Diagonal couplings of quantum Markov chains

Kümmerer, Burkhard; Schwieger, Kay

doi:10.1142/s0219025716500120

Cited by 4 publications

(4 citation statements)

References 25 publications

(25 reference statements)

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“…where 𝑐 1 and 𝑐 2 are cq statements, Φ and Ψ are relational assertions. Validity of judgments is based on the concept of coupling [32,34,43,48,49,58], which is employed to interpret judgments of relational logic in both probabilistic [6,8] and quantum [10,33,46,47] setting. We define cq-coupling as a combination of both probabilistic and quantum couplings.…”

Section: Assertions and Judgmentmentioning

confidence: 99%

“…Coupling is an essential concept to define validity of relational judgments. Probabilistic coupling, a standard abstraction from probability theory [34,43,48], is employed to interpret judgments in probabilistic relational logic [6,8]; similar concept of quantum coupling has also been proposed [32,49,58] and used in establishing quantum relational logic [10,33,46,47].…”

Section: A3 Classical-quantum Couplingmentioning

confidence: 99%

See 1 more Smart Citation

EasyPQC: Verifying Post-Quantum Cryptography

Barbosa

Barthe

Fan³

et al. 2021

Proceedings of the 2021 ACM SIGSAC Conference on Computer and Communications Security

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EasyCrypt is a formal verification tool used extensively for formalizing concrete security proofs of cryptographic constructions. However, the EasyCrypt formal logics consider only classical attackers, which means that post-quantum security proofs cannot be formalized and machine-checked with this tool. In this paper we prove that a natural extension of the EasyCrypt core logics permits capturing a wide class of post-quantum cryptography proofs, settling a question raised by (Unruh, POPL 2019). Leveraging our positive result, we implement EasyPQC, an extension of EasyCrypt for post-quantum security proofs, and use EasyPQC to verify postquantum security of three classic constructions: PRF-based MAC, Full Domain Hash and GPV08 identity-based encryption.

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Section: Assertions and Judgmentmentioning

confidence: 99%

Section: A3 Classical-quantum Couplingmentioning

confidence: 99%

EasyPQC: Verifying Post-Quantum Cryptography

Barbosa

Barthe

Fan³

et al. 2021

Proceedings of the 2021 ACM SIGSAC Conference on Computer and Communications Security

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“…Notably, we may need to reason about entangled quantum states. There are some existing proposals of analogue of probabilistic couplings in the quantum setting (see [Kümmerer and Schwieger 2016;Winter 2016]). In particular, Zhou et al [2019a] addressed this issue by developing a notion of quantum coupling, and validated their definition by showing an analogue of Strassen's theorem [Strassen 1965].…”

Section: Introductionmentioning

confidence: 99%

Relational proofs for quantum programs

Barthe

Hsu

Ying

et al. 2019

Proc. ACM Program. Lang.

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Relational verification of quantum programs has many potential applications in quantum and post-quantum security and other domains. We propose a relational program logic for quantum programs. The interpretation of our logic is based on a quantum analogue of probabilistic couplings. We use our logic to verify non-trivial relational properties of quantum programs, including uniformity for samples generated by the quantum Bernoulli factory, reliability of quantum teleportation against noise (bit and phase flip), security of quantum one-time pad and equivalence of quantum walks. 's theorem states that there exists a B-coupling between two distributions µ and µ ′ over sets X andIf ρ is a coupling for ρ 1 , ρ 2 , then they have the same trace: tr(ρ) = tr(ρ 1 ) = tr(ρ 2 ).The following are examples of quantum couplings. They are quantum generalisations of several typical examples of (discrete) probabilistic couplings (see [Barthe et al. 2019]). From these simple examples, we can see a close and natural correspondence as well as some essential differences between probabilistic coupling and their quantum counterparts. Our first example shows that couplings always exist. E3.1. Let ρ 1 ∈ D(H 1 ) and ρ 2 ∈ D(H 2 ) be density operators. The tensor product ρ ⊗ = ρ 1 ⊗ ρ 2 is a coupling for ρ 1 , ρ 2 .Just like the case for probabilistic couplings, there can be more than one quantum coupling between two operators.The coupling ρ id(B) in Example 3.3 is a witness of the lifting ρ(= sym ) # ρ, defining = sym to be the symmetrisation operator, i.e. (= sym ) ≡ 1 2 (I ⊗ I + S), where S is the SWAP operator defined by S |φ,ψ = |ψ , φ for any |φ , |ψ ∈ H together with linearity. Operator S is independent of the basis, and given any orthonormal basis {|i } of H , S has the following form: S = i j |i j | ⊗ |j i |.(4) The coupling ρ 1 ⊗ ρ 2 in Example 3.1 is a witness of the lifting ρ 1 (H 1 ⊗ H 2 ) # ρ 2 .The two operators = B and = sym in the above example represents two different kind of symmetry between two quantum systems with the same state Hilbert space H . They will be used to describe relational properties of the two quantum programs P 1 , P 2 in Example 1.1. Liftings of equality are especially interesting for verification, since they can be interpreted as relating equivalent quantum systems. The following proposition characterizes these liftings. P3.2. Let ρ 1 , ρ 2 ∈ D ≤ (H ). The following statements are equivalent: 1. ρ 1 = ρ 2 ; 2. there exists an orthonormal basis B s.t. ρ 1 (= B ) # ρ 2 ;

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“…Some aspects of noncommutative joinings also appeared in [60] and [44] related to entropy, and in [33] related to certain ergodic theorems. In [46] results closely related to joinings were presented regarding a coupling method for quantum Markov chains and mixing times.…”

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confidence: 99%

Balance Between Quantum Markov Semigroups

Duvenhage

Snyman

2018

Ann. Henri Poincaré

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The concept of balance between two state preserving quantum Markov semigroups on von Neumann algebras is introduced and studied as an extension of conditions appearing in the theory of quantum detailed balance. This is partly motivated by the theory of joinings. Balance is defined in terms of certain correlated states (couplings), with entangled states as a specific case. Basic properties of balance are derived and the connection to correspondences in the sense of Connes is discussed. Some applications and possible applications, including to non-equilibrium statistical mechanics, are briefly explored. ContentsDate: 2018-3-13.

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Diagonal couplings of quantum Markov chains

Cited by 4 publications

References 25 publications

EasyPQC: Verifying Post-Quantum Cryptography

EasyPQC: Verifying Post-Quantum Cryptography

Relational proofs for quantum programs

Balance Between Quantum Markov Semigroups

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