5 research outputs found
Axiomatic and operational connections between the -norm of coherence and negativity
Quantum coherence plays a central role in various research areas. The
-norm of coherence is one of the most important coherence measures that
are easily computable, but it is not easy to find a simple interpretation. We
show that the -norm of coherence is uniquely characterized by a few simple
axioms, which demonstrates in a precise sense that it is the analog of
negativity in entanglement theory and sum negativity in the resource theory of
magic-state quantum computation. We also provide an operational interpretation
of the -norm of coherence as the maximum entanglement, measured by the
negativity, produced by incoherent operations acting on the system and an
incoherent ancilla. To achieve this goal, we clarify the relation between the
-norm of coherence and negativity for all bipartite states, which leads to
an interesting generalization of maximally correlated states. Surprisingly, all
entangled states thus obtained are distillable. Moreover, their entanglement
cost and distillable entanglement can be computed explicitly for a qubit-qudit
system.Comment: 5+7 pages, 1 figure, published in PR
Coherifying quantum channels
Is it always possible to explain random stochastic transitions between states
of a finite-dimensional system as arising from the deterministic quantum
evolution of the system? If not, then what is the minimal amount of randomness
required by quantum theory to explain a given stochastic process? Here, we
address this problem by studying possible coherifications of a quantum channel
, i.e., we look for channels that induce the same
classical transitions , but are "more coherent". To quantify the coherence
of a channel we measure the coherence of the corresponding
Jamio{\l}kowski state . We show that the classical transition matrix
can be coherified to reversible unitary dynamics if and only if is
unistochastic. Otherwise the Jamio{\l}kowski state of
the optimally coherified channel is mixed, and the dynamics must necessarily be
irreversible. To assess the extent to which an optimal process
is indeterministic we find explicit bounds on the entropy
and purity of , and relate the latter to the unitarity of
. We also find optimal coherifications for several classes
of channels, including all one-qubit channels. Finally, we provide a
non-optimal coherification procedure that works for an arbitrary channel
and reduces its rank (the minimal number of required Kraus operators) from
to .Comment: 20 pages, 8 figures. Published versio
Using and reusing coherence to realize quantum processes
Coherent superposition is a key feature of quantum mechanics that underlies
the advantage of quantum technologies over their classical counterparts.
Recently, coherence has been recast as a resource theory in an attempt to
identify and quantify it in an operationally well-defined manner. Here we study
how the coherence present in a state can be used to implement a quantum channel
via incoherent operations and, in turn, to assess its degree of coherence. We
introduce the robustness of coherence of a quantum channel---which reduces to
the homonymous measure for states when computed on constant-output
channels---and prove that: i) it quantifies the minimal rank of a maximally
coherent state required to implement the channel; ii) its logarithm quantifies
the amortized cost of implementing the channel provided some coherence is
recovered at the output; iii) its logarithm also quantifies the zero-error
asymptotic cost of implementation of many independent copies of a channel. We
also consider the generalized problem of imperfect implementation with
arbitrary resource states. Using the robustness of coherence, we find that in
general a quantum channel can be implemented without employing a maximally
coherent resource state. In fact, we prove that \textit{every} pure coherent
state in dimension larger than , however weakly so, turns out to be a
valuable resource to implement \textit{some} coherent unitary channel. We
illustrate our findings for the case of single-qubit unitary channels.Comment: 8 pages (main text) + 9 pages (supplementary material). Comments
welcome. v2: minor edits to the introduction. v3: version accepted for
publication in Quantu
Quantifying the magic of quantum channels
To achieve universal quantum computation via general fault-tolerant schemes,
stabilizer operations must be supplemented with other non-stabilizer quantum
resources. Motivated by this necessity, we develop a resource theory for magic
quantum channels to characterize and quantify the quantum "magic" or
non-stabilizerness of noisy quantum circuits. For qudit quantum computing with
odd dimension , it is known that quantum states with non-negative Wigner
function can be efficiently simulated classically. First, inspired by this
observation, we introduce a resource theory based on completely
positive-Wigner-preserving quantum operations as free operations, and we show
that they can be efficiently simulated via a classical algorithm. Second, we
introduce two efficiently computable magic measures for quantum channels,
called the mana and thauma of a quantum channel. As applications, we show that
these measures not only provide fundamental limits on the distillable magic of
quantum channels, but they also lead to lower bounds for the task of
synthesizing non-Clifford gates. Third, we propose a classical algorithm for
simulating noisy quantum circuits, whose sample complexity can be quantified by
the mana of a quantum channel. We further show that this algorithm can
outperform another approach for simulating noisy quantum circuits, based on
channel robustness. Finally, we explore the threshold of non-stabilizerness for
basic quantum circuits under depolarizing noise.Comment: 44 pages, 7 figures; v2 close to published versio