In this work, we study the problem of testing properties of the spectrum of a mixed quantum state. Here one is given @math copies of a mixed state @math and the goal is to distinguish whether @math 's spectrum satisfies some property @math or is at least @math -far in @math -distance from satisfying @math . This problem was promoted in the survey of Montanaro and de Wolf under the name of testing unitarily invariant properties of mixed states. It is the natural quantum analogue of the classical problem of testing symmetric properties of probability distributions. Here, the hope is for algorithms with subquadratic copy complexity in the dimension @math . This is because the "empirical Young diagram (EYD) algorithm" can estimate the spectrum of a mixed state up to @math -accuracy using only @math copies. In this work, we show that given a mixed state @math : (i) @math copies are necessary and sufficient to test whether @math is the maximally mixed state, i.e., has spectrum @math ; (ii) @math copies are necessary and sufficient to test with one-sided error whether @math has rank @math , i.e., has at most @math nonzero eigenvalues; (iii) @math copies are necessary and sufficient to distinguish whether @math is maximally mixed on an @math -dimensional or an @math -dimensional subspace; and (iv) The EYD algorithm requires @math copies to estimate the spectrum of @math up to @math -accuracy, nearly matching the known upper bound. In addition, we simplify part of the proof of the upper bound. Our techniques involve the asymptotic representation theory of the symmetric group; in particular Kerov's algebra of polynomial functions on Young diagrams.
The second result comes from the work of @cite_59 . It can be thought of as a quantum analogue of Fact : Setting @math , Theorem gives a linear lower bound of @math for various properties of spectra. This is in contrast with property testing of probability distributions, in which sublinear algorithms are the main goal, with the Birthday Paradox typically precluding sub- @math -sample algorithms.
@cite_59
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In the asymptotic theory of quantum hypothesis testing, the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way. This is well known as the direct part and strong converse of quantum Stein's lemma. Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind, and hence we obtain the second-order asymptotics for quantum hypothesis testing. This actually implies quantum Stein's lemma as a special case. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have potential applications in quantum information theory. Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the optimality part in a unified fashion. This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function @math with respect to a distribution @math is @math , then any two party randomized protocol computing @math must reveal at least @math bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on @math must reveal @math bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is @math , which provides an alternative proof to the recent proof of Viola Viola11 of the @math lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight @math lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a very recent breakthrough work of kerenidis2012lower , this simulation procedure was the main building block for proving that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity. We present two general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Both methods are based on the adversary method of Ambainis. We show that they yield optimal lower bounds for several natural problems, and we challenge the reader to determine the nonadaptive quantum query complexity of the ''1-to-1 versus 2-to-1'' problem and of Hidden Translation. In addition to the results presented at Wollic 2008 in the conference version of this paper, we show that the lower bound given by the second method is always at least as good (and sometimes better) as the lower bound given by the first method. We also compare these two quantum lower bounds to probabilistic lower bounds. Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n . But using ideas from computational learning theory, we show that one can do exponentially better in a statistical setting. In particular, to predict the outcomes of most measurements drawn from an arbitrary probability distribution, one needs only a number of sample measurements that grows linearly with n . This theorem has the conceptual implication that quantum states, despite being exponentially long vectors, are nevertheless ‘reasonable’ in a learning theory sense. The theorem also has two applications to quantum computing: first, a new simulation of quantum one-way communication protocols and second, the use of trusted classical advice to verify untrusted quantum advice.