The two-qubit canonical decomposition SU(4)=[SU(2)⊗SU(2)]Δ[SU(2)⊗SU(2)] writes any two-qubit unitary operator 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 n-qubit decomposition, the concurrence canonical decomposition (CCD) SU(2n)=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any unitary in K preserves the tangle |〈φ|¯(−iσ1y)⋯(−iσny)|φ〉|2 for n even. Thus, the CCD shows that any n-qubit unitary is a composition of a unitary operator preserving this n-tangle, a unitary operator in A which applies relative phases to a set of GHZ states, and a second unitary operator which preserves the tangle. 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(22p), 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|¯(−iσ1y)⋯(−iσny)|φ〉|2 vanishes identically when the number of qubits is odd, we show that a more complicated CCD still exists in which K is a symplectic group.
The two-qubit canonical decomposition SU(4)=[SU(2)⊗SU(2)]Δ[SU(2)⊗SU(2)] writes any two-qubit unitary operator 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 n-qubit decomposition, the concurrence canonical decomposition (CCD) SU(2n)=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any unitary in K preserves the tangle |〈φ|¯(−iσ1y)⋯(−iσny)|φ〉|2 for n even. Thus, the CCD shows that any n-qubit unitary is a composition of a unitary operator preserving this n-tangle, a unitary operator in A which applies relative phases to a set of GHZ states, and a second unitary operator which preserves the tangle. 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(22p), 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|¯(−iσ1y)⋯(−iσny)|φ〉|2 vanishes identically when the number of qubits is odd, we show that a more complicated CCD still exists in which K is a symplectic group.
The two-qubit canonical decomposition SU(4) = [SU(2) \\otimes SU(2)] Delta [SU(2) \\otimes SU(2)] writes any two-qubit 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 n-qubit 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 |<phi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi>|^2 for n even. Thus, the C.C.D. shows that any n-qubit quantum computation is a composition of a computation preserving this n-tangle, 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 in A within 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=k_1 a k_2 for such an a \\in A has the same property. Finally, although |<phi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi>|^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.
On pure states of n quantum bits, the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spin-flip. The monotone vanishes for n odd, but for n even there is an explicit formula for its value on mixed states, i.e., a closed-form expression computes the minimum over all ensemble decompositions of a given density. For n even a matrix decomposition ν=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 phases a basis of entangled states, and the concurrence dynamics of u are determined by these relative phases. In this work, we provide an explicit numerical algorithm computing ν=k1ak2 for n odd. Further, in the odd case we lift the monotone to a two-argument function. The concurrence capacity of ν 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 ν⊗I2 is at least that of ν, so odd-qubit capacities have implications for even-qubit entanglement. The generalizations require considering the spin-flip 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 as the product of a time-symmetric and time-antisymmetric evolution with respect to the spin-flip. En route we observe a Kramers’ nondegeneracy: the existence of a nondegenerate eigenstate of any time reversal symmetric n-qubit Hamiltonian demands (i) n even and (ii) maximal concurrence of said eigenstate. We provide examples of how to apply this work to study the kinematics and dynamics of entanglement in spin chain Hamiltonians.
We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application, we describe a new and general method to obtain Cartan decompositions of the unitary group of evolutions of multipartite systems from Cartan decompositions on the single subsystems. The resulting decomposition, which we call of the odd-even type, contains, as a special case, the concurrence canonical decomposition (CCD) presented in the context of entanglement theory. The CCD is therefore extended from the case of a multipartite system of n qubits to the case where the component subsystems have arbitrary dimension.
The state space ?-t of n quantum bits of data is exponentially large, having dimension 2. The (pure) local states which correspond to each individual quantum bit being in an isolated one-qubit state, i.e. those which are tensor products, form a much smaller orbit of ®U(2) . 00 . . . 0) of linear dimension within the state space. Hence most states are non-local, or entangled. The concurrence function on quantum data states is one measure of entanglement, intuitively capturing an exponentially small fraction of the phenomenon. This paper reports numerical tests of how concurrence changes as one applies a quantum computation u to a pure n quantum-bit data state kl') . We make strong use of a mathematical tool for factoring u = k1 a k2 into subcomputations, namely the CCD matrix decomposition. The concurrence dynamics of a computation Ib) u) are in a certain sense localized to the a factor, and so our actual numerics concentrate on RI') '—+aI'i4'). This is a great simplification, since an arbitrary unitary evolution may vary over 4' — 1 real degrees of freedom, while the a e A of the appropriate form for the CCD matrix decomposition may vary over 2 —1 or 2n/2 as n = 2p, 2p — 1.
The state space H n of n quantum bits of data is exponentially large, having dimension 2. The (pure) local states which correspond to each individual quantum bit being in an isolated one-qubit state, i.e. those which are tensor products, form a much smaller orbit of ⊗ n 1 U(2).|00...0) of linear dimension within the state space. Hence most states are non-local, or entangled. The concurrence function on quantum data states is one measure of entanglement, intuitively capturing an exponentially small fraction of the phenomenon. This paper reports numerical tests of how concurrence changes as one applies a quantum computation u to a pure n quantum-bit data state |ψ). We make strong use of a mathematical tool for factoring u = k 1 a k 2 into subcomputations, namely the CCD matrix decomposition. The concurrence dynamics of a computation |ψ) → u|ψ) are in a certain sense localized to the a factor, and so our actual numerics concentrate on |ψ) → a|ψ). This is a great simplification, since an arbitrary unitary evolution may vary over 4 n - 1 real degrees of freedom, while the a ∈ A of the appropriate form for the CCD matrix decomposition may vary over 2 n - 1 or 2 n/2 - 1 as n = 2p, 2p - 1.
The main part of this dissertation starts with a generalization of the entangling power, which quantifies the ability of unitary operators to generate entangled states by their action on the computational basis. The entangling power has been defined for an evenly split bipartite system, whose states live in a Hilbert space of the form H1⊗ H2 where dimH1= dimH2 . I generalize this so that we can consider an odd number of qubits. Entangling power is defined in terms of the linear entropy, a linearization of the von Neumann entropy, whose polynomial form allows one to derive simpler expressions for functions of the entropy. The linear entropy measure is lifted from the state space to the operator space to measure the entanglement of operators. In particular, we focus on the three qubit case, where dimH1 = 2 and dimH2 = 4, as a step to understanding entanglement in many qubit systems. This is the content of Chapter 3.
The goal of Chapter 4 is to explore the structure of entanglement producing operators in SU(2n), with the focus on SU(8). I first review the magic matrix formalism for two-qubit operators, a well known matrix that captures the entangling capability of operators in SU(4), using only 3 of the 15 parameters of the group. I then use a recursive definition of the Cartan decomposition of SU(2n) to construct a magic matrix for operators in SU(8). Because we view the entangling operation as between the first qubit and the other two as a single subsystem, the entanglement is invariant under the action of SU(2) ⊗ SU(4), but SU(8)/SU(2) ⊗ SU(4) is not a symmetric space, and we must use the decomposition in two stages to capture enough entangling operators to account for all of SU(8). The number of parameters appearing in the entangling power is then reduced from 63 to 8, keeping track of 7 entangling operators, one of which does not commute with the others. The next chapter then presents calculations of the entangling powers of operators that generate GHZ states from the computational basis, many of which are solutions to the Yang-Baxter equation and have strong connections to topological quantum computing. We find that the Kauffman-Lomonaco two-qubit gate that produces the Bell states from the computational basis has a maximal entangling power, while the GHZ gate on three qubits does not, adding another piece to the puzzle of entanglement.