It is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p-norms for any p > 1. In this work, we show that a weaker variant of multiplicativity nevertheless holds for these channels. For any constant p > 1, given a random quantum channel ( N ) (i.e. a channel whose Stinespring representation corresponds to a random subspace S), we show that with high probability the maximum output p-norm of ( N ^ n ) decays exponentially with n. The proof is based on relaxing the maximum output ∞-norm of ( N ) to the operator norm of the partial transpose of the projector onto S, then calculating upper bounds on this quantity using ideas from random matrix theory.
Some very recent work by Collins, Fukuda and Nechita @cite_6 also uses related techniques to those which we use here to prove Theorem (e.g. calculations with Weingarten functions @cite_19 , see Section below). The goal of @cite_6 was to find the state which, when input to a tensor product of two random quantum channels (either the same channel, or a channel and its conjugate), achieves minimal output entropy. A sequence of recent papers by Collins and Nechita @cite_4 @cite_3 @cite_11 @cite_8 carries out a number of interesting quantum information-theoretic calculations using Weingarten functions. This method seems to be a powerful tool which may be expected to find many other applications in quantum information.
@cite_4 @cite_8 @cite_3 @cite_6 @cite_19 @cite_11
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In this paper, we present applications of the calculus developed in collins-nechita-1 , and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to gaussianization methods. Our main application is an in-depth study of the random matrix model introduced by Hayden and Winter and used recently by Brandao, Horodecki, Fukuda and King to refine the Hastings counterexample to the additivity conjecture in Quantum Information Theory. This model is exotic from the point of view of random matrix theory, as its eigenvalues obey to two different scalings simultaneously. We study its asymptotic behavior and obtain an asymptotic expansion for its von Neumann entropy. We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs’ eigenvalues between a quantum channel and its complementary version in the case of a non-pure input. For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p > 1, the minimum output Renyi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p = 1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2. The goal of this note is to show that the analysis of the minimum output p-Renyi entropy of a typical quantum channel essentially amounts to applying Milman’s version of Dvoretzky’s theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden–Winter, disproving the additivity conjecture for the minimal output p-Renyi entropy (for p>1). Standard security proofs of quantum-key-distribution (QKD) protocols often rely on symmetry arguments. In this paper, we prove the security of a three-state protocol that does not possess rotational symmetry. The three-state QKD protocol we consider involves three qubit states, where the first two states @math and @math can contribute to key generation, and the third state @math is for channel estimation. This protocol has been proposed and implemented experimentally in some frequency-based QKD systems where the three states can be prepared easily. Thus, by founding on the security of this three-state protocol, we prove that these QKD schemes are, in fact, unconditionally secure against any attacks allowed by quantum mechanics. The main task in our proof is to upper bound the phase error rate of the qubits given the bit error rates observed. Unconditional security can then be proved not only for the ideal case of a single-photon source and perfect detectors, but also for the realistic case of a phase-randomized weak coherent light source and imperfect threshold detectors. Our result in the phase error rate upper bound is independent of the loss in the channel. Also, we compare the three-state protocol with the Bennett-Brassard 1984 (BB84) protocol. For the single-photon source case, our result proves that the BB84 protocol strictly tolerates a higher quantum bit error rate than the three-state protocol, while for the coherent-source case, the BB84 protocol achieves a higher key generation rate and secure distance than the three-state protocol when a decoy-state method is used. We show that Trevisan's extractor and its variants [22,19] are secure against bounded quantum storage adversaries. One instantiation gives the first such extractor to achieve an output length Θ(K-b), where K is the source's entropy and b the adversary's storage, together with a poly-logarithmic seed length. Another instantiation achieves a logarithmic key length, with a slightly smaller output length Θ((K-b) Kγ) for any γ>0. In contrast, the previous best construction [21] could only extract (K b)1 15 bits. Some of our constructions have the additional advantage that every bit of the output is a function of only a polylogarithmic number of bits from the source, which is crucial for some cryptographic applications. Our argument is based on bounds for a generalization of quantum random access codes, which we call quantum functional access codes. This is crucial as it lets us avoid the local list-decoding algorithm central to the approach in [21], which was the source of the multiplicative overhead. We present a technique for proving the security of quantum-key-distribution (QKD) protocols. It is based on direct information-theoretic arguments and thus also applies if no equivalent entanglement purification scheme can be found. Using this technique, we investigate a general class of QKD protocols with one-way classical post-processing. We show that, in order to analyze the full security of these protocols, it suffices to consider collective attacks. Indeed, we give new lower and upper