The semiring of dichotomies andasymptotic relative submajorizationarXiv:2004.10587 and arXiv:2007.11258

Gergely Bunth, Christopher Perry, Peter Vrana and Albert H. Werner

Budapest University of Technology and EconomicsMTA-BME Lendulet Quantum Information Theory Research Group

Feb 4, QIP2021 Munich

Dichotomies

I quantum system with Hilbert space H

I pair of states (ρ, σ) on H

I interpretations:

1. “black-box”, possible preparations2. ρ state, σ reference

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Pair transformations

I Let ρ1, σ1 ∈ S(H) and T a channel from H to K

I T transforms (ρ1, σ1) to (T (ρ1),T (σ1)) = (ρ2, σ2)

I T linear, convenient to allow trace 6= 1

T (ρ1) = ρ2 ⇐⇒ T (cρ1) = cρ2

T (σ1) = σ2 ⇐⇒ T (cσ1) = cσ2

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Example: majorization

Let p1, p2 ∈ Rd≥0. p1 majorizes p2 if

k∑i=1

(p↓1)i ≥k∑

i=1

(p↓2)i for k = 1, 2, . . . , d , equality

if k = d .

I ρ1 = diag(p1,1, . . . , p1,d) ∈ Cd×d

I ρ2 = diag(p2,1, . . . , p2,d) ∈ Cd×d

I σ1 = σ2 = I ∈ Cd×d identity matrix

p1 majorizes p2 iff T (ρ1) = ρ2 and T (I ) = I for some channel T

Applications

I mixedness

I pure bipartite entanglement

I coherence

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Example: Gibbs preserving maps

I thermal operations model interaction with heat bath at temperature β−1

I T is Gibbs preserving if T (2−βH) = 2−βH

I (ρ1, 2−βH)→ (ρ2, 2

−βH)

I thermal operations are Gibbs preserving

I classically ρ1 7→ ρ2 possible with Gibbs preserving T iff possible with thermaloperation

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Example: hypothesis testing

Box transformations

I “black-box” either containing ρ or σ

I apply completely positive trace preserving map T to unknown state: new pair(T (ρ),T (σ))

I (ρ, σ) more distinguishable than (T (ρ),T (σ))

ρ

σ

T T (ρ)

T (σ)

(Π, I − Π)?

(T ∗(Π),T ∗(I − Π))

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Relative submajorization

I allow unnormalized ρ2, σ2, traces represent probabilities

I T completely positive trace nonincreasing such that

T (ρ1) ≥ ρ2T (σ1) ≤ σ2

relative submajorization [Ren16], notation: (ρ1, σ1) < (ρ2, σ2)

I interpretation: probabilistic transformations

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Example: hypothesis testing

I two-outcome measurement (Π, I − Π), probabilities Tr ρΠ, Tr ρ(I − Π)

I redundant: probabilities sum to 1

I test: T (ρ) = Tr(ρΠ) for some 0 ≤ Π ≤ I

I (ρ, σ) < (a, b) iff there is a test Π with

α(Π) = 1− Tr(ρΠ) ≤ 1− a type I error

β(Π) = Tr(σΠ) ≤ b type II error

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Tensor product

I (ρ1, σ1) and (ρ2, σ2) dichotomies on H1 and H2

I product(ρ1, σ1) · (ρ2, σ2) = (ρ1 ⊗ ρ2, σ1 ⊗ σ2)

dichotomy on H1 ⊗H2

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Example: independent subsystems

I H1,H2 Hamiltonians and ρ1, ρ2 states

I (ρ1, 2−βH1) and (ρ2, 2

−βH2)

I total Hamiltonian (no interaction): H = H1 ⊗ I + I ⊗ H2

I Gibbs state factorizes2−βH = 2−βH1 ⊗ 2−βH2

I independent preparation, no interaction: (ρ1 ⊗ ρ2, 2−βH1 ⊗ 2−βH2)

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Example: asymptotic hypothesis testing

I discriminate ρ⊗n and σ⊗n, test Πn

I errors αn = 1− Tr(ρ⊗nΠn) and βn = Tr(σ⊗nΠn)

