In this paper we present the following quantum compression protocol: P: Let ρ, σ be quantum states such that S(ρ σ) def = Tr(ρ log ρ − ρ log σ), the relative entropy between ρ and σ, is finite. Alice gets to know the eigen-decomposition of ρ. Bob gets to know the eigen-decomposition of σ. Both Alice and Bob know S(ρ σ) and an error parameter ε. Alice and Bob use shared entanglement and after communication of O((S(ρ σ) + 1)/ε 4 ) bits from Alice to Bob, Bob ends up with a quantum stateρ such that F(ρ,ρ) ≥ 1 − 5ε, where F(·) represents fidelity.This result can be considered as a non-commutative generalization of a result due to Braverman and Rao [2011] where they considered the special case when ρ and σ are classical probability distributions (or commute with each other) and use shared randomness instead of shared entanglement. We use P to obtain an alternate proof of a direct-sum result for entanglement assisted quantum one-way communication complexity for all relations, which was first shown by Jain, Radhakrishnan and Sen [2005,2008]. We also present a variant of protocol P in which Bob has some side information about the state with Alice. We show that in such a case, the amount of communication can be further reduced, based on the side information that Bob has.Our second result provides a quantum analogue of the widely used classical correlatedsampling protocol. For example, Holenstein [2007] used the classical correlated-sampling protocol in his proof of a parallel-repetition theorem for two-player one-round games.Relative entropy is a widely used quantity of central importance in both classical and quantum information theory. In this paper we consider the following task. The notations used below are described in section 2. P: Given a register A, Alice gets to know the eigen-decomposition of a quantum state ρ ∈ D(A). Bob gets to know the eigen-decomposition of a quantum state σ ∈ D(A) such that supp(ρ) ⊂ supp(σ). Both Alice and Bob know S(ρ σ) def = Trρ log ρ − ρ log σ, the relative entropy between ρ and σ and an error parameter ε. Alice and Bob use shared entanglement and after communication of O((S(ρ σ) + 1)/ε 4 ) bits from Alice to Bob, Bob ends up with a quantum stateρ such that F(ρ,ρ) ≥ 1 − ε, where F(·, ·) represents fidelity.This result can be considered as a non-commutative generalization of a result due to Braverman and Rao [BR11] where they considered the special case when ρ and σ are classical probability distributions and the two parties only share public random coins. Their protocol, and slightly modified versions of it, were widely used to show several direct sum and direct product results in communication complexity, for example a direct sum theorem for all relations in the bounded-round public-coin communication model [BR11], direct product theorems for all relations in the public-coin one-way and public-coin bounded-round communication models [Jai13, JPY12, BRWY13]. A direct sum result for a relation f in a model of communication (roughly) states that in order to compute k independent instanc...

We show that a quantum architecture with an error correction procedure limited to geometrically local operations incurs an overhead that grows with the system size, even if arbitrary error-free classical computation is allowed. In particular, we prove that in order to operate a quantum error correcting code in 2D at a logical error rate of δ, a space overhead of Ω( log(1/δ)) is needed for any constant depolarizing noise p > 0.

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We consider the problem of implementing two-party interactive quantum communication over noisy channels, a necessary endeavor if we wish to fully reap quantum advantages for communication. For an arbitrary protocol with n messages, designed for a noiseless qudit channel over a poly (n) size alphabet, our main result is a simulation method that fails with probability less than 2 −Θ(n ) and uses a qudit channel over the same alphabet n(1 + Θ( √ )) times, of which an fraction can be corrupted adversarially. The simulation is thus capacity achieving to leading order, and we conjecture that it is optimal up to a constant factor in the √ term. Furthermore, the simulation is in a model that does not require pre-shared resources such as randomness or entanglement between the communicating parties. Our work improves over the best previously known quantum result where the overhead is a non-explicit large constant [Brassard et al., FOCS'14] for low . Introduction Motivation The main questions.Quantum communication offers the possibility of distributed computation with extraordinary provable savings in communication as compared with classical communication (see, e.g., [RK11] and the references therein). Most often, if not always, the savings are achieved by protocols that assume access to noiseless communication channels. In practice, though, imperfection in channels is inevitable. Is it possible to make the protocols robust to noise while maintaining the advantages offered by quantum communication? If so, what is the cost of making the protocols robust, and how much noise can be tolerated? In this article, we address these questions in the context of quantum communication protocols involving two parties, in the low noise regime. Following convention, we call the two parties Alice and Bob. 1.1.2 Channel coding theory as a special case. Proposition 2.3. The Bell states φ j,k 0≤j,k≤d−1 form an orthonormal basis in A ⊗ B. Proposition 2.4. For any unitary operator U on register A, it holds that Quantum Communication ModelThe definitions for the noiseless and noisy quantum communication models are copied from Ref.[BNT + 19]. We refer the reader there for a more formal definition of the noisy quantum communication model, as well as the relationship of the noiseless quantum communication model to well-studied quantum communication complexity models such a Yao's model and the Cleve-Buhrman model. 6 Noiseless Communication ModelIn the noiseless quantum communication model that we want to simulate, there are five quantum registers: the A register held by Alice, the B register held by Bob, the C register, which is the communication register exchanged back-and-forth between Alice and Bob and initially held by Alice, the E register held by a potential adversary Eve, and finally the R register, a reference system which purifies the state of the ABCE registers throughout the protocol. The initial stateis chosen arbitrarily from the set of possible inputs, and is fixed at the outset of the protocol, but possibly unknown (totally or partial...