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 eigendecomposition of ρ. Bob gets to know the eigendecomposition 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)/ε4) bits from Alice to Bob, Bob ends up with a quantum state ̃ρ̃, such that F(ρ, ρ̃) ≥ 1-5ε, where F(·) represents fidelity. This result can be considered as a non-commutative generalization of a result due to Braverman and Rao 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? 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 et al.. We also present a variant of protocol? 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 analog of the widely used classical correlated-sampling protocol. For example, Holenstein used the classical correlated-sampling protocol in his proof of a parallel-repetition theorem for two-player one-round games.
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 eigendecomposition of ρ. Bob gets to know the eigendecomposition 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)/ε4) bits from Alice to Bob, Bob ends up with a quantum state ̃ρ̃, such that F(ρ, ρ̃) ≥ 1-5ε, where F(·) represents fidelity. This result can be considered as a non-commutative generalization of a result due to Braverman and Rao 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? 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 et al.. We also present a variant of protocol? 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 analog of the widely used classical correlated-sampling protocol. For example, Holenstein used the classical correlated-sampling protocol in his proof of a parallel-repetition theorem for two-player one-round games.
In this paper we present a new operational interpretation of relative-entropy between quantum states in the form of the following protocol.
P: Alice gets to know the eigen-decomposition of a quantum state ρ. Bob gets to know the eigen-decomposition of a quantum state σ. Both Alice and Bob know c=S(ρ∣∣σ), the relative entropy between ρ and σ and an error parameter ϵ. Alice and Bob use shared entanglement and after communication of O((c+1)/ϵ4) bits from Alice to Bob, Bob ends up with a quantum state ρ′ 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 [2011] where they considered the special case when ρ and σ are classical probability distributions. We use protocol 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 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 new operational meaning to trace distance between quantum states in the form of a protocol which can be viewed as a quantum analogue of the classical correlated-sampling protocol, which is widely used, for example by Holenstein [2007] in his proof of a parallel-repetition theorem for two-player one-round games. Recently Dinur, Steurer and Vidick [2013] have shown another version of a quantum correlated sampling protocol different from our protocol, and used it in their proof of a parallel-repetition theorem for two-prover one-round entangled projection games.
Communication complexity is the amount of communication needed to compute a function when the function inputs are distributed over multiple parties. In its simplest form, one-way communication complexity, Alice and Bob compute a function f(x,y) , where x is given to Alice and y is given to Bob, and only one message from Alice to Bob is allowed. A fundamental question in quantum information is the relationship between one-way quantum and classical communication complexities, i.e., how much shorter the message can be if Alice is sending a quantum state instead of bit strings? We make some progress towards this question with the following results.Let f:X×Y→Z∪{⊥} be a partial function and μ be a distribution with support contained in f−1(Z) . Denote d=|Z| . Let Rϵ1,μ(f) be the classical one-way communication complexity of f ; Qϵ1,μ(f) be the quantum one-way communication complexity of f and Qϵ1,μ,∗(f) be the entanglement-assisted quantum one-way communication complexity of f , each with distributional error (average error over μ ) at most ϵ . We show:1) If μ is a product distribution, η>0 and 0≤ϵ≤1−1/d , then, R2ϵ−dϵ2/(d−1)+η1,μ(f)≤2Qϵ1,μ,∗(f)+O(loglog(1/η)). 2)If μ is a non-product distribution and Z={0,1} , then ∀ϵ,η>0 such that ϵ/η+η<0.5 , R3η1,μ(f)=O(Qϵ1,μ(f)⋅CS(f)/η3), where CS(f)=maxyminz&am
We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f⊆X×Y×Z. For any 0 k ≥ 1, we show that [EQUATION] where Q1e(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error e and fk denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for the quantum communication complexity of a general relation - direct sum theorems were previously known for one-way quantum protocols for general relations, while direct product theorems were only known for special cases. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlenyi and Yao [24], and under anchored distributions due to Bavarian, Vidick and Yuen [4], as well as message compression for quantum protocols due to Jain, Radhakrishnan and Sen [29]. In particular, we show that a direct product theorem holds for the distributional one-way quantum communication complexity of f under any distribution q on X × Y that is anchored on one side, i.e., there exists a y* such that q(y*) is constant and q(x\y*) = q(x) for all x. This allows us to show a direct product theorem for general distributions, since for any relation f and any distribution p on its inputs, we can define a modified relation [EQUATION] which has an anchored distribution q close to p, such that a protocol that fails with probability at most e for [EQUATION] under q can be used to give a protocol that fails with probability at most e + ζ for f under p. Our techniques also work for entangled non-local games which have input distributions anchored on any one side, i.e., either there exists a y* as previously specified, or there exists an x* such that q(x*) is constant and q(y\x*) = q(y) for all y. In particular, we show that for any game G = (q,X×Y, A×B, V) where q is a distribution on X × Y anchored on any one side with constant anchoring probability, then [EQUATION] where ω* (G) represents the entangled value of the game G. This is a generalization of the result of [4], who proved a parallel repetition theorem for games anchored on both sides, i.e., where both a special x* and a special y* exist, and potentially a simplification of their proof.
