quantum sinkhorn's theorem: applications in entanglement ...hhlee/filippov_maqit_2019.pdf ·...

Quantum Sinkhorn's theorem:

Applications in entanglement dynamics,

channel capacities, and compatibility theory

Sergey Filippov

1Moscow Institute of Physics and Technology (National Research University)2Steklov Mathematical Institute of Russian Academy of Sciences

Mathematical Aspects in Current Quantum Information Theory 2019

Seoul National University, Korea

May 21, 2019

1. Sinkhorn's theorem for matrices

2. Quantum Sinkhorn's theorem

3. Lower and upper bounds on classical capacity

4. Entanglement robustness

5. Compatibility of trace decreasing operations

Sinkhorn's theorem

Theorem (1)

If X is an n× n matrix with strictly positive elements, then there

exist diagonal matrices D1 and D2 with strictly positive diagonal

elements such that D1XD2 is doubly stochastic.

1R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices,

Ann. Math. Statist. 35, 876879 (1964).

3 · · ·· · ·· · ·

3 · · ·1 · ·· · ·

3 · 1 ·1 · ·· · ·

3 · 1 ·1 · ·1 · ·

3 · 1 ·1 · ·1 · 1

3 · 1 ·1 · 11 · 1

3 · 1 ·1 1 11 · 1

3 · 2 ·1 1 11 · 1

3 1 2 ·1 1 11 · 1

3 1 2 ·2 1 11 · 1

3 1 2 ·2 1 11 1 1

3 1 2 12 1 11 1 1

3 1 2 13 1 11 1 1

3 1 2 13 1 11 2 1

3 1 2 13 1 11 2 2

P1(t+ ∆t)P2(t+ ∆t)P3(t+ ∆t)

p1→1 p2→1 p3→1

p1→2 p2→2 p3→2

p1→3 p2→3 p3→3

P1(t)P2(t)P3(t)

transition matrix

p1→1 p2→1 p3→1

p1→2 p2→2 p3→2

p1→3 p2→3 p3→3

1 2 13 1 11 2 2

Premise: transition matrix Y is bistochastic (uniform distribution is

a xed point)

1 2 13 1 11 2 2

1/5 0 00 1/5 00 0 1/4

0.2 0.4 0.250.6 0.2 0.250.2 0.4 0.5

left stochastic, but not right stochastic

D1XD2 =

0.85−1 0 00 1.05−1 00 0 1.1−1

0.2 0.4 0.250.6 0.2 0.250.2 0.4 0.5

= 0.235 0.471 0.2940.572 0.190 0.2280.182 0.364 0.454

right stochastic, but not left stochastic

and so on ...

Sinkhorn's theorem

Theorem (2)

If X is an n× n matrix with strictly positive elements, then there

exist diagonal matrices D1 and D2 with strictly positive diagonal

elements such that D1XD2 is doubly stochastic.

2R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices,

Ann. Math. Statist. 35, 876879 (1964).

Let A and B be operators acting on Hd. Denote

ΦA[X] = AXA†

ΦB[X] = BXB†

Theorem (3)

Let Φ : B(Hd) 7→ B(Hd) be a linear map which belongs to the

interior of the cone of positive maps. Then there exist

positive-denite operators A and B such that Υ = ΦA Φ ΦB is

bistochastic.

3G. Aubrun, S.J. Szarek, Two proofs of Størmer's theorem, arXiv:1512.03293 (2015)

Alternative discussions of the relation Υ = ΦA Φ ΦB:

I L. Gurvits, Classical complexity and quantum entanglement, J.

Comput. System Sci. 69, 448484 (2004).

For maps Φ s.t. infdetΦ[X]|X > 0,detX = 1 > 0.

I T. T. Georgiou, M. Pavon, Positive contraction mappings for

classical and quantum Schrodinger systems, J. Math. Phys.

56, 033301 (2015).

For so-called positivity improving CPT maps with the property

Φ†[ρ] > 0 for all ρ.

Proof.

Υ[I] = AΦ[B2]A = I ⇐⇒ (Φ[B2])−1 = A2

Υ†[I] = BΦ†[A2]B = I ⇐⇒ (Φ†[A2])−1 = B2

Φ†[S])−1])−1

A = S1/2

B =(Φ†[S]

)−1/2

S is a xed point of the map F [X] =

Φ†[X])−1])−1

f [X] =F [X]

tr[F [X]]

By Brouwer's xed-point theorem there exists a density operator %such that f [%] = % and hence F [%] = α%, where α = tr[F [%]] > 0.If we choose A = %1/2 and B = (Φ†[%])−1/2, then Υ is trace

preserving and satises Υ[I] = αI. Therefore, if α = 1, then % is a

xed point of F that we needed to conclude the proof.

Applications

The main idea is to translate known properties of bistochastic

channels into new properties of nonunital channels or operations.

Classical capacity

i ∈ 1, . . . , NEncoder: i→ %

n is the number of qubits

Classical capacity

Quantum channel Φ is a CPT map for individual qubit

Map Φ⊗n for n qubits

The output state of n qubits is Φ⊗n[%(n)i ]

Classical capacity

Decoder: POVM, which assigns a positive-semidenite operator

M(n)j (acting on 2n-dimensional Hilbert space) to each observed

outcome j ∈ 1, . . . , N

p(n)(j|i) = tr[%(n)i M

(n)j ]

Condition∑N

j=1M(n)j = I guarantees

∑Nj=1 p

(n)(j|i) = 1.

perr(n,N) = maxj=1,...,N

(1− p(n)(j|j)

Classical capacity

R is called an achievable rate of information transmission if

limn→∞

perr(n, 2nR) = 0

Classical capacity:

C(Φ) = supR : lim

n→∞perr(n, 2

nR) = 0

Holevo4SchumacherWestmoreland5 theorem:

C(Φ) = limn→∞

nCχ(Φ⊗n)

Cχ(Ψ) = suppk,ρk

pkΨ[ρk]

)−∑k

pkS(Ψ[ρk])

S(ρ) = −tr(ρlog2ρ)4A. S. Holevo, IEEE Trans. Inf. Theory 44, 269 (1998).

5B. Schumacher, M. Westmoreland, Phys. Rev. A 56, 131 (1997).

Classical capacity

Additivity property

Cχ(Φ⊗n) = nCχ(Φ)

holds for a limited classes of channels only (depolarizing channels6,

entanglement breaking channels7, unital qubit channels8).

Υ is unital if Υ[I] = I

Unital qubit channel:

Υ[X] =1

(tr[X]I +

3∑k=1

tr[Xσk]λkσk

6C. King, IEEE Trans. Inf. Theory 49, 221 (2003).

7P. W. Shor, J. Math. Phys. 43, 4334 (2002).

8C. King, J. Math. Phys. 43, 4641 (2002).

Classical capacity of unital qubit channels

C(Υ) = Cχ(Υ) = 1− h(

(1− max

i=1,2,3|λi|))

h(x) = −xlog2x− (1− x)log2(1− x)

Optimal encodings

and decodings are known!

Message

i → binary form 0,1,0,0,1,1,. . .

%(n)i = |0〉〈0| ⊗ |1〉〈1| ⊗ |0〉〈0| ⊗|0〉〈0| ⊗ |1〉〈1| ⊗ |1〉〈1| ⊗ . . .M

(n)j =

∑x: g(x)=j

⊗nk=1M

(1)xk ,

M(1)xk ∈ |0〉〈0|, |1〉〈1|

Classical capacity of nonunital qubit channels

Φ[I] 6= I

What is the capacity

of a nonunital qubit channel?

Nobody knows

Bounds:

I X. Wang, W. Xie, R. Duan, Semidenite programming strong

converse bounds for classical capacity, IEEE Trans. Inf. Theory

64, 640 (2018).

I F. Leditzky, E. Kaur, N. Datta, M. M. Wilde, Approaches for

approximate additivity of the Holevo information of quantum

channels, Phys. Rev. A 97, 012332 (2018).

Bounds on capacity

Proposition9. Suppose Φ is a channel such that Ψ = ΦA Φ ΦB

is a channel too. Then C(Φ) > C(Ψ)− 2 log2(‖A‖‖B‖).Proof. Let %(n)

i ,M(n)i Ni=1 be the optimal code of size N = 2nRΨ

for the composite channel Ψ⊗n s.t. limn→∞ perr Ψ(n, 2nRΨ) = 0.Modied input states:

%(n)i =

B⊗n%(n)i (B†)⊗n

tr[B⊗n%(n)i (B†)⊗n]

Modied positive operator-valued measure j → M(n)j Nj=0:

M(n)0 = I −

N∑j=1

M(n)j , M

(n)j =

(A†)⊗nM(n)j A⊗n

‖A‖2n, j = 1, . . . , N,

‖X‖ = ‖X‖∞ = maxψ:〈ψ|ψ〉=1〈ψ|X†X|ψ〉 is the operator norm.9S. N. Filippov, Rep. Math. Phys. 82, 149 (2018)

Bounds on capacity

Using the modied code, let each qubit be transmitted through the

channel Φ. Then the probability to observe outcome j 6= 0 provided

input message i equals

p(n)(j|i)=tr[%

(n)i M

trA⊗nΦ⊗n

[B⊗n%

(n)i (B†)⊗n

](A†)⊗nM

tr[B⊗n%

(n)i (B†)⊗n]‖A‖2n

Since ΦA Φ ΦB = Ψ, we get

p(n)(j|i) =tr

Ψ⊗n[%(n)i ]M

tr[B⊗n%

(n)i (B†)⊗n]‖A‖2n

=p(n)(j|i)

tr[B⊗n%(n)i (B†)⊗n]‖A‖2n

where p(n)(j|i) is the probability to get outcome j ∈ 1, . . . , Nfor the input message i ∈ 1, . . . , N in the original optimal

protocol for channel Ψ⊗n.

Bounds on capacity

Observation of the outcome j = 0 in the modied protocol would

be treated as unsuccessful event, whereas observation of the

outcome j ∈ 1, . . . , N leads to a successful identication of the

message because p(n)(j|i)→ δij if n→∞.

The probability to observe nonzero outcome j equals

P (n) =

N∑j=1

p(n)(j|i) =1

(‖A‖‖B‖)2n

One can transmit information in the case of successful events j 6= 0,the average number of successfully transmitted messages N is

N = P (n)N = P (n)2nRΨ > 2n(RΨ−2 log2(‖A‖‖B‖))

Therefore, the considered protocol enables one to achieve the rate

R > RΨ − 2 log2(‖A‖‖B‖)

Bounds on capacity

If RΨ 6 C(Ψ) and one observes the successful event (j 6= 0), thanthe maximum error probability in the modied protocol

perr(n, N) = maxj=1,...,N

(1− p(n)(j|j)

)= max

j=1,...,N

(1− p(n)(j|j)

n→∞0.

Taking supremum on both sides of R > RΨ− 2 log2(‖A‖‖B‖) withrequirement limn→∞ perr(n, N) = 0, we get

C(Φ) > C(Ψ)− 2 log2(‖A‖‖B‖)

Q.E.D.

Bounds on capacity

Υ = ΦA Φ ΦB

Φ = ΦA−1 Υ ΦB−1

Corollary (10)

Let Φ be a positivity-improving qubit channel, then there exist

positive denite operators A and B acting on H2 such that the

map Υ = ΦA Φ ΦB is a unital channel and

C(Υ)− 2 log2(‖A‖‖B‖) 6 C(Φ) 6 C(Υ) + 2 log2(‖A−1‖‖B−1‖).

10S. N. Filippov, Rep. Math. Phys. 82, 149 (2018)

4-parameter nonunital qubit channels

Nonunital qubit channel

Φ[X] = 12

(tr[X](I + t3σ3) +

∑3j=1 λjtr[σj%]σj

|t3|+ |λ3| < 1

A = diag

√(1− t3)2 − λ2

√(1 + t3)2 − λ2

(1− t3)2 − λ23 − 4

√(1 + t3)2 − λ2

)2)−1/2

(1− t3)2 − λ23

(1 + t3)2 − λ23

×diag

(√(1 + t3 − λ3)

√(1− t3)2 − λ2

3 + (1− t3 + λ3)4

√(1 + t3)2 − λ2

3,√(1 + t3 + λ3)

√(1− t3)2 − λ2

3 + (1− t3 − λ3)4

√(1 + t3)2 − λ2

4-parameter nonunital qubit channels

Unital qubit channel Υ has parameters11

λ1 =2λ1√

(1 + λ3)2 − t23 +√

(1− λ3)2 − t23,

λ2 =2λ2√

(1 + λ3)2 − t23 +√

(1− λ3)2 − t23,

λ3 =4λ3(√

(1 + λ3)2 − t23 +√

(1− λ3)2 − t23)2 .

11S. N. Filippov, V. V. Frizen, D. V. Kolobova, Phys. Rev. A 97, 012322 (2018).

Example

Following Ref. 12, consider a one-parameter qubit channel

Φmix = pAp + (1− p)Dp,

where 0 ≤ p ≤ 1,Ap[X] = K1XK

†1 +K2XK

†2 is the qubit amplitude damping

channel with K1 = |0〉〈0|+√

1− p|1〉〈1| and K2 =√p|0〉〈1|,

Dp is the qubit depolarizing channel given by

Dp[X] = (1− p)X + p3(σxXσx + σyXσy + σzXσz).

