2,183 research outputs found
The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)
We describe stabilizer states and Clifford group operations using linear
operations and quadratic forms over binary vector spaces. We show how the
n-qubit Clifford group is isomorphic to a group with an operation that is
defined in terms of a (2n+1)x(2n+1) binary matrix product and binary quadratic
forms. As an application we give two schemes to efficiently decompose Clifford
group operations into one and two-qubit operations. We also show how the
coefficients of stabilizer states and Clifford group operations in a standard
basis expansion can be described by binary quadratic forms. Our results are
useful for quantum error correction, entanglement distillation and possibly
quantum computing.Comment: 9 page
Four qubits can be entangled in nine different ways
We consider a single copy of a pure four-partite state of qubits and
investigate its behaviour under the action of stochastic local quantum
operations assisted by classical communication (SLOCC). This leads to a
complete classification of all different classes of pure states of four-qubits.
It is shown that there exist nine families of states corresponding to nine
different ways of entangling four qubits. The states in the generic family give
rise to GHZ-like entanglement. The other ones contain essentially 2- or 3-qubit
entanglement distributed among the four parties. The concept of concurrence and
3-tangle is generalized to the case of mixed states of 4 qubits, giving rise to
a seven parameter family of entanglement monotones. Finally, the SLOCC
operations maximizing all these entanglement monotones are derived, yielding
the optimal single copy distillation protocol
Local unitary versus local Clifford equivalence of stabilizer states
We study the relation between local unitary (LU) equivalence and local
Clifford (LC) equivalence of stabilizer states. We introduce a large subclass
of stabilizer states, such that every two LU equivalent states in this class
are necessarily LC equivalent. Together with earlier results, this shows that
LC, LU and SLOCC equivalence are the same notions for this class of stabilizer
states. Moreover, recognizing whether two given stabilizer states in the
present subclass are locally equivalent only requires a polynomial number of
operations in the number of qubits.Comment: 8 pages, replaced with published versio
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The role of right and left parietal lobes in the conceptual processing of numbers
Neuropsychological and functional imaging studies have associated the conceptual processing of numbers with bilateral parietal regions (including intraparietal sulcus). However, the processes driving these effects remain unclear because both left and right posterior parietal regions are activated by many other conceptual, perceptual, attention, and response-selection processes. To dissociate parietal activation that is number-selective from parietal activation related to other stimulus or response-selection processes, we used fMRI to compare numbers and object names during exactly the same conceptual and perceptual tasks while factoring out activations correlating with response times. We found that right parietal activation was higher for conceptual decisions on numbers relative to the same tasks on object names, even when response time effects were fully factored out. In contrast, left parietal activation for numbers was equally involved in conceptual processing of object names. We suggest that left parietal activation for numbers reflects a range of processes, including the retrieval of learnt facts that are also involved in conceptual decisions on object names. In contrast, number selectivity in right parietal cortex reflects processes that are more involved in conceptual decisions on numbers than object names. Our results generate a new set of hypotheses that have implications for the design of future behavioral and functional imaging studies of patients with left and right parietal damage
The Lorentz singular value decomposition and its applications to pure states of 3 qubits
All mixed states of two qubits can be brought into normal form by the action of SLOCC operations of the kind . These normal forms can be obtained by considering a Lorentz singular value decomposition on a real parameterization of the density matrix. We show that the Lorentz singular values are variationally defined and give rise to entanglement monotones, with as a special case the concurrence. Next a necessary and sufficient criterion is conjectured for a mixed state to be convertible into another specific one with a non-zero probability. Finally the formalism of the Lorentz singular value decomposition is applied to tripartite pure states of qubits. New proofs are given for the existence of the GHZ- and W-class of states, and a rigorous proof for the optimal distillation of a GHZ-state is derived
Number-space associations in synaesthesia are not influenced by finger-counting habits
In many cultures, one of the earliest representations of number to be learned is a finger-counting system. Although most children stop using their fingers to count as they grow more confident with number, traces of this system can still be seen in adulthood. For example, an individual's finger-counting habits appear to affect the ways in which numbers are implicitly associated with certain areas of space, as inferred from the spatial–numerical association of response codes (SNARC) effect. In this study, we questioned the finger-counting habits of 98 participants who make explicit, idiosyncratic associations between number and space, known as number-space synaesthesia. Unexpectedly, neither handedness nor finger-counting direction (left-to-right or right-to-left) was associated with the relative positions of 1 and 10 in an individual's number-space synaesthesia. This lack of association between finger-counting styles and number-space synaesthesia layout may result from habitual use of synaesthetic space rather than fingers when learning to count; we offer some testable hypotheses that could assess whether this is the case
Lorentz singular-value decomposition and its applications to pure states of three qubits
All mixed states of two qubits can be brought into normal form by the action of local operations and classical communication operations of the kind rho'=(AxB) rho(AxB)dagger. These normal forms can be obtained by considering a Lorentz singular-value decomposition on a real parametrization of the density matrix. We show that the Lorentz singular values are variationally defined and give rise to entanglement monotones, with as a special case the concurrence. Next a necessary and sufficient criterion is conjectured for a mixed state to be convertible into another specific one with a nonzero probability. Finally the formalism of the Lorentz singular-value decomposition is applied to tripartite pure states of qubits. New proofs are given for the existence of the Greenberger-Horne-Zeilinger (GHZ) class and W class of states, and a rigorous proof for the optimal distillation of a GHZ state is derived
Agrammatic but numerate
A central question in cognitive neuroscience concerns the extent to
which language enables other higher cognitive functions. In the
case of mathematics, the resources of the language faculty, both
lexical and syntactic, have been claimed to be important for exact
calculation, and some functional brain imaging studies have shown
that calculation is associated with activation of a network of
left-hemisphere language regions, such as the angular gyrus and
the banks of the intraparietal sulcus. We investigate the integrity
of mathematical calculations in three men with large left-hemisphere
perisylvian lesions. Despite severe grammatical impairment
and some difficulty in processing phonological and orthographic
number words, all basic computational procedures were intact
across patients. All three patients solved mathematical problems
involving recursiveness and structure-dependent operations (for
example, in generating solutions to bracket equations). To our
knowledge, these results demonstrate for the first time the remarkable
independence of mathematical calculations from language
grammar in the mature cognitive system
Bunge’s Mathematical Structuralism Is Not a Fiction
In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involved in mathematical knowledge, in particular its dependence on mental/brain states and material objects
Lateralized neural responses to letters and digits in first graders.
The emergence of visual cortex specialization for culturally acquired characters like letters and digits, both arbitrary shapes related to specific cognitive domains, is yet unclear. Here, 20 young children (6.12 years old) were tested with a frequency-tagging paradigm coupled with electroencephalogram recordings to assess discrimination responses of letters from digits and vice-versa. One category of stimuli (e.g., letters) was periodically inserted (1/5)
in streams of the other category (e.g., digits) presented at a fast rate (6 Hz). Results show clear right-lateralized discrimination
responses at 6 Hz/5 for digits within letters, and a trend for left-lateralization for letters. These results support an early developmental emergence of ventral occipito-temporal cortex specialization for visual recognition of digits and letters, potentially in relation with relevant coactivated brain networks
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