Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Additivity questions and tensor powers ofrandom quantum channels
Motohisa Fukuda

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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

1

Introduction to additivity violation
Set-upAdditivity violation and its consequences
Open problems

2

Understanding proofs on additivity violation
Intuitive explanation
Aubrun-Szarek-Werner approach
On various proofs of additivity violation

3Tensor powers of quantum channels“Best inputs” for tensor powers
Bounds on tensor powers of quantum channels
Short discussion

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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Quantum states and channels
A quantum state ρ (in finite dimension) is a positive
semi-definite Hermitian operator with trace one on a Hilbert
space Cn .
A channel can be written as
Φ(ρ) = TrCk [V ρV ∗ ]Here, V : Cl → Ck ⊗ Cn is a partial isometry. This meansthat a channel is completely positive and trace preserving.

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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Complementary channels
When the input ρ = |xihx| is a rank-one projection the following
two matrices share the same non-zero eigenvalues.
TrCk [V |xihx|V ∗ ] ∼ TrCn [V |xihx|V ∗ ](∼ diag(r1 , . . . , rd ))Indeed, V |xi ∈ Ck ⊗ Cn has the Schmidt decomposition:
V |xi =dX
√ri |ui i ⊗ |vi ii=1

where ri > 0 is a probability distribution, and {ui }, {vi } are
orthonormal in Ck and Cn .
We define the complementary channel of Φ by 1Φc (ρ) = TrCn [V ρV ∗ ]1[Holevo][King, Matsumoto, Nathanson, Ruskai]
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Minimum output entropy (MOE)The minimal output entropy of channel Φ is defined bySmin (Φ) = min S(Φ(ρ))ρwhere ρ are input states. [King, Ruskai]
Here, the von Neumann entropy S(·) of quantum state ρ is:S(ρ) = − Tr[ρ log ρ] = −dXλi log λii=1where λi are eigenvalues of ρ. Note thatS(ρ ⊗ σ) = S(ρ) + S(σ)
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Holevo capacity (HC)Holevo capacity of channel Φ is defined as:"#XXpi σi )) −pi S(Φ(σi ))χ(Φ) = max S(Φ(pi ,σiii

where {pi , σi } is an ensemble. [Holevo][Schumacher,Westmoreland]
We have an easy bound: χ(Φ) ≤ log d − Smin (Φ)
The above bound is saturated when, for example,
Φ(Ug ρUg∗ ) = Ug Φ(ρ)Ug∗
where g 7→ Ug · Ug∗ is an irreducible representation. [Holevo]
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Remarks on MOE and HC
MOE measures purity of channels by considering optimal
output while HC is connected to the capacity C (·):
C (Φ) = lim

1n→∞ nχ(Φ⊗n )Without entangled inputs or if additivity of χ holds, thenC (Φ) = χ(Φ)
Since von Neumann entropy is concave, MOE is achieved bypure input states. This means,
Smin (Φ) = Smin (Φc )To calculate HC, we need to know about more than just one
output state, and in generalχ(Φ) 6= χ(Φc )7 / 23

Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Additivity violation
Write quantum channels:
Φ(ρ) = TrCk [V ρV ∗ ]and their complex conjugate channels:hiΦ̄(ρ) = TrCk V̄ ρV T
Then, with high probability we have additivity violation 2 :Smin (Φ ⊗ Φ̄) < Smin (Φ) + Smin (Φ̄)
Note that, for any channels Φ and Ω,min S((Φ ⊗ Ω)(ρ ⊗ σ)) = min S(Φ(ρ)) + min S(Ω(σ))ρ⊗σ2ρσ[Hastings]: more precisely, another model was used.
8 / 23Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Hastings proved:
Smin (Φ) + Smin (Φ̄) − Smin (Φ ⊗ Φ̄) ∼log kkby using a “random random unitary channel” with 1  k  n:Φ(ρ) =kXri Ui ρUi∗

i=1

whereUi ∈ U(n) are i. i. d.
, 2nk2ni
XXri ∼Xj2
Xj2
j=2n(i−1)+1

j=1

where Xi are i. i. d. normal distributions.
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Entangled inputs can improve the capacity - sketchy
1We know that there is a channel such that
Smin (Φ ⊗ Φ̄) = Smin (Φ) + Smin (Φ̄) − 2where>0This implies 3 that Ω = Φ ⊕ Φ̄ gives
Smin (Ω⊗2 ) = 2Smin (Ω) − Then, there exists 4 a channel Ψ such thatχ(Ψ ⊗ Ψ) = 2χ(Ψ) + 

3So, we have11· χ Ψ⊗2n ≥ lim · χ Ψ⊗2 = χ(Ψ) +n→∞ 2nn→∞ 2
2C (Ψ) = lim

I.e.,entangled inputs improve the classical capacity: C (·).
34

[Fukuda, Wolf][Shor]
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Additivity question for regularized quantities
Classical capacity:?