Let H be a finite dimensional Hilbert space and ρ, σ ∈ D(H) be quantum states in H \nsuch that S(ρ||σ) is finite. In this thesis, we consider the following communication task involving two parties Alice and Bob. Suppose that Alice is given a classical description of the states ρ and σ. Given an unlimited number of copies of an entangled state whose marginal on Bob’s side is σ, Alice’s goal is to help Bob output a single copy of the state ρ by sending a single message to Bob in a one-way LOCC (short for local operation and classical communication) protocol. We propose a class of one-way LOCC protocols for this task which we refer to as quantum rejection sampling protocols. Inspired by the classical rejection sampling protocol of Harsha, Jain, McAllester, and Radhakrishnan [25] for a similar classical communication task, we introduce the Greedy Quantum Rejection Sampler. We characterize the expected communication cost of the protocol in terms of max-relative entropy of ρ with respect to σ, in the case where the state ρ is a pure state and prove that the Greedy Quantum Rejection Sampler is an optimal quantum rejection sampling protocol in this case. We describe an optimal quantum rejection sampling protocol in terms of a semidefinite program and we find general lower bounds and upper bounds on the expected communication cost of the optimal protocol. We propose an LOCC compression protocol based on the Greedy Quantum Rejection Sampler protocol, for lossless compression of an arbitrary pure state quantum source in the visible compression model and we show an upper bound on the average length of this encoding. The upper bound is always less than or equal to the Shannon entropy of the quantum source and the gap between the two quantities can be arbitrary large. Finally, we introduce a high-entanglement deterministic exact protocol for remote preparation of an arbitrary ensemble of quantum states. Our protocol is based on a quantum rejection sampling protocol which uses a prefix-free encoding for communication of the messages. We establish an upper bound on the expected communication cost of this protocol for the worst case choice of the target state in terms of the max-information in Bob’s output register at the end of the protocol about Alice’s classical input register. Furthermore, this protocol can be used as a non-oblivious universal protocol for exact remote preparation of an arbitrary d-dimensional state at an expected communication cost of at most lg(d) +O (lg(lg(d))).
We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f ⊆ 𝒳×𝒴×𝒵. For any 0 < ε < δ < 1/2 and any k≥1, we show that Q¹_{1-(1-ε)^{Ω(k/log|𝒵|)}}(f^k) = Ω(k⋅Q¹_{δ}(f)), where Q¹_{ε}(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and f^k denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for the quantum communication complexity of a general relation - direct sum theorems were previously known for one-way quantum protocols for general relations, while direct product theorems were only known for special cases. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao [Rahul Jain et al., 2014], and under anchored distributions due to Bavarian, Vidick and Yuen [Bavarian et al., 2017], as well as message compression for quantum protocols due to Jain, Radhakrishnan and Sen [Rahul Jain et al., 2005]. In particular, we show that a direct product theorem holds for the distributional one-way quantum communication complexity of f under any distribution q on 𝒳×𝒴 that is anchored on one side, i.e., there exists a y^* such that q(y^*) is constant and q(x|y^*) = q(x) for all x. This allows us to show a direct product theorem for general distributions, since for any relation f and any distribution p on its inputs, we can define a modified relation f̃ which has an anchored distribution q close to p, such that a protocol that fails with probability at most ε for f̃ under q can be used to give a protocol that fails with probability at most ε + ζ for f under p. Our techniques also work for entangled non-local games which have input distributions anchored on any one side, i.e., either there exists a y^* as previously specified, or there exists an x^* such that q(x^*) is constant and q(y|x^*) = q(y) for all y. In particular, we show that for any game G = (q, 𝒳×𝒴, 𝒜×ℬ, 𝖵) where q is a distribution on 𝒳×𝒴 anchored on any one side with constant anchoring probability, then ω^*(G^k) = (1 - (1-ω^*(G))⁵) ^{Ω(k/(log(|𝒜|⋅|ℬ|)))} where ω^*(G) represents the entangled value of the game G. This is a generalization of the result of [Bavarian et al., 2017], who proved a parallel repetition theorem for games anchored on both sides, i.e., where both a special x^* and a special y^* exist, and potentially a simplification of their proof.
We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f⊆X×Y×Z. For any ε,ζ>0 and any k≥1, we show that Q1−(1−ε)Ω(ζ6k/log∣Z∣)1(fk)=Ω(k(ζ5⋅Qε+12ζ1(f)−loglog(1/ζ))), where Qε1(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and fk denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for quantum communication. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao, and under anchored distributions due to Bavarian, Vidick and Yuen, as well as message-compression for quantum protocols due to Jain, Radhakrishnan and Sen. Our techniques also work for entangled non-local games which have input distributions anchored on any one side. In particular, we show that for any game G=(q,X×Y,A×B,V) where q is a distribution on X×Y anchored on any one side with anchoring probability ζ, then ω∗(Gk)=(1−(1−ω∗(G))5)Ω(log(∣A∣⋅∣B∣)ζ2k) where ω∗(G) represents the entangled value of the game G. This is a generalization of the result of Bavarian, Vidick and Yuen, who proved a parallel repetition theorem for games anchored on both sides, and potentially a simplification of their proof.