Φmix is a partial case of the 4-parameter channel discussed before:

λ1 = λ2 = p√

1− p+ (1− p)(

1− 4p

)λ3 = (1− p)

(1− p

)t3 = p2.

12F. Leditzky, E. Kaur, N. Datta, M. M. Wilde, Phys. Rev. A 97, 012332 (2018)

Example

0 0.1 0.2 0.3 0.4 0.50

I X. Wang, W. Xie, R. Duan, IEEE Trans. Inf. Theory 64, 640 (2018), upper bound

I F. Leditzky, E. Kaur, N. Datta, M. M. Wilde, Phys. Rev. A 97, 012332 (2018), upper bound

I S. N. Filippov, Rep. Math. Phys. 82, 149 (2018), upper and lower bounds

I · · · · ·· Cχ(Φmix), lower bound

Improvement of bounds

Room for improvement:

I M(n)j =

(A†)⊗nM(n)j A⊗n

‖A‖2n .

Since M(n)j =

⊗nk=1Mjk and we know A explicitly, we can

replace ‖A‖2 by maxj,k‖AMjkA

†‖

I P (n) =∑N

j=1 p(n)(j|i) = 1

> 1(‖A‖‖B‖)2n

Since %(n)i =

⊗nk=1 %ik and we know B explicitly, we can

replace ‖B‖2 by maxi,k‖B%ikB†‖

This approach works for improvement of lower bound.

Improvement of bounds

0 0.1 0.2 0.3 0.4 0.50

I X. Wang, W. Xie, R. Duan, IEEE Trans. Inf. Theory 64, 640 (2018), upper boundI F. Leditzky, E. Kaur, N. Datta, M. M. Wilde, Phys. Rev. A 97, 012332 (2018), upper boundI S. N. Filippov, Rep. Math. Phys. 82, 149 (2018), upper and lower boundsI · · · · ·· Cχ(Φmix), lower boundI improved lower bound

0.0 0.2 0.4 0.6 0.8 1.0γ

1.0N = 0.1

0.0 0.2 0.4 0.6 0.8 1.0γ

1.0N = 0.2

0.0 0.2 0.4 0.6 0.8 1.0γ

1.0N = 0.3

0.0 0.2 0.4 0.6 0.8 1.0γ

1.0N = 0.4

0.0 0.2 0.4 0.6 0.8 1.0γ

1.0N = 0.45

0.0 0.2 0.4 0.6 0.8 1.0γ

1.0N = 0.5

χ(Aγ,N) Cβ(Aγ,N) CUBcov CUB

EB CUBFil CE(Aγ,N)

S. Khatri, K. Sharma, M. M. Wilde. Information-theoretic aspects of the generalized amplitude dampingchannel. arXiv:1903.07747 [quant-ph]

entanglement

(local)

F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

entanglement

(local)

F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

entanglement

(local)

F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

entanglement

(local)

F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

entanglement

(local) F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

F FÄ1 2

entanglement

(local) F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

F FÄ1 2

entanglement

(local) F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

F FÄ1 2

entanglement

(local) F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

F FÄ1 2

%out = Φ1 ⊗ Φ2[%in]

entanglement

(local) F FÄ1 2

FA BA B

Global noise:

(local)

External noises

thermal

F FÄ1 2

%⊗nout −→purication

(|ψ+〉〈ψ+|)⊗m, |ψ+〉 =1√2

(|00〉+ |11〉)

Entanglement of purication13

Ep = limn→∞

For two qubit states %out

Ep > 0 i %out is entangled14

The noise Φ1 ⊗ Φ2 is admissible if there exists an input state %in

such that Φ1 ⊗ Φ2[%in] is entangled.

DenitionThe channel Φ1 ⊗ Φ2 is called entanglement annihilating if

Φ1 ⊗ Φ2[%in] is separable for all input states %in.15

13B.M. Terhal, M. Horodecki, D.W. Leung, D.P. DiVincenzo, J. Math. Phys. 43 4286 (2002)

14M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 78 574 (1997)

15L. Moravcikova and M. Ziman, J. Phys. A: Math. Theor. 43 275306 (2010)

Proposition

The local two-qubit unital map Υ⊗Υ is entanglement annihilating

if and only if Υ2 is entanglement breaking, i.e. λ21 + λ2

2 + λ23 6 1.16

Proposition

The maximally entangled state |ψ+〉 = 1√2(|00〉+ |11〉) is the most

robust to the loss of entanglement in the case of general local

unital dynamical maps Υt ⊗Υt.

