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Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
K 2004 Canonical decompositions of nqubit quantum computations and concurrence
"... The twoqubit canonical decomposition SU(4) = [SU(2) ⊗ SU(2)]∆[SU(2) ⊗ SU(2)] writes any twoqubit quantum computation as a composition of a local unitary, a relative phasing of Bell states, and a second local unitary. Using Lie theory, we generalize this to an nqubit decomposition, the concurre ..."
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Cited by 4 (1 self)
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The twoqubit canonical decomposition SU(4) = [SU(2) ⊗ SU(2)]∆[SU(2) ⊗ SU(2)] writes any twoqubit quantum computation as a composition of a local unitary, a relative phasing of Bell states, and a second local unitary. Using Lie theory, we generalize this to an nqubit decomposition, the concurrence canonical decomposition (C.C.D.) SU(2 n) = KAK. The group K fixes a bilinear form related to the concurrence, and in particular any computation in K preserves the tangle 〈φ ∗ (−iσ y 1)···(−iσy n)φ〉  2 for n even. Thus, the C.C.D. shows that any nqubit quantum computation is a composition of a computation preserving this ntangle, a computation in A which applies relative phases to a set of GHZ states, and a second computation which preserves it. As an application, we study the extent to which a large, random unitary may change concurrence. The result states that for a randomly chosen a ∈ A ⊂ SU(2 2p), the probability that a carries a state of tangle 0 to a state of maximum tangle approaches 1 as the even number of qubits approaches infinity. Any v = k1ak2 for such an a ∈ A has the same property. Finally, although 〈φ ∗ (−iσ y 1)···(−iσy n)φ〉  2 vanishes identically when the number of qubits is odd, we show that a more complicated C.C.D. still exists in which K is a symplectic group. I.
Time reversal and nqubit canonical decomposition
 J. Math. Phys
, 2005
"... On pure states of n quantum bits, the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spinflip. The monotone vanishes for n odd, but for n even there is an explicit formula for its value on mixed states, i.e. a closedform expression computes the min ..."
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On pure states of n quantum bits, the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spinflip. The monotone vanishes for n odd, but for n even there is an explicit formula for its value on mixed states, i.e. a closedform expression computes the minimum over all ensemble decompositions of a given density. For n even a matrix decomposition v = k1ak2 of the unitary group is explicitly computable and allows for study of the monotone’s dynamics. The side factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the a factor. This unitary a is diagonal on a basis of GHZlike states, with dynamics of the entire state space depending on the relative phases within a. In this work, we provide an explicit numerical algorithm computing v = k1ak2 for n odd. Further, in the odd case we lift the monotone to a twoargument function. The concurrence capacity of v according to the double argument lift may be nontrivial for n odd and reduces to the usual concurrence capacity in the literature for n even. The generalization may also be studied using the CCD, leading again to maximal capacity for most unitaries. The capacity of v ⊗ I2 is at least that of v, so oddqubit capacities have implications for evenqubit entanglement. The generalizations require considering the spinflip as a time reversal symmetry operator in Wigner’s axiomatization, and the original Lie algebra homomorphism defining the CCD may be restated entirely in terms of this time reversal. The polar decomposition related to the CCD then writes any unitary evolution
Submitted to: New J. Phys.
, 710
"... Creation of resilient entangled states and a resource for measurementbased quantum computation with optical superlattices ..."
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Creation of resilient entangled states and a resource for measurementbased quantum computation with optical superlattices
1 Distinguishability of Quantum States by Separable Operations
, 705
"... Abstract — We study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for ..."
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Abstract — We study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of (D − 1) pure states, where D is the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of 2 ⊗ 2 separable operations not being realizable by local operations and classical communication. Before our work only a class of 3 ⊗ 3 nonlocal separable operations was known [Bennett et al, Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of 1 or 2 orthogonal product states, i.e., has an orthogonal Schmidt number not less than 3, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp. three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.