C (Φ ⊗ Ω) = C (Φ) + C (Ω)

for Φ 6= ΩRegularized minimum output entropy:?

Ŝmin (Φ ⊗ Ω) = Ŝmin (Φ) + Ŝmin (Ω)
Here,for Φ 6= Ω1· Smin (Φ⊗N )N→∞ NŜmin (Φ) = lim

Remark: Non-additivity of Ŝmin (·) implies non-additivity of C (·).

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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Set-upAdditivity violation and its consequences
Open problems

Finding counterexamples
Concrete counterexamples for 1 ≤ p ≤ 2 are still open.
Remark:Concrete counterexamples for the following additivity violation
were found [Grudka, M. Horodecki,Pankowski]:Sp,min (Φ ⊗ Φ) < Sp,min (Φ) + Sp,min (Φ)

p>2Here,Sp,min (Φ) = min Sp (Φ(ρ))ρplog kσkp .
1−pIrreducible subspaces of group representations are being
investigated by Brannan and Collins.

where Sp is the Renyi p-entropy: Sp (σ) =12 / 23

Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Intuitive explanation
Aubrun-Szarek-Werner approach

Tensor of “conjugate pair” has rather small entropy
Suppose we have a quantum channel
Φ(ρ) = TrCn [V ρV ∗ ]whereV : Cl → Cn ⊗ Ck
is an isometry. Then, for |bi a Bell state,
lhbk | Φ ⊗ Φ̄(|bl ihbl |) |bk i ≥kn
This means that Φ ⊗ Φ̄ has an output with a large eigenvalue.
Additivity violation for 1 < p ≤ ∞ was shown via this trick 5 , andfor p = 1 later.
5[Hayden, Winter]13 / 23

Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Intuitive explanation
Aubrun-Szarek-Werner approach

Tensor of conjugate pair - Example
The idea behind is:
U ⊗ Ū |bm i = |bm i
for U ∈ U(m).
For example, take a random unitary channel:
k1X
Ψ(ρ) =Ui ρUi∗ki=1

so thatΨ ⊗ Ψ̄(|bihb|) =1
1 X|bihb| + 2(Ui ⊗ Ūj ) |bihb| (Ui∗ ⊗ UjT )kk
i6=j

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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Intuitive explanation
Aubrun-Szarek-Werner approach

Single channel has rather large entropy
What are typical outputs for randomly selected channels like?
|aiha| 7→ V |aiha|V ∗ = |w ihw | 7→ TrCn [|w ihw |] = WW ∗|ai is a fixed vector in Cl .
V |ai is a random vector in Ck ⊗ Cn .
WW ∗ is the normalized Wishart matrix.
The probability density of WW ∗ is proportional to:
XYYδ 1 −pi (pi − pj )2pin−k1≤i≤k1≤i<j≤k1≤i≤k

The last factor shows that n  k implies concentration ofeigenvalues. So, typical outputs have rather large entropy.
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels

Intuitive explanation
Aubrun-Szarek-Werner approach

Aubrun-Szarek-Werner approach for p > 1
Define a random quantum channel Φ by the random isometry:
V : Cn1+1/p→ Cn ⊗ Cn .

Based on the previous argument, Φ ⊗ Φ̄ has a large output
eigenvalue larger than n−1+1/p .
By using Dvoretzky’s theorem,Sp,min (Φ) ∼ Sp,min (Φ ⊗ Φ̄).
Of course then we have violation for large n.
Sp,min (Φ) + Sp,min (Φ̄) > Sp,min (Φ ⊗ Φ̄).
They later showed additivity violation for p = 1 by a similartechnique.
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels“Best inputs” for tensor powers
Bounds on tensor powers of quantum channels
Short discussion

What are candidates for optimal inputs for Φ ⊗ Φ̄?6Take a random quantum channels defined byΦn (ρ) = TrCn [V ρV ∗ ]withV : Cl → Ckn
where l = tkn, k ∈ N, t ∈ (0, 1) are fixed and n → ∞.
Then, we investigated the asymptotic behavior (as n → ∞) ofoutput eigenvalues ofZn = Φn ⊗ Φ̄n (|an ihan |)where (an )n∈N is a fixed sequence of unit vectors.
6[Collins, F, Nechita]
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels“Best inputs” for tensor powersBounds on tensor powers of quantum channels
Short discussion

