Multiqubit entanglement of a general input state

Abstract

Measurement of entanglement remains an important problem for quantum information. We present the design and simulation of an experimental method for an entanglement indicator for a general multiqubit state. The system can be in a pure or a mixed state, and it need not be ``close'' to any particular state. The system contains information about its own entanglement; we use dynamic learning methods to map this information onto a single experimental measurement which is our entanglement indicator. Our method does not require prior state reconstruction or lengthy optimization. An entanglement witness emerges from the learning process, beginning with two-qubit systems, and extrapolating this to three, four, and five qubit systems where the entanglement is not well understood. Our independently learned measures for three-qubit systems compare favorably with known entanglement measures. As the size of the system grows the amount of additional training necessary diminishes, raising hopes for applicability to large computational systems.

Learning quantum annealing

“…In general the amount of additional training diminishes with increasing size, which raises hopes for the applicability of our technique to systems even of hundreds of qubits. The exception was in the training from three qubits to four (possibly for symmetry reasons -we saw this phenomenon also with the entanglement indicator [17]), though even here, the initial error for the fourqubit calculation was a tenth the size when starting from the three-qubit functions, compared with starting from scratch. And, while direct training by symmetry occurs along a pathway whose progress is not experimentally accessible, we have shown that it is possible to specify a pathway along which the average spin is nonzero and thus checkable.…”

“…For large N, we expect the W states to bootstrap more easily than the GHZ states, because in that limit, the coupling probably only needs to be nearest neighbor. Our earlier work [17] seems to show that for pairwise functions in N qubit systems, there is exponentially less training necessary to go from (N-1) to N qubits, but we have yet to do these calculations on the annealed systems.…”

“…Multiqubit entanglement is common in nature but, depending on the physical setup, it is not always easy to produce it in a controlled manner [7,8,9] or to maintain it [9,10,11,12,13]. In previous work, we have developed an entanglement indicator [14], corrected it for anomalous oscillation [15], and extended it to multiqubit systems [16,17]. As the size of the system grows the amount of additional training necessary diminishes, a kind of "bootstrapping" effect [18,19].…”

“…In a series of papers [14,15,16,17,29] we have explored the possibility of using one or more physical measurements on an N-qubit quantum system as outputs for the QNN; specifically, measurements, at the final time, that would estimate the entanglement of the initial (input) state. For a pure product N-qubit input state (the minimum flexibility in QNN training) it is easy to show that each output of this type can be written as a sum of quadratics in the amplitudes of the input state |ψ(0) , with linearly independent coefficients, plus sums and products of cosines and sines in each of the phase angles in additional cross terms of the amplitudes.…”

“…Because each of the parameter functions can be taken to be time varying (as we do here), this essentially means that any single measurement output can have the complexity of almost any reasonably well-behaved function, a kind of "quadratic spline" in the amplitudes of the elements of the density matrix, and enough nonlinear cross terms to approximate their phase dependence. And, indeed, we were quite successful in showing that this mapping, for the entanglement indicator, is relatively easy [14], that it bootstraps well to larger systems with a difficulty that decreases with size [17], and that it is robust to both noise and decoherence, with increasing robustness as the system gets larger [29].…”