On entanglement-assisted classical capacity

This paper is essentially a lecture from the author's course on quantum information theory, which is devoted to the result of C. H. Bennett, P. W. Shor, J. A. Smolin and A. V. Thapliyal (quant-ph/0106052) concerning entanglement-assisted classical capacity of a quantum channel. A modified proof of this result is given and relation between entanglement-assisted and unassisted classical capacities is discussed.Comment: 10 pages, LATE

Transition probabilities between quasifree states

We obtain a general formula for the transition probabilities between any state of the algebra of the canonical commutation relations (CCR-algebra) and a squeezed quasifree state. Applications of this formula are made for the case of multimode thermal squeezed states of quantum optics using a general canonical decomposition of the correlation matrix valid for any quasifree state. In the particular case of a one mode CCR-algebra we show that the transition probability between two quasifree squeezed states is a decreasing function of the geodesic distance between the points of the upper half plane representing these states. In the special case of the purification map it is shown that the transition probability between the state of the enlarged system and the product state of real and fictitious subsystems can be a measure for the entanglement.Comment: 13 pages, REVTeX, no figure

Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding

In the simple quantum hypothesis testing problem, upper bound with asymmetric setting is shown by using a quite useful inequality by Audenaert et al, quant-ph/0610027, which was originally invented for symmetric setting. Using this upper bound, we obtain the Hoeffding bound, which are identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. Our upper bound improves the bound by Ogawa-Hayashi, and also provides a simpler proof of the direct part of the quantum Stein's lemma. Further, using this bound, we obtain a better exponential upper bound of the average error probability of classical-quantum channel coding

Information gain versus coupling strength in quantum measurements

We investigate the relationship between the information gain and the interaction strength between the quantum system and the measuring device. A strategy is proposed to calculate the information gain of the measuring device as the coupling strength is a variable. For qubit systems, we prove that the information gain increases monotonically with the coupling strength. It is obtained that the information gain of the projective measurement along the x-direction reduces with the increasing of the measurement strength along the z-direction, and a complementarity of information gain in the measurements along those two directions is presented.Comment: 7 pages, 1 figure

Substituting a qubit for an arbitrarily large number of classical bits

We show that a qubit can be used to substitute for an arbitrarily large number of classical bits. We consider a physical system S interacting locally with a classical field phi(x) as it travels directly from point A to point B. The field has the property that its integrated value is an integer multiple of some constant. The problem is to determine whether the integer is odd or even. This task can be performed perfectly if S is a qubit. On the otherhand, if S is a classical system then we show that it must carry an arbitrarily large amount of classical information. We identify the physical reason for such a huge quantum advantage, and show that it also implies a large difference between the size of quantum and classical memories necessary for some computations. We also present a simple proof that no finite amount of one-way classical communication can perfectly simulate the effect of quantum entanglement.Comment: 8 pages, LaTeX, no figures. v2: added result on entanglement simulation with classical communication; v3: minor correction to main proof, change of title, added referenc

Shadow Tomography of Quantum States

We introduce the problem of *shadow tomography*: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability that $E_{i}$ accepts $\rho$, to within additive error $\varepsilon$, for each of the $M$ measurements. How many copies of $\rho$ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only $\widetilde{O}\left( \varepsilon^{-4}\cdot\log^{4} M\cdot\log D\right)$ copies. This means, for example, that we can learn the behavior of an arbitrary $n$-qubit state, on all accepting/rejecting circuits of some fixed polynomial size, by measuring only $n^{O\left( 1\right)}$ copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Brand\~ao et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.Comment: 29 pages, extended abstract appeared in Proceedings of STOC'2018, revised to give slightly better upper bound (1/eps^4 rather than 1/eps^5) and lower bounds with explicit dependence on the dimension

Fidelity for Multimode Thermal Squeezed States

In the theory of quantum transmission of information the concept of fidelity plays a fundamental role. An important class of channels, which can be experimentally realized in quantum optics, is that of Gaussian quantum channels. In this work we present a general formula for fidelity in the case of two arbitrary Gaussian states. From this formula one can get a previous result (H. Scutaru, J. Phys. A: Mat. Gen {\bf 31}, 3659 (1998)), for the case of a single mode; or, one can apply it to obtain a closed compact expression for multimode thermal states.Comment: 5 pages, RevTex, submitted to Phys. Rev.

Using post-measurement information in state discrimination

We consider a special form of state discrimination in which after the measurement we are given additional information that may help us identify the state. This task plays a central role in the analysis of quantum cryptographic protocols in the noisy-storage model, where the identity of the state corresponds to a certain bit string, and the additional information is typically a choice of encoding that is initially unknown to the cheating party. We first provide simple optimality conditions for measurements for any such problem, and show upper and lower bounds on the success probability. For a certain class of problems, we furthermore provide tight bounds on how useful post-measurement information can be. In particular, we show that for this class finding the optimal measurement for the task of state discrimination with post-measurement information does in fact reduce to solving a different problem of state discrimination without such information. However, we show that for the corresponding classical state discrimination problems with post-measurement information such a reduction is impossible, by relating the success probability to the violation of Bell inequalities. This suggests the usefulness of post-measurement information as another feature that distinguishes the classical from a quantum world.Comment: 10 pages, 4 figures, revtex, v2: published version, minor change

Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing

In the asymptotic setting, the optimal test for hypotheses testing of the maximally entangled state is derived under several locality conditions for measurements. The optimal test is obtained in several cases with the asymptotic framework as well as the finite-sample framework. In addition, the experimental scheme for the optimal test is presented

A Stronger Subadditivity of Entropy

The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[\rho]= - Tr (\rho \ln \rho) of a density matrix \rho_{123} on the product of three Hilbert spaces satisfies S[\rho_{123}] - S[\rho_{23}] \leq S[\rho_{12}]- S[\rho_2]. We strengthen this to S[\rho_{123}] - S[\rho_{12}] \leq \sum_\alpha n^\alpha (S[\rho_{23}^\alpha ] - S[\rho_2^\alpha ]), where the n^\alpha are weights and the \rho_{23}^\alpha are partitions of \rho_{23}. Correspondingly, there is a strengthening of the theorem that the map A -> Tr \exp[L + \ln A] is concave. As applications we prove some monotonicity and convexity properties of the Wehrl entropy and entropy inequalities for quantum gases.Comment: LaTeX2e, 24 page