bounds on the secret-key rate which only involve entropies of two-qubit density operators and which are thus easy to compute. As an illustration of our results, we analyze the Bennett-Brassard 1984, the six-state, and the Bennett 1992 protocols with one-way error correction and privacy amplification. Surprisingly, the performance of these protocols is increased if one of the parties adds noise to the measurement data before the error correction. In particular, this additional noise makes the protocols more robust against noise in the quantum channel. In this paper we obtain new bounds for the minimum output entropies of random quantum channels. These bounds rely on random matrix techniques arising from free probability theory. We then revisit the counterexamples developed by Hayden and Winter to get violations of the additivity equalities for minimum output Renyi entropies. We show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the @math -Renyi entropy. For some sequences of random quantum channels, we compute almost surely the limit of their Schatten @math norms. The highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1−exp[−nE(R)+o(n)] for some function E(R) on noisy quantum channels that are subject to not necessarily independent errors. The E(R) is positive below some threshold R0, which implies R0 is a lower bound on the quantum capacity. This work is an extension of the author’s previous works [M. Hamada, Phys. Rev. A, 65, 052305 (2002); quant-ph 0109114, LANL (2001); M. Hamada, quant-ph 0112103, LANL (2001)], which presented the bound for channels subject to independent errors, or channels modeled as tensor products of copies of a completely positive linear map. The relation of the channel class treated in this article to those in the previous works is similar to that of Markov chains to sequences of independent identically distributed random variables. In the near future, there will likely be special-purpose quantum computers with 40--50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by the Quantum AI group at Google. We show that there's a natural average-case hardness assumption, which has nothing to do with sampling, yet implies that no polynomial-time classical algorithm can pass a statistical test that the quantum sampling procedure's outputs do pass. Compared to previous work -- for example, on BosonSampling and IQP -- the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled. Second, in an attempt to refute our hardness assumption, we give a new algorithm, inspired by Savitch's Theorem, for simulating a general quantum circuit with n qubits and depth d in polynomial space and dO(n) time. We then discuss why this and other known algorithms fail to refute our assumption. Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem -- of the form "if approximate quantum sampling is classically easy, then the polynomial hierarchy collapses"-- must be non-relativizing. This sharply contrasts with the situation for exact sampling. Fourth, refuting a conjecture by Aaronson and Ambainis, we show that there is a sampling task, namely Fourier Sampling, with a 1 versus linear separation between its quantum and classical query complexities. Fifth, in search of a "happy medium" between black-box and non-black-box arguments, we study quantum supremacy relative to oracles in P poly. Previous work implies that, if one-way functions exist, then quantum supremacy is possible relative to such oracles. We show, conversely, that some computational assumption is needed: if SampBPP = SampBQP and NP ⊆ BPP, then quantum supremacy is impossible relative to oracles with small circuits. The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R∘N)(ρ), where R is a recovery map with the property that σ=(R∘N)(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction. Consider a two-party quantum communication protocol for computing some function f : 0, 1 sup n spl times 0, 1 sup n spl rarr Z. We show that the first message of P can be compressed to 0(k) classical bits using prior entanglement if it carries at most k bits of information about the sender's input. This implies a general direct sum result for one-round and simultaneous quantum protocols. It also implies a new round elimination lemma in quantum communication, which allows us to extend recent classical lower bounds on the cell probe complexity of some data structure problems, e.g. approximate nearest neighbor searching on the Hamming cube 0, 1 sup n , to the quantum setting. We then show an optimal tradeoff between the privacy losses of Alice and Bob in computing f in terms of the one-round quantum communication complexity of f with prior entanglement. This tradeoff is independent of the number of rounds of communication. The above message compression and privacy tradeoff results use a lot of qubits of prior entanglement, leading one to wonder how much prior entanglement is really required by a quantum protocol. We show that Newman's [1991] technique of reducing the number of public coins in a classical protocol cannot be lifted to the quantum setting. We do this by defining a general notion of black-box reduction of prior entanglement that subsumes Newman's technique. Intuitively, a black-box reduction does not change the unitary transforms of Alice and Bob; it only decreases the amount of entanglement of the prior entangled state. We prove that such a black-box reduction is impossible for quantum protocols by exhibiting a particular one-round quantum protocol for the equality function where the black-box technique fails to reduce the amount of prior entanglement by more than a constant factor. Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of “very strong” and “strong” interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case. Here, we introduce and study the quantum interference channel, a natural generalization of the interference channel to the setting of quantum information theory. We restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results (though generalizations of our results to channels with quantum inputs are straightforward). We are able to determine the exact classical capacity of this channel in the settings of “very strong” and “strong” interference, by exploiting Winter's successive decoding strategy and a novel two-sender quantum simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi strategy is achievable with Holevo information rates, up to a conjecture regarding the existence of a three-sender quantum simultaneous decoder. This conjecture holds for a special class of quantum multiple-access channels with average output states that commute, and we discuss some other variations of the conjecture that hold. Finally, we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates. Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that BQP qpoly is contained in PP poly, where BQP qpoly is the class of problems solvable in quantum polynomial time, given a polynomial-size "quantum advice state" that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Underlying our complexity result is a general new relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which NP is not contained in BQP qpoly. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has other applications; for example, it can be used to fix a flawed result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on quantum one-way communication complexity. Using this method, we obtain optimal quantum lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols. We construct a practically implementable classical processing for the Bennett-Brassard 1984 (BB84) protocol and the six-state protocol that fully utilizes the accurate channel estimation method, which is also known as the quantum tomography. Our proposed processing yields at least as high a key rate as the standard processing by Shor and Preskill. We show two examples of quantum channels over which the key rate of our proposed processing is strictly higher than the standard processing. In the second example, the BB84 protocol with our proposed processing yields a positive key rate even though the so-called error rate is higher than the 25 limit. In this paper, we consider the problem of multifunctional compression with side information. The problem is how we can compress a source X so that the receiver is able to compute some deterministic functions f1(X, Y1), ..., fm(X, Ym), where Yi, 1 ≤ i ≤ m, are available at the receiver as side information. In [1], Wyner and Ziv considered this problem for the special case of m = 1 and f1(X, Y1) = X and derived a rate-distortion function. Yamamoto extended this result in [2] to the case of having one general function f1(X, Y1). Both of these results were in terms of an auxiliary random variable. For the case of zero distortion, in [3], Orlitsky and Roche gave an interpretation of this variable in terms of properties of the characteristic graph which led to a particular coding scheme. This result was extended in [4] by providing an achievable scheme based on colorings of the characteristic graph. In a recent work, reference [5] has considered this problem for a general tree network where intermediate nodes are allowed to perform some computations. These previous works only considered the case where the receiver only wants to compute one function (m=1). Here, we want to consider the case in which the receiver wants to compute several functions with different side information random variables and zero distortion. Our results do not depend on the fact that all functions are desired in one receiver and one can apply them to the case of having several receivers with different desired functions (i.e., functions are separable). We define a new concept named the multi-functional graph entropy which is an extension of the graph entropy defined by Korner in [6]. We show that the minimum achievable rate for this problem is equal to the conditional multi-functional graph entropy of random variable X given side informations. We also propose a coding scheme based on graph colorings to achieve this rate. We present a simple proof of the strong converse for identification via discrete memoryless quantum channels, based on a novel covering lemma. The new method is a generalization to quantum communication channels of Ahlswede's (1979, 1992) approach to classical channels. It involves a development of explicit large deviation estimates to the case of random variables taking values in self-adjoint operators on a Hilbert space. This theory is presented separately in an appendix, and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error @math in trace distance required @math copies for a @math -dimensional density matrix of rank @math . Here, we give a theoretical measurement scheme (POVM) that requires @math copies to estimate @math to error @math in infidelity, and a matching lower bound up to logarithmic factors. This implies @math copies suffice to achieve error @math in trace distance. We also prove that for independent (product) measurements, @math copies are necessary in order to achieve error @math in infidelity. For fixed @math , our measurement can be implemented on a quantum computer in time polynomial in @math . In this paper we consider a three-state variant of the BB84 quantum key distribution (QKD) protocol. We derive a new lower-bound on the key rate of this protocol in the asymptotic scenario and use mismatched measurement outcomes to improve the channel estimation. Our new key rate bound remains positive up to an error rate of @math , exactly that achieved by the four-state BB84 protocol. We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a, b distributed according to some pre-specified joint distribution p(a, b|x, y). Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa. By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication.