Two-message quantum interactive proofs and the quantum separability problem

Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unitary interactions followed by a partial trace, generates a quantum state shared between two parties. One might then wonder, does this quantum circuit produce a state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of quantum computation. We begin by exhibiting a two-message quantum interactive proof system that can decide the answer to a promise version of the question. We then prove that the promise problem is hard for the class of promise problems with "quantum statistical zero knowledge" (QSZK) proof systems by demonstrating a polynomial-time Karp reduction from the QSZK-complete promise problem "quantum state distinguishability" to our quantum separability problem. By exploiting Knill's efficient encoding of a matrix description of a state into a description of a circuit to generate the state, we can show that our promise problem is NP-hard with respect to Cook reductions. Thus, the quantum separability problem (as phrased above) constitutes the first nontrivial promise problem decidable by a two-message quantum interactive proof system while being hard for both NP and QSZK. We also consider a variant of the problem, in which a given polynomial-time mixed-state quantum circuit accepts a quantum state as input, and the question is to decide if there is an input to this circuit which makes its output separable across some bipartite cut. We prove that this problem is a complete promise problem for the class QIP of problems decidable by quantum interactive proof systems. Finally, we show that a two-message quantum interactive proof system can also decide a multipartite generalization of the quantum separability problem.Comment: 34 pages, 6 figures; v2: technical improvements and new result for the multipartite quantum separability problem; v3: minor changes to address referee comments, accepted for presentation at the 2013 IEEE Conference on Computational Complexity; v4: changed problem names; v5: updated references and added a paragraph to the conclusion to connect with prior work on separability testin

Quantum trade-off coding for bosonic communication

The trade-off capacity region of a quantum channel characterizes the optimal net rates at which a sender can communicate classical, quantum, and entangled bits to a receiver by exploiting many independent uses of the channel, along with the help of the same resources. Similarly, one can consider a trade-off capacity region when the noiseless resources are public, private, and secret key bits. In [Phys. Rev. Lett. 108, 140501 (2012)], we identified these trade-off rate regions for the pure-loss bosonic channel and proved that they are optimal provided that a longstanding minimum output entropy conjecture is true. Additionally, we showed that the performance gains of a trade-off coding strategy when compared to a time-sharing strategy can be quite significant. In the present paper, we provide detailed derivations of the results announced there, and we extend the application of these ideas to thermalizing and amplifying bosonic channels. We also derive a "rule of thumb" for trade-off coding, which determines how to allocate photons in a coding strategy if a large mean photon number is available at the channel input. Our results on the amplifying bosonic channel also apply to the "Unruh channel" considered in the context of relativistic quantum information theory.Comment: 20 pages, 7 figures, v2 has a new figure and a proof that the regions are optimal for the lossy bosonic channel if the entropy photon-number inequality is true; v3, submission to Physical Review A (see related work at http://link.aps.org/doi/10.1103/PhysRevLett.108.140501); v4, final version accepted into Physical Review

Leggett-Garg inequalities and the geometry of the cut polytope

The Bell and Leggett-Garg tests offer operational ways to demonstrate that non-classical behavior manifests itself in quantum systems, and experimentalists have implemented these protocols to show that classical worldviews such as local realism and macrorealism are false, respectively. Previous theoretical research has exposed important connections between more general Bell inequalities and polyhedral combinatorics. We show here that general Leggett-Garg inequalities are closely related to the cut polytope of the complete graph, a geometric object well-studied in combinatorics. Building on that connection, we offer a family of Leggett-Garg inequalities that are not trivial combinations of the most basic Leggett-Garg inequalities. We then show that violations of macrorealism can occur in surprising ways, by giving an example of a quantum system that violates the new "pentagon" Leggett-Garg inequality but does not violate any of the basic "triangle" Leggett-Garg inequalities.Comment: 5 pages, 1 figur

