Showing papers by "Andreas Winter published in 2013"
Posted Content•
07 Jun 2013
265 citations
TL;DR: In this article, the authors studied the quantum channel version of Shannon's zero-error capacity problem, and proposed a quantum version of Lovasz' famous ϑ function on general operator systems as the norm-completion (or stabilization) of a "naive" generalization of ϑ.
Abstract: We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain subspace of operators (so-called operator systems) as the quantum generalization of the adjacency matrix, in terms of which the zero-error capacity of a quantum channel, as well as the quantum and entanglement-assisted zero-error capacities can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovasz' famous ϑ function on general operator systems, as the norm-completion (or stabilization) of a “naive” generalization of ϑ. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite program, whose dual we write down explicitly, and that it is multiplicative with respect to the tensor product of operator systems (corresponding to the tensor product of channels). We explore various other properties of the new quantity, which reduces to Lovasz' original ϑ in the classical case, give several applications, and propose to study the operator systems associated with channels as “noncommutative graphs,” using the language of Hilbert modules.
155 citations
TL;DR: It is shown that a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former) that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical Capacity.
Abstract: A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a "sandwiched" Renyi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.
83 citations
TL;DR: It is demonstrated here that entropy vectors for stabiliser states satisfy, in addition to the classic inequalities, a type of linear rank inequalities associated with the combinatorial structure of normal subgroups of certain matrix groups.
Abstract: We investigate the universal linear inequalities that hold for the von Neumann entropies in a multi-party system, prepared in a stabiliser state. We demonstrate here that entropy vectors for stabiliser states satisfy, in addition to the classic inequalities, a type of linear rank inequalities associated with the combinatorial structure of normal subgroups of certain matrix groups.
In the 4-party case, there is only one such inequality, the so-called Ingleton inequality. For these systems we show that strong subadditivity, weak monotonicity and Ingleton inequality exactly characterize the entropy cone for stabiliser states.
42 citations
TL;DR: It is shown that it is possible to clone quantum states to arbitrary accuracy in the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity converging to one in the limit as the dimension of the CTC system becomes large--thus resolving an open conjecture.
Abstract: We show that it is possible to clone quantum states to arbitrary accuracy in the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity converging to one in the limit as the dimension of the CTC system becomes large--thus resolving an open conjecture [Brun et al, Phys Rev Lett 102, 210402 (2009)] This result follows from a D-CTC-assisted scheme for producing perfect clones of a quantum state prepared in a known eigenbasis, and the fact that one can reconstruct an approximation of a quantum state from empirical estimates of the probabilities of an informationally complete measurement Our results imply more generally that every continuous, but otherwise arbitrarily nonlinear map from states to states, can be implemented to arbitrary accuracy with D-CTCs Furthermore, our results show that Deutsch's model for closed timelike curves is in fact a classical model, in the sense that two arbitrary, distinct density operators are perfectly distinguishable (in the limit of a large closed timelike curve system); hence, in this model quantum mechanics becomes a classical theory in which each density operator is a distinct point in a classical phase space
41 citations
TL;DR: The result here is that the regularized entanglement of formation characterizes the quantum rate-distortion function, extending earlier work of Devetak and Berger and combining this bound with theEntanglement-assisted bound from their prior work to obtain the best known bounds on the quantum Rate-Distortion function for an isotropic qubit source.
Abstract: We extend quantum rate-distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression. The first setting we consider is that of quantum rate-distortion coding with the help of a classical side channel. Our result here is that the regularized entanglement of formation characterizes the quantum rate-distortion function, extending earlier work of Devetak and Berger. We also combine this bound with the entanglement-assisted bound from our prior work to obtain the best known bounds on the quantum rate-distortion function for an isotropic qubit source. The second setting we consider is that of quantum rate-distortion coding with quantum side information (QSI) available to the receiver. In order to prove results in this setting, we first state and prove a quantum reverse Shannon theorem with QSI (for tensor-power states), which extends the known tensor-power quantum reverse Shannon theorem. The achievability part of this theorem relies on the quantum state redistribution protocol, while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver's QSI. This quantum reverse Shannon theorem with QSI naturally leads to quantum rate-distortion theorems with QSI, with or without entanglement assistance.
