73 research outputs found
Permutation Symmetry Determines the Discrete Wigner Function
The Wigner function provides a useful quasiprobability representation of
quantum mechanics, with applications in various branches of physics. Many nice
properties of the Wigner function are intimately connected with the high
symmetry of the underlying operator basis composed of phase point operators:
any pair of phase point operators can be transformed to any other pair by a
unitary symmetry transformation. We prove that, in the discrete scenario, this
permutation symmetry is equivalent to the symmetry group being a unitary
2-design. Such a highly symmetric representation can only appear in odd prime
power dimensions besides dimensions 2 and 8. It suffices to single out a unique
discrete Wigner function among all possible quasiprobability representations.
In the course of our study, we show that this discrete Wigner function is
uniquely determined by Clifford covariance, while no Wigner function is
Clifford covariant in any even prime power dimension.Comment: 5+2 pages, connection with unitary 2-designs added, accepted by Phys.
Rev. Lett. as Editors' Suggestio
Quasiprobability representations of quantum mechanics with minimal negativity
Quasiprobability representations, such as the Wigner function, play an
important role in various research areas. The inevitable appearance of
negativity in such representations is often regarded as a signature of
nonclassicality, which has profound implications for quantum computation.
However, little is known about the minimal negativity that is necessary in
general quasiprobability representations. Here we focus on a natural class of
quasiprobability representations that is distinguished by simplicity and
economy. We introduce three measures of negativity concerning the
representations of quantum states, unitary transformations, and quantum
channels, respectively. Quite surprisingly, all three measures lead to the same
representations with minimal negativity, which are in one-to-one correspondence
with the elusive symmetric informationally complete measurements. In addition,
most representations with minimal negativity are automatically covariant with
respect to the Heisenberg-Weyl groups. Furthermore, our study reveals an
interesting tradeoff between negativity and symmetry in quasiprobability
representations.Comment: 5.2+3 pages; accepted by Phys. Rev. Let
Multiqubit Clifford groups are unitary 3-designs
Unitary -designs are a ubiquitous tool in many research areas, including
randomized benchmarking, quantum process tomography, and scrambling. Despite
the intensive efforts of many researchers, little is known about unitary
-designs with in the literature. We show that the multiqubit
Clifford group in any even prime-power dimension is not only a unitary
2-design, but also a 3-design. Moreover, it is a minimal 3-design except for
dimension~4. As an immediate consequence, any orbit of pure states of the
multiqubit Clifford group forms a complex projective 3-design; in particular,
the set of stabilizer states forms a 3-design. In addition, our study is
helpful to studying higher moments of the Clifford group, which are useful in
many research areas ranging from quantum information science to signal
processing. Furthermore, we reveal a surprising connection between unitary
3-designs and the physics of discrete phase spaces and thereby offer a simple
explanation of why no discrete Wigner function is covariant with respect to the
multiqubit Clifford group, which is of intrinsic interest to studying quantum
computation.Comment: 7 pages, published in Phys. Rev.
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