Towards efficient decoding of classical-quantum polar codes

Known strategies for sending bits at the capacity rate over a general channel with classical input and quantum output (a cq channel) require the decoder to implement impractically complicated collective measurements. Here, we show that a fully collective strategy is not necessary in order to recover all of the information bits. In fact, when coding for a large number N uses of a cq channel W, N I(W_acc) of the bits can be recovered by a non-collective strategy which amounts to coherent quantum processing of the results of product measurements, where I(W_acc) is the accessible information of the channel W. In order to decode the other N (I(W) - I(W_acc)) bits, where I(W) is the Holevo rate, our conclusion is that the receiver should employ collective measurements. We also present two other results: 1) collective Fuchs-Caves measurements (quantum likelihood ratio measurements) can be used at the receiver to achieve the Holevo rate and 2) we give an explicit form of the Helstrom measurements used in small-size polar codes. The main approach used to demonstrate these results is a quantum extension of Arikan's polar codes.Comment: 21 pages, 2 figures, submission to the 8th Conference on the Theory of Quantum Computation, Communication, and Cryptograph

Quantum interactive proofs and the complexity of separability testing

We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by proving that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.Comment: v2: 43 pages, 5 figures, completely rewritten and in Theory of Computing (ToC) journal forma

Trade-off capacities of the quantum Hadamard channels

Coding theorems in quantum Shannon theory express the ultimate rates at which a sender can transmit information over a noisy quantum channel. More often than not, the known formulas expressing these transmission rates are intractable, requiring an optimization over an infinite number of uses of the channel. Researchers have rarely found quantum channels with a tractable classical or quantum capacity, but when such a finding occurs, it demonstrates a complete understanding of that channel's capabilities for transmitting classical or quantum information. Here, we show that the three-dimensional capacity region for entanglement-assisted transmission of classical and quantum information is tractable for the Hadamard class of channels. Examples of Hadamard channels include generalized dephasing channels, cloning channels, and the Unruh channel. The generalized dephasing channels and the cloning channels are natural processes that occur in quantum systems through the loss of quantum coherence or stimulated emission, respectively. The Unruh channel is a noisy process that occurs in relativistic quantum information theory as a result of the Unruh effect and bears a strong relationship to the cloning channels. We give exact formulas for the entanglement-assisted classical and quantum communication capacity regions of these channels. The coding strategy for each of these examples is superior to a naive time-sharing strategy, and we introduce a measure to determine this improvement.Comment: 27 pages, 6 figures, some slight refinements and submitted to Physical Review

Options for International Financing of Climate Change Mitigation in Developing Countries

This paper provides a model-based analysis of the potential macro-economic impacts of different options for international financing of climate change mitigation in developing countries. The model used is the multi-region and multi-sector climate change version of the WorldScan model. Following the outcome of the UNFCCC conference in Copenhagen, it makes no specific assumptions about the future international climate regime. The analysis shows that the environmental prospects systematically improve in a transition from the Clean Development Mechanism projects towards a global carbon market, while the opposite is foreseen for the economic costs. The more of a carbon market we have when moving from the project-based CDM to sectoral crediting mechanisms and internationally linked cap-and-trade, the more finance the carbon market will channel to developing countries.european union eu annex I non-annex I climate conference in Copenhagen climate change mitigation clean development mechanism emission trading system the US brazil china india own participation of developing countries sectoral crediting mechanisms hayden Veenendaal Zarnic

The Burgeoning “Biorights Movement”: Its Legal Basis, What’s at Stake, and How to Respond

The advent of genetic and genomic technologies has the power to transform the understanding, prevention, and treatment of disease on a scale unprecedented in modern medicine. The promise of the era of precision medicine risks being tempered by the emergence of what is increasingly being referred to as the “biorights movement.” Of particular concern is the growing trend of individuals refusing to contribute their biological material to research studies absent some form of monetary compensation. Recently announced, but yet to be implemented, regulations seek to mitigate some of the potentially harmful and progress-impeding positions advanced by the biorights movement. The proposed changes to the legal and regulatory framework, however, do not sufficiently address the opportunities and challenges of the rapidly evolving patient-consumer landscape as it relates to personal genetic testing. Never before have patients been able to know so much about their genetic profile and the potentially valuable information their DNA contains from both a research and commercial perspective. Bolstered by numerous public policy justifications, this Note argues that legislative action needs to be taken that proscribes the ability of individuals to sell their biological material for research purposes

