7,364 research outputs found
Short proofs of the Quantum Substate Theorem
The Quantum Substate Theorem due to Jain, Radhakrishnan, and Sen (2002) gives
us a powerful operational interpretation of relative entropy, in fact, of the
observational divergence of two quantum states, a quantity that is related to
their relative entropy. Informally, the theorem states that if the
observational divergence between two quantum states rho, sigma is small, then
there is a quantum state rho' close to rho in trace distance, such that rho'
when scaled down by a small factor becomes a substate of sigma. We present new
proofs of this theorem. The resulting statement is optimal up to a constant
factor in its dependence on observational divergence. In addition, the proofs
are both conceptually simpler and significantly shorter than the earlier proof.Comment: 11 pages. Rewritten; included new references; presented the results
in terms of smooth relative min-entropy; stronger results; included converse
and proof using SDP dualit
Resource theory of asymmetric distinguishability
This paper systematically develops the resource theory of asymmetric
distinguishability, as initiated roughly a decade ago [K. Matsumoto,
arXiv:1010.1030 (2010)]. The key constituents of this resource theory are
quantum boxes, consisting of a pair of quantum states, which can be manipulated
for free by means of an arbitrary quantum channel. We introduce bits of
asymmetric distinguishability as the basic currency in this resource theory,
and we prove that it is a reversible resource theory in the asymptotic limit,
with the quantum relative entropy being the fundamental rate of resource
interconversion. The distillable distinguishability is the optimal rate at
which a quantum box consisting of independent and identically distributed
(i.i.d.) states can be converted to bits of asymmetric distinguishability, and
the distinguishability cost is the optimal rate for the reverse transformation.
Both of these quantities are equal to the quantum relative entropy. The exact
one-shot distillable distinguishability is equal to the min-relative entropy,
and the exact one-shot distinguishability cost is equal to the max-relative
entropy. Generalizing these results, the approximate one-shot distillable
distinguishability is equal to the smooth min-relative entropy, and the
approximate one-shot distinguishability cost is equal to the smooth
max-relative entropy. As a notable application of the former results, we prove
that the optimal rate of asymptotic conversion from a pair of i.i.d. quantum
states to another pair of i.i.d. quantum states is fully characterized by the
ratio of their quantum relative entropies.Comment: v3: 28 page
The operational meaning of min- and max-entropy
We show that the conditional min-entropy Hmin(A|B) of a bipartite state
rho_AB is directly related to the maximum achievable overlap with a maximally
entangled state if only local actions on the B-part of rho_AB are allowed. In
the special case where A is classical, this overlap corresponds to the
probability of guessing A given B. In a similar vein, we connect the
conditional max-entropy Hmax(A|B) to the maximum fidelity of rho_AB with a
product state that is completely mixed on A. In the case where A is classical,
this corresponds to the security of A when used as a secret key in the presence
of an adversary holding B. Because min- and max-entropies are known to
characterize information-processing tasks such as randomness extraction and
state merging, our results establish a direct connection between these tasks
and basic operational problems. For example, they imply that the (logarithm of
the) probability of guessing A given B is a lower bound on the number of
uniform secret bits that can be extracted from A relative to an adversary
holding B.Comment: 12 pages, v2: no change in content, some typos corrected (including
the definition of fidelity in footnote 8), now closer to the published
versio
Entropy of a quantum channel
The von Neumann entropy of a quantum state is a central concept in physics
and information theory, having a number of compelling physical interpretations.
There is a certain perspective that the most fundamental notion in quantum
mechanics is that of a quantum channel, as quantum states, unitary evolutions,
measurements, and discarding of quantum systems can each be regarded as certain
kinds of quantum channels. Thus, an important goal is to define a consistent
and meaningful notion of the entropy of a quantum channel. Motivated by the
fact that the entropy of a state can be formulated as the difference of
the number of physical qubits and the "relative entropy distance" between
and the maximally mixed state, here we define the entropy of a channel
as the difference of the number of physical qubits of the channel
output with the "relative entropy distance" between and the
completely depolarizing channel. We prove that this definition satisfies all of
the axioms, recently put forward in [Gour, IEEE Trans. Inf. Theory 65, 5880
(2019)], required for a channel entropy function. The task of quantum channel
merging, in which the goal is for the receiver to merge his share of the
channel with the environment's share, gives a compelling operational
interpretation of the entropy of a channel. We define Renyi and min-entropies
of a channel and prove that they satisfy the axioms required for a channel
entropy function. Among other results, we also prove that a smoothed version of
the min-entropy of a channel satisfies the asymptotic equipartition property.Comment: v2: 29 pages, 1 figur
Toward physical realizations of thermodynamic resource theories
Conventional statistical mechanics describes large systems and averages over
many particles or over many trials. But work, heat, and entropy impact the
small scales that experimentalists can increasingly control, e.g., in
single-molecule experiments. The statistical mechanics of small scales has been
quantified with two toolkits developed in quantum information theory: resource
theories and one-shot information theory. The field has boomed recently, but
the theorems amassed have hardly impacted experiments. Can thermodynamic
resource theories be realized experimentally? Via what steps can we shift the
theory toward physical realizations? Should we care? I present eleven
opportunities in physically realizing thermodynamic resource theories.Comment: Publication information added. Cosmetic change
Approximate reversibility in the context of entropy gain, information gain, and complete positivity
There are several inequalities in physics which limit how well we can process
physical systems to achieve some intended goal, including the second law of
thermodynamics, entropy bounds in quantum information theory, and the
uncertainty principle of quantum mechanics. Recent results provide physically
meaningful enhancements of these limiting statements, determining how well one
can attempt to reverse an irreversible process. In this paper, we apply and
extend these results to give strong enhancements to several entropy
inequalities, having to do with entropy gain, information gain, entropic
disturbance, and complete positivity of open quantum systems dynamics. Our
first result is a remainder term for the entropy gain of a quantum channel.
This result implies that a small increase in entropy under the action of a
subunital channel is a witness to the fact that the channel's adjoint can be
used as a recovery map to undo the action of the original channel. Our second
result regards the information gain of a quantum measurement, both without and
with quantum side information. We find here that a small information gain
implies that it is possible to undo the action of the original measurement if
it is efficient. The result also has operational ramifications for the
information-theoretic tasks known as measurement compression without and with
quantum side information. Our third result shows that the loss of Holevo
information caused by the action of a noisy channel on an input ensemble of
quantum states is small if and only if the noise can be approximately corrected
on average. We finally establish that the reduced dynamics of a
system-environment interaction are approximately completely positive and
trace-preserving if and only if the data processing inequality holds
approximately.Comment: v3: 12 pages, accepted for publication in Physical Review
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