22 research outputs found
Sequential decoding of a general classical-quantum channel
Since a quantum measurement generally disturbs the state of a quantum system,
one might think that it should not be possible for a sender and receiver to
communicate reliably when the receiver performs a large number of sequential
measurements to determine the message of the sender. We show here that this
intuition is not true, by demonstrating that a sequential decoding strategy
works well even in the most general "one-shot" regime, where we are given a
single instance of a channel and wish to determine the maximal number of bits
that can be communicated up to a small failure probability. This result follows
by generalizing a non-commutative union bound to apply for a sequence of
general measurements. We also demonstrate two ways in which a receiver can
recover a state close to the original state after it has been decoded by a
sequence of measurements that each succeed with high probability. The second of
these methods will be useful in realizing an efficient decoder for fully
quantum polar codes, should a method ever be found to realize an efficient
decoder for classical-quantum polar codes.Comment: 12 pages; accepted for publication in the Proceedings of the Royal
Society
Performance of polar codes for quantum and private classical communication
We analyze the practical performance of quantum polar codes, by computing
rigorous bounds on block error probability and by numerically simulating them.
We evaluate our bounds for quantum erasure channels with coding block lengths
between 2^10 and 2^20, and we report the results of simulations for quantum
erasure channels, quantum depolarizing channels, and "BB84" channels with
coding block lengths up to N = 1024. For quantum erasure channels, we observe
that high quantum data rates can be achieved for block error rates less than
10^(-4) and that somewhat lower quantum data rates can be achieved for quantum
depolarizing and BB84 channels. Our results here also serve as bounds for and
simulations of private classical data transmission over these channels,
essentially due to Renes' duality bounds for privacy amplification and
classical data transmission of complementary observables. Future work might be
able to improve upon our numerical results for quantum depolarizing and BB84
channels by employing a polar coding rule other than the heuristic used here.Comment: 8 pages, 6 figures, submission to the 50th Annual Allerton Conference
on Communication, Control, and Computing 201
Polar codes for private classical communication
We construct a new secret-key assisted polar coding scheme for private
classical communication over a quantum or classical wiretap channel. The
security of our scheme rests on an entropic uncertainty relation, in addition
to the channel polarization effect. Our scheme achieves the symmetric private
information rate by synthesizing "amplitude" and "phase" channels from an
arbitrary quantum wiretap channel. We find that the secret-key consumption rate
of the scheme vanishes for an arbitrary degradable quantum wiretap channel.
Furthermore, we provide an additional sufficient condition for when the secret
key rate vanishes, and we suspect that satisfying this condition implies that
the scheme requires no secret key at all. Thus, this latter condition addresses
an open question from the Mahdavifar-Vardy scheme for polar coding over a
classical wiretap channel.Comment: 11 pages, 2 figures, submission to the 2012 International Symposium
on Information Theory and its Applications (ISITA 2012), Honolulu, Hawaii,
US
Magic state distillation with punctured polar codes
We present a scheme for magic state distillation using punctured polar codes.
Our results build on some recent work by Bardet et al. (ISIT, 2016) who
discovered that polar codes can be described algebraically as decreasing
monomial codes. Using this powerful framework, we construct tri-orthogonal
quantum codes (Bravyi et al., PRA, 2012) that can be used to distill magic
states for the gate. An advantage of these codes is that they permit the
use of the successive cancellation decoder whose time complexity scales as
. We supplement this with numerical simulations for the erasure
channel and dephasing channel. We obtain estimates for the dimensions and error
rates for the resulting codes for block sizes up to for the erasure
channel and for the dephasing channel. The dimension of the
triply-even codes we obtain is shown to scale like for the binary
erasure channel at noise rate and for the dephasing
channel at noise rate . The corresponding bit error rates drop to
roughly for the erasure channel and for
the dephasing channel respectively.Comment: 18 pages, 4 figure
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
Decoupling with random quantum circuits
Decoupling has become a central concept in quantum information theory with
applications including proving coding theorems, randomness extraction and the
study of conditions for reaching thermal equilibrium. However, our
understanding of the dynamics that lead to decoupling is limited. In fact, the
only families of transformations that are known to lead to decoupling are
(approximate) unitary two-designs, i.e., measures over the unitary group which
behave like the Haar measure as far as the first two moments are concerned.
Such families include for example random quantum circuits with O(n^2) gates,
where n is the number of qubits in the system under consideration. In fact, all
known constructions of decoupling circuits use \Omega(n^2) gates.
Here, we prove that random quantum circuits with O(n log^2 n) gates satisfy
an essentially optimal decoupling theorem. In addition, these circuits can be
implemented in depth O(log^3 n). This proves that decoupling can happen in a
time that scales polylogarithmically in the number of particles in the system,
provided all the particles are allowed to interact. Our proof does not proceed
by showing that such circuits are approximate two-designs in the usual sense,
but rather we directly analyze the decoupling property.Comment: 25 page
Bounds on Information Combining With Quantum Side Information
"Bounds on information combining" are entropic inequalities that determine
how the information (entropy) of a set of random variables can change when
these are combined in certain prescribed ways. Such bounds play an important
role in classical information theory, particularly in coding and Shannon
theory; entropy power inequalities are special instances of them. The arguably
most elementary kind of information combining is the addition of two binary
random variables (a CNOT gate), and the resulting quantities play an important
role in Belief propagation and Polar coding. We investigate this problem in the
setting where quantum side information is available, which has been recognized
as a hard setting for entropy power inequalities.
Our main technical result is a non-trivial, and close to optimal, lower bound
on the combined entropy, which can be seen as an almost optimal "quantum Mrs.
Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the
concavity of von Neumann entropy, which is tight in the regime of low pairwise
state fidelities; (2) the quantitative improvement of strong subadditivity due
to Fawzi-Renner, in which we manage to handle the minimization over recovery
maps; (3) recent duality results on classical-quantum-channels due to Renes et
al. We furthermore present conjectures on the optimal lower and upper bounds
under quantum side information, supported by interesting analytical
observations and strong numerical evidence.
We finally apply our bounds to Polar coding for binary-input
classical-quantum channels, and show the following three results: (A) Even
non-stationary channels polarize under the polar transform. (B) The blocklength
required to approach the symmetric capacity scales at most sub-exponentially in
the gap to capacity. (C) Under the aforementioned lower bound conjecture, a
blocklength polynomial in the gap suffices.Comment: 23 pages, 6 figures; v2: small correction