Adaptations for growing wheat in a drying climate

Declining rainfall in the winter months in southwest Australia could have large impacts on wheat production in the area, particularly in those parts where production is historically limited by water supply.It is expected that the climate in southwest Australia will become drier, particularly in the winter months. These months have historically received the most rainfall in southwest Australia and make an important contribution to the in-season water supply to spring wheat (Triticum aestivum). Published simulation studies of wheat yield in the medium rainfall (325 to 450 mm annual rainfall) parts of the southwest Australian wheatbelt suggested that reductions in winter rainfall after 1975 have not reduced grain yields because soil water supply during these months often exceeds crop demand. However, the effect of recent changes in rainfall distribution on wheat production in more water-limited production areas (≤ 325 mm annual rainfall) is not known. This study aims to investigate the effects of changes in rainfall distribution on wheat yield in marginal parts of the southwest Australian wheatbelt.Two field experiments were conducted at Merredin Research Station, Western Australia (31.50°S, 118.22°E, mean annual rainfall 313 mm) in 2008 and 2009. Water supply to ‘Rainfall Distribution’ treatments were partly controlled with the use of rainout shelters and irrigation. The experiments investigated the effects and interactions of rainfall distribution, row spacing (23 cm and 60 cm), genotype and timing of nitrogen on growth, water use and grain yield of spring wheat. Results from the experiments showed that wheat yields in out of season dominant rainfall varied according to water storage at sowing. Widening row spacing reduced biomass and slowed water use but did not increase grain yield due to increase evaporation in season and residual water left in the soil after maturity. Data from the 2009 experiment was used to calibrate the Agricultural Production Systems Simulator (APSIM) crop model, which in turn was used to investigate the effects of recent and projected climate change on wheat yield in marginal wheat growing areas of southwest Australia. Row spacing, nitrogen rates and timing, and phenology treatments were included in the simulations to test for interactions with changes in climate. The simulation was run for eight locations that span the fringe of the southwest Australian wheatbelt and compared the 35 years preceding 1975 with the 35 years following 1975.In northern parts there were large reductions in June and July rainfall after 1975 and the slight increase in out of season rainfall didn’t result in sufficient water storage before seeding to compensate for it. In central parts, the decrease in growing season rainfall after 1975 was almost equivalent to the increase in out of season rainfall but grain yield declined. In eastern parts there was little change to growing season rainfall and grain yield benefitted by having additional stored soil water at seeding from increased out of season rainfall. In southern parts there was little change in rainfall pre and post 1975 and subsequently little change in grain yield. Two climate change scenarios from the online climate change scenario generator OzClim were applied to historical climatic data to create two plausible future climates (‘optimistic’ and ‘pessimistic’) for the year 2030. Both future climates resulted in reduced grain yields for each location when compared to the 1975 to 2009 time period.The worrying thing for producers was that none of the strategies tested aided wheat yield in the predicted changing climate. It was hypothesised that widening row spacing would slow biomass accumulation and water use by the crop, which would result in more post-anthesis water use and grain yield when the crop was growing largely from stored soil water at seeding. The results showed that widening row spacing did indeed slow water use and provide more water post-anthesis, but wheat yields were consistently lower due to lack of head density, increased evaporation from the soil surface and the inability of the crop to use all the available water before maturity.Findings in this thesis suggest that there is scope to breed wheat genotypes that are better suited to wide rows, and would allow growers to exploit the practical advantages of wide rows while minimizing the disadvantages. Field experiments suggested that early vigour and tillering (and head density) were important to grain yield in wide rows and it is suggested that a genotype suited to wide rows would include early vigour to reduce evaporation and increase competition with weeds, heavy tillering so that grain yield isn’t limited by sink-size and a vigorous root system with a good lateral spread to access all available soil water.In addition to highlighting the sensitivity of wheat production to changes in climate in marginal southwest Australia, and identifying the potential for row spacing and genetic interactions to be exploited, this study has identified gaps in knowledge that will benefit wheat producers if filled. Firstly there are some key unknown factors about the effects of climate change on Australian wheat production such as the effect of the interaction between elevated atmospheric CO2, increased temperatures and water deficit on wheat growth and yield. Also, the likely effect of climate change on the frequency, timing and severity of frosts is unknown. Both of these are fundamental to the successful development of strategies to adapt to climate change, and maintain grain production in a region that is very important to the prosperity of Australia’s rural industries