1,289 research outputs found
Estimating operator norms using covering nets
We present several polynomial- and quasipolynomial-time approximation schemes
for a large class of generalized operator norms. Special cases include the
norm of matrices for , the support function of the set of
separable quantum states, finding the least noisy output of
entanglement-breaking quantum channels, and approximating the injective tensor
norm for a map between two Banach spaces whose factorization norm through
is bounded.
These reproduce and in some cases improve upon the performance of previous
algorithms by Brand\~ao-Christandl-Yard and followup work, which were based on
the Sum-of-Squares hierarchy and whose analysis used techniques from quantum
information such as the monogamy principle of entanglement. Our algorithms, by
contrast, are based on brute force enumeration over carefully chosen covering
nets. These have the advantage of using less memory, having much simpler proofs
and giving new geometric insights into the problem. Net-based algorithms for
similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in
each case with a run-time that is exponential in the rank of some matrix. We
achieve polynomial or quasipolynomial runtimes by using the much smaller nets
that exist in spaces. This principle has been used in learning theory,
where it is known as Maurey's empirical method.Comment: 24 page
Quantum de Finetti Theorems under Local Measurements with Applications
Quantum de Finetti theorems are a useful tool in the study of correlations in
quantum multipartite states. In this paper we prove two new quantum de Finetti
theorems, both showing that under tests formed by local measurements one can
get a much improved error dependence on the dimension of the subsystems. We
also obtain similar results for non-signaling probability distributions. We
give the following applications of the results:
We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the
exponential time hypothesis.
We show that the maximum winning probability of free games can be estimated
in polynomial time by linear programming. We also show that 3-SAT with m
variables can be reduced to obtaining a constant error approximation of the
maximum winning probability under entangled strategies of O(m^{1/2})-player
one-round non-local games, in which the players communicate O(m^{1/2}) bits all
together.
We show that the optimization of certain polynomials over the hypersphere can
be performed in quasipolynomial time in the number of variables n by
considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy
of semidefinite programs. As an application to entanglement theory, we find a
quasipolynomial-time algorithm for deciding multipartite separability.
We consider a result due to Aaronson -- showing that given an unknown n qubit
state one can perform tomography that works well for most observables by
measuring only O(n) independent and identically distributed (i.i.d.) copies of
the state -- and relax the assumption of having i.i.d copies of the state to
merely the ability to select subsystems at random from a quantum multipartite
state.
The proofs of the new quantum de Finetti theorems are based on information
theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor
improvements. v3: added some explanations, mostly about Theorem 1 and
Conjecture 5. STOC version. v4, v5. small improvements and fixe
Shadow Tomography of Quantum States
We introduce the problem of *shadow tomography*: given an unknown
-dimensional quantum mixed state , as well as known two-outcome
measurements , estimate the probability that
accepts , to within additive error , for each of the
measurements. How many copies of are needed to achieve this, with high
probability? Surprisingly, we give a procedure that solves the problem by
measuring only copies. This means, for example, that we can learn the behavior of an
arbitrary -qubit state, on all accepting/rejecting circuits of some fixed
polynomial size, by measuring only 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
Infiltrating Paradigms: Examining Current Stormwater Management Practices within the Township of Keswick and Barriers to Implementing Innovative Water Management Strategies
The increasing demand for drinking water, due to a growing population, is in competition with agriculture, industry, and economic uses. Changes in freshwater can be seen in the dropping water levels of the Great Lakes region yet is not evident in Lake Simcoe, due to the regulation of water levels after 1918. Therefore, to understand the changes in freshwater quantity, stormwater infiltration must be observed. Infiltration of water through plants and soil increases water health, storage, and baseflow to streams; subsequently creating future supplies, ensuring resilience in a changing environment. Water moves through society and nature within a watershed, creating a socio-ecohydrological landscape. Keswick, Ontario and its watershed, the Maskinonge River, have poor stormwater infiltration and low vegetative cover, which effects flooding in urban centres and the health of Lake Simcoe. This requires stormwater management (SWM); and a key factor affecting SWM is vegetative cover. Vegetation has many ecological functions (e.g. holding water, preventing erosion), as well as improving physical and mental health. A way to increase vegetative cover and infiltration is through low-impact development (LID), which creates stability between the built and natural environments. However, there are barriers to LID, and through conversations in specialized interviews I explored the