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A Lossy Bosonic Quantum Channel with Non-Markovian Memory O. V. Pilyavets, V. G. Zborovskii and S. Mancini Universit` a di Camerino (Italy) & P. N. Lebedev Physical Institute (Russia) August 30, 2008 Introduction Quantum channels

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SLIDE 1

A Lossy Bosonic Quantum Channel with Non-Markovian Memory

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Universit` a di Camerino (Italy) & P. N. Lebedev Physical Institute (Russia)

August 30, 2008

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Introduction

Quantum channels

Gaussian channels

Lossy bosonic Gaussian channels

Characteristics of quantum channel

Quantum capacity Classical capacity Rates

Homodyne rate Heterodyne rate

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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General definitions

Any state can be labeled as ρ or V as all states are Gaussian.

Vin - covariance matrix for input state Venv - covariance matrix for environment state Vcl - covariance matrix for classical distribution of coherent amplitude α. Vout - covariance matrix for output state of the channel V out - covariance matrix for output state of the channel avaraged over classical distribution (encoding of information) Vcl

Capacities and rates

Cn = maxstates 1

nχn - classical capacity for n uses of channel

C = maxn→∞ Cn - classical capacity on infinite amount of channel uses

Conjectures

Capacity for lossy bosonic channel can be achieved on Gaussian states Maximizing of Holevo χ leads to capacity for memory channel too

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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Scheme for 1 use of lossy bosonic channel

ρout(α)=Tr

env(ρout tot )

ρin(α) input

  • utput

environment ρin

tot

ρout

tot

η ρenv Beam splitter action: Energy restriction: encoding of classical information decoding (measurement) ρin(α)=D(α)ρ0D+(α) e.g. heterodyne: <α|ρout|α> P(α)↔ Vcl ρtot

  • ut=U(ρin⊗ρenv)U+=Uρtot

in U+

Tr(V

in+Vcl)/(2n)=N+1/2

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: notations and known results

It is sufficient to consider only diagonal matrices (!). Abitrary covariance matrices in diagonal form for 1 use: Venv = (Nenv + 1/2) es e−s

  • Vin = (Nin + 1/2)

er e−r

  • Already known capacities:

If environment is in ground (vacuum) state: C = g[ηN] If environment is in termal state: C = g[ηN + (1 − η)Nenv] − g[(1 − η)Nenv]

Environment in termal and squeezed state: C = g

  • ηN + (1 − η)
  • (Nenv + 1/2) cosh(s) − 1/2
  • − g[(1 − η)Nenv]

g(x) = (x + 1) log2 (x + 1) − x log2 x

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: classical capacity

2 4 6 8 10 12 1 2 3 Classical capacity C(s) [parameters: Nenv=2, N=3] s η=0.99 η=1 η=0.05 Step by η is 0.05

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: classical capacity

2 4 6 8 10 12 1 2 3 Classical capacity C(s) [parameters: Nenv=2, N=3] s η=0.99 η=1 η=0.05

Will they continue to grow?

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: classical capacity

2 4 6 8 10 12 1 2 3 Classical capacity C(s) [parameters: Nenv=2, N=3] s η=0.99 η=1 η=0.05 Step by η is 0.05

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: classical capacity

2 4 6 8 10 12 1 2 3 Classical capacity C(s) [parameters: Nenv=2, N=3] s η=0.99 η=1 η=0.05 Step by η is 0.05

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: classical capacity

2 4 6 8 10 12 1 2 3 Classical capacity C(s) [parameters: Nenv=2, N=3] s η=0.99 η=1 η=0.05 Step by η is 0.05

why?

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: classical capacity

2 4 6 0.005 0.01 Classical capacity C(s) [parameters: Nenv=0.005, N=0.001] s η=0.1 η=1 η=0.9 Step by η is 0.1

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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SLIDE 12

1 use capacity: classical capacity

2 4 6 8 10 12 0.5 1 1.5 2 Optimal value ropt [parameters: Nenv=2, N=3, η=0.7] s 2 4 6 8 10 12 1 2 3 Classical capacity C(s) [parameters: Nenv=2, N=3] s η=0.99 η=1 η=0.05 Step by η is 0.05

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: classical capacity

2 4 6 8 10 12 0.5 1 1.5 2 Optimal value ropt [parameters: Nenv=2, N=3, η=0.7] s

kink? ropt = s

2 4 6 8 10 12 1 2 3 Classical capacity C(s) [parameters: Nenv=2, N=3] s η=0.99 η=1 η=0.05 Step by η is 0.05

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: complete analytical solution

Suppose that eigenvalues of Venv matrix are e1, e2. Then, eigenvalues

  • f matrix Vcl (which are c1, c2) and Vin (which are i1, i2) can be

found from the following relations if both c1 and c2 are positive: c1 = N + 1 2 − 1 2 e1 e2 + 1 2

  • 1 − 1

η

  • (e1 − e2)

c2 = N + 1 2 − 1 2 e2 e1 + 1 2

  • 1 − 1

η

  • (e2 − e1)

i1 = 1 2 e1 e2 , i2 = 1 2 e2 e1 In this case capacity can be expressed in explicit form and is equal to C = g

  • ηN + (1 − η)
  • (Nenv + 1/2) cosh(s) − 1/2
  • − g[(1 − η)Nenv]
  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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1 use capacity: complete analytical solution

If ck (according to previous relations) is negative, then ck = 0, cm = 2N + 1 − ik − 1/(4ik), im = 1/(4ik), and ik is a solution of the following transcedental equation ({k, m} = {1, 2} or {k, m} = {2, 1}): am − ak √amak log2 √amak + 1/2 √amak − 1/2 = omik − okim ik √omok log2 √omok + 1/2 √omok − 1/2 where

  • 1 = ηi1 + (1 − η)e1

a1 = η(i1 + c1) + (1 − η)e1

  • 2 = ηi2 + (1 − η)e2

a2 = η(i2 + c2) + (1 − η)e2 No explicit relation for capacity.

