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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
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Plan
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
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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).
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1
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1
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3 1 2 13 1 11 2 2
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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
Y =
p1→1 p2→1 p3→1
p1→2 p2→2 p3→2
p1→3 p2→3 p3→3
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X =
1 2 13 1 11 2 2
Premise: transition matrix Y is bistochastic (uniform distribution is
a xed point)
XD2 =
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 ...
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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).
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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)
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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.
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Proof.
Υ[I] = AΦ[B2]A = I ⇐⇒ (Φ[B2])−1 = A2
Υ†[I] = BΦ†[A2]B = I ⇐⇒ (Φ†[A2])−1 = B2
(Φ[(
Φ†[S])−1])−1
= S
A = S1/2
B =(Φ†[S]
)−1/2
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S is a xed point of the map F [X] =
(Φ[(
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.
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Applications
The main idea is to translate known properties of bistochastic
channels into new properties of nonunital channels or operations.
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Classical capacity
n
...
i
i ∈ 1, . . . , NEncoder: i→ %
(n)i
n is the number of qubits
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Classical capacity
n
...
n
...
F
i
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 ]
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Classical capacity
t
n
...
n...
F
i
j
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)
)
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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→∞
1
nCχ(Φ⊗n)
Cχ(Ψ) = suppk,ρk
[S
(∑k
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).
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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
l
l
3
1
t
l
l
3
1
l
l
3
1
Unital qubit channel:
Υ[X] =1
2
(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).
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Classical capacity of unital qubit channels
C(Υ) = Cχ(Υ) = 1− h(
1
2
(1− max
i=1,2,3|λi|))
h(x) = −xlog2x− (1− x)log2(1− x)
l
l
3
1
t
l
l
3
1
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|
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Classical capacity of nonunital qubit channels
t
l
l
3
1
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).
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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)
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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
(n)j
]=
trA⊗nΦ⊗n
[B⊗n%
(n)i (B†)⊗n
](A†)⊗nM
(n)j
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
(n)j
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.
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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
tr[B⊗n%(n)i (B†)⊗n]‖A‖2n
>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‖)
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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)
P (n)
)= 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.
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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)
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4-parameter nonunital qubit channels
Nonunital qubit channel
Φ[X] = 12
(tr[X](I + t3σ3) +
∑3j=1 λjtr[σj%]σj
)with
|t3|+ |λ3| < 1
A = diag
(4
√(1− t3)2 − λ2
3 ,4
√(1 + t3)2 − λ2
3
)
B =
√2
(4−
(4√
(1− t3)2 − λ23 − 4
√(1 + t3)2 − λ2
3
)2)−1/2
4√
(1− t3)2 − λ23
4√
(1 + t3)2 − λ23
×diag
(√(1 + t3 − λ3)
4
√(1− t3)2 − λ2
3 + (1− t3 + λ3)4
√(1 + t3)2 − λ2
3,√(1 + t3 + λ3)
4
√(1− t3)2 − λ2
3 + (1− t3 − λ3)4
√(1 + t3)2 − λ2
3
)
C(Υ)− 2 log2(‖A‖‖B‖) 6 C(Φ) 6 C(Υ) + 2 log2(‖A−1‖‖B−1‖).
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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 .
C(Υ)− 2 log2(‖A‖‖B‖) 6 C(Φ) 6 C(Υ) + 2 log2(‖A−1‖‖B−1‖).
11S. N. Filippov, V. V. Frizen, D. V. Kolobova, Phys. Rev. A 97, 012322 (2018).
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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
3
)λ3 = (1− p)
(1− p
3
)t3 = p2.
12F. Leditzky, E. Kaur, N. Datta, M. M. Wilde, Phys. Rev. A 97, 012332 (2018)
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Example
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
p
C
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
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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
tr[B⊗n%(n)i (B†)⊗n]‖A‖2n
> 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.
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Improvement of bounds
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
p
C
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
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0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.1
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.2
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.3
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.4
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.45
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
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]
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entanglement
entanglement
(local)
AA
BB
F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
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entanglement
entanglement
(local)
AA
BB
F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
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entanglement
entanglement
(local)
AA
BB
F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
AA
BB
IdId
IdId
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entanglement
entanglement
(local)
AA
BB
F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
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entanglement
entanglement
(local) F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
AA
BB
AA
BB
IdId
IdId
F FÄ1 2
AA BB
AA
BB
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entanglement
entanglement
(local) F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
AA
BB
AA
BB
IdId
IdId
F FÄ1 2
AA BB
AA
BB
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entanglement
entanglement
(local) F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
AA
BB
AA
BB
IdId
IdId
F FÄ1 2
AA BB
AA
BBF1
F2
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entanglement
entanglement
(local) F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
AA
BB
AA
BB
IdId
IdId
F FÄ1 2
AA BB
AA
BBF1
F2
AA
BBF1
F2
%in
%out = Φ1 ⊗ Φ2[%in]
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entanglement
entanglement
(local) F FÄ1 2
AA BB
FA BA B
Global noise:
(local)
Alice
Bob
External noises
Alice
Bob
Alice
Bob
Alice
Bob
vac
thermal
AA
BB
AA
BB
AA
BB
AA
BB
IdId
IdId
F FÄ1 2
AA BB
AA
BB
AA
BBF1
F2
AA
BBF1
F2
BB
BB
BB
AA
AA
AA
...
...
...
...
...
...
%in
%in
%in
%in
%⊗nout −→purication
(|ψ+〉〈ψ+|)⊗m, |ψ+〉 =1√2
(|00〉+ |11〉)
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Entanglement of purication13
Ep = limn→∞
m
n
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)
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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)
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Υt = ΦAt Φt ΦBt
Φt = ΦA−1tΥt ΦB−1
t
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).
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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
2(t).
Solving equation λ21(t) + λ2
2(t) + λ23(t) = 1, we nd the maximal
entanglement lifetime:
τ =1
2γln
4(√
2 + 1)w(1− w)
1+4(√
2+1)w(1−w)−√
1+8(√
2+1)w(1−w)
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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〉
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0 0.5 g ty+g tè
g t
0.1
0.2
0.3
0.4
0.5N
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.
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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)
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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)
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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.
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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.
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Thank you for attention!