (

] 2 , [ * , * ,

∫ ∫

∈ ∈

⊗ Ν = l l k k P

c i i l k c

⊗ = ∑ ∑

= = +

( ) ( )

∑ ∑

= =

⊗ =

c i c i c

i i i i R

This bound only includes a finite dimensional SDP. can be also derived by a numerical calculation.

c c P

Pη

Relation with Miguel’s talk

Suppose , where is the set of all PPT - extendible positive operators. Then, we derive and . However, from our experience, when our memory is limited, by increasing , we can decrease more than by increasing . Practically, we should choose . ( is the set of all PPT positive operators.)

( )

{ }

) ( 1 , | sup

, ) ( , c c A B ppt N c c R Ran B N c

P P Tr I Tr S P P Tr F

c

η η − + = Ω ∈ Ω Ω =

∈ Ω

F F N

c

≥

, N c N c

F F

, ,lim ∞ →

=

ppt N

S ,

N

c

N c

F ,

N 1 = N

ppt

S ,

1

Numerical result 1

When and , our ensemble coincides Hammerer et al.’s ensemble of coherent states. We can compare them.

1 = s

( )

{ }

A B c c R Ran B finite

I Tr P P Tr F

c

= Ω ≥ Ω ≥ Ω Ω =

Γ ∈ Ω

, , | sup

) (

η

( )

c c finite def infinite

P P Tr F F F η − + = ≤ 1

( )

2

exp ) ( ξ α π α ξ − = q

Our method efficiently recover’s the known result for coherent states!

( We chose .)

35 = c

Numerical result 2:

The numerical results for and . A noisy channel is chosen as an attenuation channel:

( )

2

exp ) ( ξ α π α ξ − = q

8 = s

Ν

I

N

λ γ γ

ρ ρ

+ =

) ( ρ ρ

λ d d N =

) (

In the case of squeezed states, can be less than for a finite displacement.

F

2 1

( We chose .)

30 = c

Proof of the main theorem

First, we use the following well-known lemma about the Jamiolkowski isomorphism of EB channels: Lemma A channel on is entanglement breaking, if and only if there exist a unique separable positive operator on such that and for all and .

H H H ⊗

Γ ( )

T Ω

( ) ( )

B B

I = Γ Ω Tr

( ) ( ) ( ) ( )

B A Tr A B Tr ⊗ Γ Ω = Γ

( )

H

1

C ∈ A

( )

H B ∈ B

We can reduce an optimization over entanglement breaking channels to an optimization over separable states.

Proof of the main theorem

By using the lemma, we derive: , where . We succeeded to reduce the problem to an infinite dimensional SDP. However, we could not numerically solve an infinite dim. SDP. So, then, we reduce the above infinite dim. SDP to a finite dim. SDP.

( ) { } ( )

{ }

A B A B

0, 0, | sup Sep, | sup I I = Ω ≥ Ω ≥ Ω Ω ≤ = Ω ∈ Ω Ω =

Γ ∈ Ω ∈ Ω

Tr Tr Tr Tr F η η

) B( ) B( H H

( )

ξ π θ ρ ρ ξ η

π θ ξ θ ξ θ θ ξ θ

d d U U U U q

C s s def

2 ) (

] 2 , [ * , * ,

∫ ∫

∈ ∈

⊗ Ν =

Proof of the main theorem

We need the following lemma:

• Lemma. If a positive separable operator satisfies ,

then, . By using the above lemma, we derive , where and . In the third line, we use ; this can be seem from .

( )

H H ⊗ ∈ Ω B

A B

I = Ω Tr 1 ≤ Ω op

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

c c c c c c

• p

c c tr c

• p

c c c c c c

P P Tr P P Tr P P Tr P P Tr P P P P Tr P P Tr P P Tr Tr η η η η η η η η η η η η − + Ω ≤ − ⋅ Ω + Ω = − ⋅ Ω + Ω ≤ − Ω + Ω = Ω 1

∑ =

=

c i c c

Q P

∑

= +

⊗ =

c l k c

l l k k Q

∑

∞ + =

≥ = −

1 c i i i c c

Q Q P P η η η ( ) (

)

( )

∑ ∫

∞ = ∈

⊗ Ν =

, , i i s s i

Q d q Q

C ξ ξ ξ

ξ ρ ρ ξ η We used the rotationally covariance of .

Ν

Proof of the main theorem

Finally, by using the previous inequality, we derive , where . We have completed the proof!

( ) { } ( ) ( ) { } ( ) ( ) { } ( ) ( )

{ }

( )

c c c c R c c c c R c c c c c c c c c c

P P Tr Tr P P Tr P P Tr Tr P P Tr P P Tr Tr P P R R Tr P P Tr Tr P P Tr F

c c

η η η η η η η η − + ≤ Ω ≥ Ω ≥ Ω Ω ≤ − + ≤ Ω ∈ Ω Ω ≤ − + = Ω ∈ Ω Ω = − + = Ω ∈ Ω Ω ≤

Γ ∈ Ω ∈ Ω ∈ Ω ∈ Ω

1 0, 0, | sup 1 Sep, | sup 1 Sep, | sup 1 Sep, | sup

A B supp A B supp A B A B

I I I I

) B( ) B( ) B( ) B( H H

( ) ( )

∑ ∑

= =

⊗ =

c i c i c

i i i i R

Summary

Under the two restriction,

States became mixed by a fixed rotationally covariant noisy channel. The ensemble is rotationally invariant.

We derived the following results for an ensemble of squeezed states:

• 1. For input states with uniform rotations and displacement,

we derived an analytical formula:

• 2. For input states with uniform rotations and general displacement,

we derived an upper-bound of described as a finite dimensional SDP. So, we can efficiently calculate it by a numerical calculation.

2 / 1

2 1 1 2 1 ) (

−

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = s s s F η η F

Future works

Now Copenhagen Group is running an experiment of an atomic ensemble quantum memory for a rotationally invariant set of squeezed states. The experimental result may appear soon.