41 citations
TL;DR: In this paper, the universal inequalities relating the -Rnyi entropies of the marginals of a multi-partite quantum state were investigated, in analogy to the same question for the Shannon and von Neumann Entropies.
Abstract: We investigate the universal inequalities relating the -Rnyi entropies of the marginals of a multi-partite quantum state. This is in analogy to the same question for the Shannon and von Neumann ent...
38 citations
Posted Content•
07 Jun 2013
TL;DR: This paper presents a probabilistic simulation of the response of the immune system to quantum entanglement and shows clear patterns in response to the proton-proton collision.
Abstract: Mark M. Wilde, Andreas Winter, 3, 4 and Dong Yang 5 Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA ICREA – Institucio Catalana de Recerca i Estudis Avancats, Pg. Lluis Companys 23, ES-08010 Barcelona, Spain Fisica Teorica: Informacio i Fenomens Quantics, Universitat Autonoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom Laboratory for Quantum Information, China Jiliang University, Hangzhou, Zhejiang 310018, China
32 citations
TL;DR: It is shown that the entanglement-assisted independence number is bounded by the Schrijver number: α*(G) ≤ ϑ-(G).
Abstract: We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs $G$ and $H$. Such vectors exist if and only if $\vartheta(\overline{G}) \le \vartheta(\overline{H})$ where $\vartheta$ represents the Lovasz number. We also obtain similar inequalities for the related Schrijver $\vartheta^-$ and Szegedy $\vartheta^+$ numbers.
These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: $\alpha^*(G) \le \vartheta^-(G)$. Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovasz number. Beigi introduced a quantity $\beta$ as an upper bound on $\alpha^*$ and posed the question of whether $\beta(G) = \lfloor \vartheta(G) \rfloor$. We answer this in the affirmative and show that a related quantity is equal to $\lceil \vartheta(G) \rceil$. We show that a quantity $\chi_{\textrm{vect}}(G)$ recently introduced in the context of Tsirelson's conjecture is equal to $\lceil \vartheta^+(\overline{G}) \rceil$.
In an appendix we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.
32 citations
TL;DR: In this paper, it was shown that the distinguishability norm associated to a tensor product of sufficiently symmetric measurements is essentially equivalent to a multi-partite generalisation of the non-commutative -norm (aka Hilbert-Schmidt norm), in comparing the two, the constants of domination depend only on the number of parties but not on the Hilbert spaces dimensions.
Abstract: We analyze the distinguishability norm on the states of a multi-partite system, defined by local measurements. Concretely, we show that the norm associated to a tensor product of sufficiently symmetric measurements is essentially equivalent to a multi-partite generalisation of the non-commutative \({\ell_2}\) -norm (aka Hilbert-Schmidt norm): in comparing the two, the constants of domination depend only on the number of parties but not on the Hilbert spaces dimensions.
30 citations
TL;DR: A single-letter formula is derived for the minimum rate of classical communication needed for quantum-to-classical rate distortion coding in this setting, in which a sender Alice has many copies of a quantum information source.
Abstract: We establish a theory of quantum-to-classical rate distortion coding. In this setting, a sender Alice has many copies of a quantum information source. Her goal is to transmit a classical description of the source, obtained by performing a measurement on it, to a receiver Bob, up to some specified level of distortion. We derive a single-letter formula for the minimum rate of classical communication needed for this task. We also evaluate this rate in the case in which Bob has some quantum side information about the source. Our results imply that, in general, Alice's best strategy is a non-classical one, in which she performs a collective measurement on successive outputs of the source.
