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 ?-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 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.
Local unitary invariants allow one to test whether multipartite states are equivalent up to local basis changes. Equivalently, they specify the geometry of the obtained by factoring out local unitary action from the state space. This space is of interest because of its intimate relationship to entanglement. Unfortunately, the dimension of the orbit space grows exponentially with the number of subsystems, and the number of invariants needed to characterise orbits grows at least as fast. This makes the study of entanglement via local unitary invariants seem very daunting. I point out here that there is a simplifying principle: Invariants fall into families related by the tracing-out of subsystems, and these families grow exponentially with the number of subsystems. In particular, in the case of pure qubit systems, there is a family whose size is about half the dimension of orbit space. These invariants are closely related to cumulants and to multipartite separability. Members of the family have been repeatedly discovered in the literature, but the fact that they are related to cumulants and constitute a family has apparently not been observed.
Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.