16S.N. Filippov, T. Rybar, M. Ziman, Phys. Rev. A 85 012303 (2012)

Υt = ΦAt Φt ΦBt

Φt = ΦA−1tΥt ΦB−1

Proposition

Local non-unital map Φt ⊗ Φt is entanglement annihilating if and

only if λ21(t) + λ2

2(t) + λ23(t) 6 1.

Solving equation λ21 + λ2

2 + λ23 = 1, we nd entanglement lifetime τ .

Proposition

The most robust entangled state w.r.t. local non-unital noises

Φt ⊗ Φt is17

|ψ〉 =Bτ ⊗Bτ |ψ+〉√

〈ψ+|B†τBτ ⊗B†τBτ |ψ+〉

17S. N. Filippov, V. V. Frizen, D. V. Kolobova, Ultimate entanglement robustness of two-qubit

states against general local noises, Phys. Rev. A 97, 012322 (2018).

Generalized amplitude-damping noise

λ1 = λ2 =√λ3 = e−γt, t3 = (2w − 1)(1− e−2γt),

where w, 1− w are the populations of ground and excited levels in

thermal equilibrium, i.e. w = 11+exp(−∆E/kT )

λ1(t) = λ2(t) = e−γt√

w(1− w)(1− e−2γt)

[1− w(1− e−2γt)][w + e−2γt(1− w)]−1

and λ3(t) = λ21(t) = λ2

Solving equation λ21(t) + λ2

2(t) + λ23(t) = 1, we nd the maximal

entanglement lifetime:

2 + 1)w(1− w)

1+4(√

2+1)w(1−w)−√

1+8(√

2+1)w(1−w)

Generalized amplitude-damping noise

The most robust entangled state:

|ψ〉 =

√(1− w)[1− (1− w)(1− e−2γτ )]

1− (1− 2w + 2w2)(1− e−2γτ )|0〉 ⊗ |0〉

√w[1− w(1− e−2γτ )]

1− (1− 2w + 2w2)(1− e−2γτ )|1〉 ⊗ |1〉

0 0.5 g ty+g tè

If w → 0, then τ /τψ+ → 2, i.e. the use of the ultimately robust

state allows to prolong entanglement lifetime twice as compared

with the entanglement lifetime of the maximally entangled state.

Trace decreasing operations

To deal with a nite dimensional Hilbert space, we here consider a

situation where the transmitted state is postselected in the basis of

the injected qubit state, |−l0,−l0〉, |−l0, l0〉, |l0,−l0〉, |l0, l0〉.Since such postselection entails the decay of the output state, the

decaying output biphoton state needs to be renormalized by its

trace before we can quantify the entanglement evolution in

turbulence by the concurrence. N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner. Universal entanglement decay of photonicorbital-angular-momentum qubit states in atmospheric turbulence. Phys. Rev. A 91, 012345 (2015)

Compatibility of channels19

Φ is 2-selfcompatible if and only if the Choi state ΩΦ is symmetric

extendable.

For qubits, Ω is symmetric extendable if and only if18

tr[Ω2B] ≥ tr[Ω2

AB]− 4√

detΩAB.

18J. Chen, Z. Ji, D. Kribs, N. Lutkenhaus, and B. Zeng. Symmetric extension of two-qubit states.

Phys. Rev. A 90, 032318 (2014)19

T. Heinosaari and T. Miyadera. Incompatibility of quantum channels. J. Phys. A: Math. Theor. 50,135302 (2017)

Summary

We have reviewed the quantum analogue of Sinkhorn's theorem

and found the explicit decomposition in the case of qubit maps.

As applications of it we have considered estimation of capacity for

nonunital qubit channels and entanglement robustness.

I We have obtained new lower and upper bounds on classical

capacities of nonunital qubit channels.

I The obtained result holds true for the regularized version of

χ-capacity.

I We have illustrated our ndings by 4-parameter family of

nonunital channels and, in particular, a mixture of amplitude

damping and depolarizing channels.

Summary

The most robust states to loss of entanglement under local noises:

I |ψ+〉 for two qubit unital Υ⊗Υ

I |ψ〉 ∝ B⊗B|ψ+〉 for two qubit non-unital Φ⊗ Φ

Our proofs are based on the relation between unital and nonunital

qubit channels. Such a relation may turn out to be productive in

other research areas as well.

Thank you for attention!

quantum sinkhorn's theorem: applications in entanglement ...hhlee/filippov_maqit_2019.pdf ·...

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