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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Is there something new for n uses of the channel? Is there new “physics” there? Can we say that entanglemet is useful for information transmission for many uses of the channel? Let us see...

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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Scheme for n uses of lossy bosonic channel

ENVIRONMENT INPUT OUTPUT

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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n uses capacity: analytics

In the case of our type of memory it is sufficient to consider only commuting matrices (!). Suppose that eigenvalues of Venv matrix are eqk, epk, k = 1, ..., n. Then, eigenvalues of matrix Vcl (which are cqk, cpk) and Vin (which are iqk, ipk) can be found from the following relations if for all k both cqk and cpk are positive: cqk = N + 1 2 − 1 2 eqk epk + 1 − η η Tr Venv 2n − eqk

  • cpk = N + 1

2 − 1 2 epk eqk + 1 − η η Tr Venv 2n − epk

  • iqk = 1

2 eqk epk , ipk = 1 2 epk eqk

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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n uses capacity: analytics

In this case the capacity can be expressed in explicit form. It is equal to

Cn = g

  • ηN + (1 − η)

Tr Venv 2n − 1 2

  • − 1

n

n

  • k=1

[(1 − η)(√eqkepk − 1/2)] C = g

  • ηN + (1 − η)
  • lim

n→∞

Tr Venv 2n − 1 2

  • − lim

n→∞

1 n

n

  • k=1

[(1 − η)(√eqkepk − 1/2)]

If cqk or cpk is negative we don’t have explicit relation for the capacity Capacity is always achieved on states Vin minimizing uncertainty relation (!)

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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n uses capacity: Ω-model of channel memory

Environment matrix: Venv = 1 2 exp(sΩ) exp(−sΩ)

  • where

Ω =            1 . . . . . . . . . . . . 1 1 . . . . . . . . . . 1 1 . . . . . . . . . ... ... ... . . . . . . ... ... 1 . . . . . . . 1           

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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n uses capacity: Ω-model of channel memory

Correlations decay exponentially over channel uses, it is quite “realistic” (correlations are non-Markovian) Energy constraint: Tr(Vin + Vcl) 2n = N + 1 2 Ω-model allows us to test entanglement Capacity for Ω-model: C = g

  • ηN + 1

2(1 − η)(I0(2s) − 1)

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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n uses capacity: Ω-model, maximization over set

Input state: Vin = 1 2 exp(rΩ) exp(−rΩ)

  • Covariance matrix for classical distribution used to encode an

information: Vcl := 2nN(1 − θn) Tr(Y ) Y where Y = 1 2 exp(yΩ) exp(−yΩ)

  • θn := Tr(Vin) − n

2nN Region of possible values of r is restricted by input energy constraint (by roots of equation θn = 1)

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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Rates

Heterodyne rate: I[Z : A] = H[Z] − H[Z|A] = 1 2 log2 det

  • V out + 1

2 Vout + 1 2 −1 Homodyne rate (measurement of quadratures q in all modes): I[ℜZ : ℜA] = H[ℜZ] − H[ℜZ|ℜA] = 1 2 log2 det

  • V

(11)

  • ut

V (11)

  • ut

−1

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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SLIDE 24

Capacities and rates: maximization over set

1 2 3 4 5 0.5 1 1.5 2 s ropt

1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 s C F(hom) F(het)

Figure: On the left, the optimal value of r for the quantities C (solid line with circles),

F (heterodyne rate, line with circles only) and F (homodyne rate, pure solid line) is shown versus s. The values of the other parameters are N = 8, η = 0.7. On the right, the quantities C (solid lines with circles), F (heterodyne rate, lines with circles only) and F (homodyne rate, pure solid lines) are plotted versus s for values of η going from 0.1 (bottom curve) to 1 (top curve) with step 0.1. The value of the other parameter is N = 8.

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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Heterodyne rates for 1 use of the channel

1 2 3 4 5 0.2 0.4 0.6 0.8 1 Optimal value ropt for het. rate [Nenv=2, N=3, η=0.7] s 2 4 6 8 0.5 1 1.5 2 Heterodyne rate [parameters: Nenv=2, N=3] s η=0.1 η=0.9 η=1 Step by η is 0.1

  • O. V. Pilyavets, V. G. Zborovskii and S. Mancini

Lossy Bosonic Channel with Non-Markovian Memory

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Conclusions

Squeezing can enhance the capacity in some cases Optimal input squeezing is related to the environment squeezing Entanglemet is useful for information transmission as it always comes with squeezing Almost all interesting “physics” (behavior) can be found in already 1 use of the channel

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Conclusions

THANK YOU!