01 Nov 2013
TL;DR: In this paper, the authors investigated the universal linear inequalities that hold for the von Neumann entropies in a multi-party system, prepared in a stabiliser state, and showed that strong subadditivity, weak monotonicity and Ingleton inequality exactly characterize the entropy cone for stabiliser states.
Abstract: We investigate the universal linear inequalities that hold for the von Neumann entropies in a multi-party system, prepared in a stabiliser state. We demonstrate here that entropy vectors for stabiliser states satisfy, in addition to the classic inequalities, a type of linear rank inequalities associated with the combinatorial structure of normal subgroups of certain matrix groups. In the 4-party case, there is only one such inequality, the so-called Ingleton inequality. For these systems we show that strong subadditivity, weak monotonicity and Ingleton inequality exactly characterize the entropy cone for stabiliser states.
TL;DR: It is proved that there exists a trade-off between communication rate and error probability if one imposes only a mean photon number constraint on the channel inputs, and that a small modification of the well-known coherent-state coding scheme meets this more demanding constraint.
Abstract: This paper strengthens the interpretation and understanding of the classical capacity of the pure-loss bosonic channel, first established in [Giovannetti et al., Physical Review Letters 92, 027902 (2004), arXiv:quant-ph/0308012]. In particular, we first prove that there exists a trade-off between communication rate and error probability if one imposes only a mean-photon number constraint on the channel inputs. That is, if we demand that the mean number of photons at the channel input cannot be any larger than some positive number N_S, then it is possible to respect this constraint with a code that operates at a rate g(\eta N_S / (1-p)) where p is the code's error probability, \eta\ is the channel transmissivity, and g(x) is the entropy of a bosonic thermal state with mean photon number x. We then prove that a strong converse theorem holds for the classical capacity of this channel (that such a rate-error trade-off cannot occur) if one instead demands for a maximum photon number constraint, in such a way that mostly all of the "shadow" of the average density operator for a given code is required to be on a subspace with photon number no larger than n N_S, so that the shadow outside this subspace vanishes as the number n of channel uses becomes large. Finally, we prove that a small modification of the well-known coherent-state coding scheme meets this more demanding constraint.
TL;DR: In this article, the authors show that all degradable channels obey what they call a "pretty strong" converse: when the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 0.707, asymptotically.
Abstract: We exhibit a possible road towards a strong converse for the quantum capacity of degradable channels. In particular, we show that all degradable channels obey what we call a "pretty strong" converse: When the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 0.707, asymptotically. A similar result can be shown for the private (classical) capacity. Furthermore, we can show that if the strong converse holds for symmetric channels (which have quantum capacity zero), then degradable channels obey the strong converse: The above-mentioned asymptotic jump of the fidelity at the quantum capacity is then from 1 down to 0.
01 Jan 2013
TL;DR: In this article, the authors review the development of the quantum version of Ahlswede and Dueck's theory of identification via channels and present a summary overview of conceptual points and open problems.
Abstract: We review the development of the quantum version of Ahlswede and Dueck's theory of identification via channels. As is often the case in quantum probability, there is not just one but several quantizations: we know at least two different concepts of identification of classical information via quantum channels, and three different identification capacities for quantum information.
In the present summary overview we concentrate on conceptual points and open problems, referring the reader to the small set of original articles for details.
07 Jul 2013
TL;DR: It is shown that all degradable channels obey what is called a “pretty strong” converse: when the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 1/√2, asymptotically.
Abstract: We exhibit a possible road towards a strong converse for the quantum capacity of degradable channels. In particular, we show that all degradable channels obey what we call a “pretty strong” converse: When the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 1/√2, asymptotically. A similar result can be shown for the private (classical) capacity. Furthermore, we can show that if the strong converse holds for symmetric channels (which have quantum capacity zero), then degradable channels obey the strong converse: The above-mentioned asymptotic jump of the fidelity at the quantum capacity is then from 1 down to 0.