question, what are the pros and cons to alternative stormwater management practices, as well as the barriers to implementing these solutions? According to my research, the pros to LID are improved water filtration, phosphorus reduction, aesthetics, and increased storage of stormwater. The only cons were difficulty infiltrating in places with poor drainage, and not wishing to infiltrate salts and oils into the ground where present. Today, SWM ponds are not working as needed, and on site water conveyance has long been suggested. Keswick has no examples of LID, despite education of its benefits. My research revealed status quo, cost and permitting as the barriers to implementation, however these are symptoms; the main barriers are entrenched paradigms. For example, status quo and cost are perceived as why 'others' are not changing; although everyone was willing to change once something happened (e.g. new guidelines, stormwater credit). Developers have seen cost benefits with LID; and Operations find the cost of the current system already unreasonable. As well, society's perception of natural as 'wild' has created opposition to naturalized solutions. These are perceptions based in fear, and not unwillingness. As for permitting, the dilemma is the strict use of guidelines. The Conservation Authority has an entrenched idea that regulations will be abused, and municipalities fear liability. LID is not one-size-fits-all, so revisions will always be necessary, resulting in challenges and delays enervating the process. Unfortunately, there is much time and money wasted in a back and forth battle of wills concluding with a SWM pond, which the municipality will need to remove later. My recommendation is for communication rather than revision. Allow for innovative designs, using guidelines as guides, and open a dialogue between all invested parties to inspire collaboration. The outcome will be a compromise, born out of a desire to move forward
Efficient Quantum Pseudorandomness
Randomness is both a useful way to model natural systems and a useful tool
for engineered systems, e.g. in computation, communication and control. Fully
random transformations require exponential time for either classical or quantum
systems, but in many case pseudorandom operations can emulate certain
properties of truly random ones. Indeed in the classical realm there is by now
a well-developed theory of such pseudorandom operations. However the
construction of such objects turns out to be much harder in the quantum case.
Here we show that random quantum circuits are a powerful source of quantum
pseudorandomness. This gives the for the first time a polynomialtime
construction of quantum unitary designs, which can replace fully random
operations in most applications, and shows that generic quantum dynamics cannot
be distinguished from truly random processes. We discuss applications of our
result to quantum information science, cryptography and to understanding
self-equilibration of closed quantum dynamics.Comment: 6 pages, 1 figure. Short version of http://arxiv.org/abs/1208.069
Superdense coding of quantum states
We describe a method to non-obliviously communicate a 2l-qubit quantum state
by physically transmitting l+o(l) qubits of communication, and by consuming l
ebits of entanglement and some shared random bits. In the non-oblivious
scenario, the sender has a classical description of the state to be
communicated. Our method can be used to communicate states that are pure or
entangled with the sender's system; l+o(l) and 3l+o(l) shared random bits are
sufficient respectively.Comment: 5 pages, revtex
Entanglement-swapping boxes and their communication properties
We pose the fundamental question of communication properties of primitives
irrespectively of their implementation. To illustrate the idea we introduce the
concept of entanglement-swapping boxes, i.e. we consider any quantum operations
which perform entanglement swapping, not necessarily via simple quantum
teleportation. We ask a question about the properties of such boxes., i.e. what
local operations and how much classical communication are needed to perform
them. We also ask if any box which performs entanglement swapping can be used
to establish classical communication. We show that each box needs at least two
bits of classical communication to perform it. It is also shown that each box
can be used for classical communication and, most importantly, that there exist
boxes which allow to communicate at most one bit. Surprisingly we find basic
irreversibility in the process of entanglement swapping with respect to its
communication properties.Comment: Accepted for publication in Phys. Rev. A as a Rapid Communicatio
Entanglement can completely defeat quantum noise
We describe two quantum channels that individually cannot send any
information, even classical, without some chance of decoding error. But
together a single use of each channel can send quantum information perfectly
reliably. This proves that the zero-error classical capacity exhibits
superactivation, the extreme form of the superadditivity phenomenon in which
entangled inputs allow communication over zero capacity channels. But our
result is stronger still, as it even allows zero-error quantum communication
when the two channels are combined. Thus our result shows a new remarkable way
in which entanglement across two systems can be used to resist noise, in this
case perfectly. We also show a new form of superactivation by entanglement
shared between sender and receiver.Comment: 4 pages, 1 figur
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