We found that the empirical eigenvalue distribution of the matrix
Zn converges almost surely, as n → ∞, to:1 2+(k−1)δ
dx
δ
λλ
2
1
k2
where the Dirac masses are located at
1 − t|m|2
1 − t|m|2
λ1 = t|m|2 +andλ=.
2k2k2if 
1Tr [An ]√=m+O
n2lHere, |an i ↔ An is the correspondence Cl ⊗ Cl ↔ Ml (C).
Conclusion: The Bell state is best. Examine, for example,
X
a=αi |ii ⊗ |iiiRemark. Nothing interesting with Φ ⊗ Φ, Φ ⊗ ΦT or Φ ⊗ Φ∗ .
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels“Best inputs” for tensor powers
Bounds on tensor powers of quantum channels
Short discussion

How about tensor powers (Φ ⊗ Φ̄)⊗r ?7
Our calculation shows that tensor-products of Bell states are best.
Suppose we have a random quantum channel:
12r

1̂

2̂r̂

Φ ⊗ Φ ⊗ ··· ⊗ Φ ⊗ Φ ⊗ Φ ⊗ ··· ⊗ Φwhere best inputs are|bπ(1),1̂ i ⊗ |bπ(2),2̂ i ⊗ · · · ⊗ |bπ(r ),r̂ iwhere π ∈ Sr . Here, |bi,j i is a Bell state over the i-th space for Φand j-th space for Φ̄.
Remark. Hastings conjectured that violation of additivity happens
only within each conjugate pair.
7[F, Nechita]
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels“Best inputs” for tensor powers
Bounds on tensor powers of quantum channels
Short discussion

How about tensor powers Φ⊗2r , where Φ is orthogonal ? 8This time, we generate random channels by orthogonal matrices
instead of unitary ones. So, Φ̄ = Φ.
12rr +1r +22r

Φ ⊗ Φ ⊗ ··· ⊗ Φ ⊗ Φ ⊗ Φ ⊗ ··· ⊗ Φwhere best inputs areO|bc ic∈πwhere π is a paring of 2r elements. Here, |bc i is a Bell state overthe i-th and j-th spaces when c = (i, j).
We conjecture that typically for orthogonal caseSmin (Φ⊗2r ) = r Smin (Φ⊗2 )or, we can make it weaker:
11lim Smin (Φ⊗r ) = Smin (Φ⊗2 )r →∞ r28[F, Nechita]

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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels“Best inputs” for tensor powers
Bounds on tensor powers of quantum channels
Short discussion

Montanaro’s multiplicative boundrkΦ⊗r k1→∞ ≤ k(V V ∗ )Γ k∞
where V is the isometry defining Φ.
F-Nechita’s multiplicative boundrkΦ⊗r k1→2 ≤ kCΦΓ k∞where CΦΓ is the partially transposed Choi matrix of Φ.
Then the bounds lead to the following weak additivity respectively
for p = ∞, 2: typically under random choice of channels
rSp,min (Φ⊗r ) ≥ Sp,min (Φ)
2Montanaro first described it as “weakly multiplicative”, in terms ofmaximum output p-norms.
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels“Best inputs” for tensor powers
Bounds on tensor powers of quantum channels
Short discussion

F-Gour’s multiplicative bound (no random here)
For a unital quantum channel: Mn (C) → Mk (C),kΦ⊗r k1→2 ≤ (γΦ )r /2 .
Here,
11kDΦ DΦ∗ k∞γΦ = + 1 −kn
where DΦ is the dynamical matrix of Φ restricted on trace-less
Hermitian matrices.
We also got an upper bound for the classical capacity:C (Φ) ≤ log k + log γΦ .
This bound is saturated by the Werner-Holevo channel.
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Introduction to additivity violation
Understanding proofs on additivity violation
Tensor powers of quantum channels“Best inputs” for tensor powers
Bounds on tensor powers of quantum channels
Short discussionSummary
Additivity violation may be a special phenomena only for
conjugate pairs.
Perhaps, additivity violation typically does not hold for Φ⊗nwhen Φ is generated by unitary group.
Otherwise, we need to know how fast non-additivity grows
and how much contribution it makes for regularized quantity.

Thank you very much.

Acknowledgement: JSPS KAKENHI Grant Number JP16K00005

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