I quantum Stein’s lemma [HP91]: αn → 0 possible if βn ≥ 2−nD(ρ‖σ)+o(n)

Strong converse

I if βn → 0 faster, then αn → 1

I exponential convergence, βn = 2−rn+o(n) and αn = 1− 2−Rn+o(n)

R∗(r) = supα>1

α− 1

α

[r − Dα(ρ‖σ)

]strong converse exponent [MO15]

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Asymptotic relative submajorization

I compare many copies (ρ⊗n1 , σ⊗n1 ) and (2−Rn+o(n)ρ⊗n2 , 2−rnσ⊗n2 )

I ordered by relative submajorization

I can be made exact in second component if Tr σ1 ≥ 2−r Tr σ2

Goal

Characterize trade-off between R and r .

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Result: characterization of asymptotic transformations

Theorem

Let (ρ1, σ1), (ρ2, σ2) be pairs of states such that supp ρi ⊆ suppσi . Given R, r ,(ρ⊗n1 , σ⊗n1 ) relative submajorizes (2−Rn+o(n)ρ⊗n2 , 2−rnσ⊗n2 ) for every n iff

R ≥ R∗(r) = supα>1

α− 1

α

[r − Dα(ρ1‖σ1) + Dα(ρ2‖σ2)

]

Dα(ρ‖σ) =1

α− 1log Tr(σ

1−α2α ρσ

1−α2α )α

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Overview of proof

Dual approaches:

I constructions: channels T such that T (ρ⊗n1 ) ≥ ρ⊗n2 and T (σ⊗n1 ) ≤ σ⊗n2

I obstructions: monotone quantities that disprove (ρ1, σ1) & (ρ2, σ2)

Strategy

1. show that special obstructions suffice

2. classify these obstructions

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Preordered semirings

I preordered semiring: (S ,+, ·, 0, 1,4), operations and preorder behave like in Nin detail:

I +,· associative, commutative, distributiveI 0 and 1 neutral elements for + and ·I 4 reflexive, transitiveI x 4 y implies x + z 4 y + z and x · z 4 y · z

I asymptotic preorder &: for suitable u ∈ S , define x & y as uo(n) · xn < yn

I if f : S → R≥0 multiplicative, <-monotone, then also &-monotone:

uo(n) · xn < yn =⇒ f (uo(n) · xn) ≥ f (y) =⇒ f (u)o(1)︸ ︷︷ ︸→1

f (x) ≥ f (y)

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Duality

I spectrum: ∆(S) = {f : S → R≥0|f monotone homomorphism}in detail:

I f (1) = 1I f (x + y) = f (x) + f (y)I f (x · y) = f (x)f (y)I x < y =⇒ f (x) ≥ f (y)

I x & y =⇒ ∀f ∈ ∆(S) : f (x) ≥ f (y)

Theorem (informal, see precise forms in [Str88] and [Vra20b])

Let S be preordered semiring satisfying additional conditions. Then

x & y ⇐⇒ ∀f ∈ ∆(S) : f (x) ≥ f (y)

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Examples of preordered semirings

semiring dual char spectrum

2-tensors yes trivialk-tensors [Str88] yes hard (but see recent progress [CVZ18])graphs [Zui19] yes hard (but see recent progress [Vra19])graphs, q/ea [LZ20] yes hardnc graphs [LZ20] ? hardnc graphs, ea [LZ20] yes hardbipartite LOCC [JV19] yes nontrivial but solvedk-partite LOCC [JV19] yes hard (but see recent progress [Vra20a])dichotomies [PVW20] yes [Vra20b] nontrivial but solved (this talk)multiple states [BV20] yes [Vra20b] nontrivial but solved (this talk)

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Semiring of dichotomies

I elements: (equivalence classes of) pairs (ρ, σ) where ρ, σ ≥ 0, supp ρ ⊆ suppσ

I operations: direct sum, tensor product

(ρ1, σ1) + (ρ2, σ2) =

([ρ1 00 ρ2

],

[σ1 00 σ2

])(ρ1, σ1) · (ρ2, σ2) = (ρ1 ⊗ ρ2, σ1 ⊗ σ2)

I preorder: relative submajorization

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Characterization of sandwiched Renyi divergences

need to find spectrum, i.e. functions f (ρ, σ) that satisfy

(i) f (ρ1 ⊗ ρ2, σ1 ⊗ σ2) = f (ρ1, σ1)f (ρ2, σ2) (multiplicativity)

(ii) f (ρ1 ⊕ ρ2, σ1 ⊕ σ2) = f (ρ1, σ1) + f (ρ2, σ2) (additivity)

(iii) f (In, In) = n (normalization)

(iv) f (T (ρ),T (σ)) ≤ f (ρ, σ) when T is a completely positive trace-nonincreasingmap (data processing inequality)

(v) f is increasing in the first and decreasing in the second argument (with respect tothe semidefinite partial order). (monotonicity)

Theorem

These arefα(ρ, σ) = Qα(ρ, σ) = Tr(σ

1−α2α ρσ

1−α2α )α

with α ≥ 1.

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Summary

I characterization of asymptotic relative submajorization

I extends operational interpretation of sandwiched Renyi divergences

I axiomatic characterization of sandwiched Renyi divergences

I new technique for studying resource theories directly in the asymptotic regime

I robust tools, may be suitable for studying similar problems, e.g. channeldiscrimination, restricted transformations

I would be interesting to extend method to different asymptotic regimes(approximate, vanishing error, direct region, etc.)

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[BV20] Gergely Bunth and Peter Vrana.Asymptotic relative submajorization of multiple-state boxes.2020.arXiv:2007.11258.

[CVZ18] Matthias Christandl, Peter Vrana, and Jeroen Zuiddam.Universal points in the asymptotic spectrum of tensors.In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory ofComputing, pages 289–296. ACM, 2018.arXiv:1709.07851, doi:10.1145/3188745.3188766.

[HP91] Fumio Hiai and Denes Petz.The proper formula for relative entropy and its asymptotics in quantumprobability.Communications in mathematical physics, 143(1):99–114, 1991.doi:10.1007/BF02100287.

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[JV19] Asger Kjærulff Jensen and Peter Vrana.The asymptotic spectrum of LOCC transformations.IEEE Transactions on Information Theory, 66(1):155–166, Jan 2019.arXiv:1807.05130, doi:10.1109/TIT.2019.2927555.

[LZ20] Yinan Li and Jeroen Zuiddam.Quantum asymptotic spectra of graphs and non-commutative graphs, andquantum shannon capacities.IEEE Transactions on Information Theory, 67(1):416–432, 2020.arXiv:1810.00744, doi:10.1109/TIT.2020.3032686.

[MO15] Milan Mosonyi and Tomohiro Ogawa.Quantum hypothesis testing and the operational interpretation of thequantum Renyi relative entropies.Communications in Mathematical Physics, 334(3):1617–1648, 2015.arXiv:1309.3228, doi:10.1007/s00220-014-2248-x.

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[PVW20] Christopher Perry, Peter Vrana, and Albert H Werner.The semiring of dichotomies and asymptotic relative submajorization.2020.arXiv:2004.10587.

[Ren16] Joseph M Renes.Relative submajorization and its use in quantum resource theories.Journal of Mathematical Physics, 57(12):122202, 2016.arXiv:1510.03695, doi:10.1063/1.4972295.

[Str88] Volker Strassen.The asymptotic spectrum of tensors.Journal fur die reine und angewandte Mathematik, 384:102–152, 1988.doi:10.1515/crll.1988.384.102.

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[Vra19] Peter Vrana.Probabilistic refinement of the asymptotic spectrum of graphs.2019.arXiv:1903.01857.

[Vra20a] Peter Vrana.A family of multipartite entanglement measures.2020.arXiv:2008.11108.

[Vra20b] Peter Vrana.A generalization of Strassen’s spectral theorem.2020.arXiv:2003.14176.

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[Zui19] Jeroen Zuiddam.The asymptotic spectrum of graphs and the Shannon capacity.Combinatorica, 2019.arXiv:1807.00169, doi:10.1007/s00493-019-3992-5.

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