Bipartite sampling method: QPtomographer.bistates¶
This module implements the bipartite state sampling method for the reliable diamond norm estimation.
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QPtomographer.bistates.run(...)¶ The main execution routine for the bipartite state sampling method. This function behaves analogously to
tomographer.tomorun.tomorun(), but in the setting described in arXiv:1808.00358 and calculating the diamond norm to a reference channel as figure of merit.As described in the paper, the log-likelihood function is calculated from the matrices Emn as follows:
\[\ln\mathcal{L}(\rho_{AB}\mid E) = \sum_k \texttt{Nm[k]} \cdot \ln \mbox{tr}\bigl(\rho_{AB}\, \texttt{Emn[k]}\bigr)\ ,\]Hence,
Emn[k]must be the POVM effect which was observed on the joint \(A\otimes B\) systems (see paper, and see the function argument Emn= below).All arguments should be specified as keyword arguments. Accepted arguments are:
dimX: dimension of input system (required)
dimY: dimension of output system (required)
Emn: a list of POVM effects on \(X\otimes Y\). Should be a python list where each item is a matrix provided as a complex \(d_X d_Y\times d_X d_Y\) NumPy array.
Nm: the corresponding list of frequency counts. This is a list of integers, of the same length as Emn, where Nm[k] is the number of times the outcome described by the POVM effect Emn[k] was observed.
fig_of_merit: which figure of merit to use. May be one of ‘diamond-distance’ (the default), ‘entanglement-fidelity’, ‘worst-entanglement-fidelity’, or a custom callable Python object. Refer to section Figures of Merit for Channels.
The callable will be invoked with as argument a square complex matrix \(T\) of shape (d_X*dY, d_X*d_Y) given as a NumPy array, and where the process matrix, a normalized bipartite state, can be simply computed as \(T T^\dagger\). If you conjugate by the inverse square root of the reduced state on the first system, you get the Choi matrix of the process. The following code computes this:
def custfigofmerit(T): # compute average entanglement fidelity manually with custom # callable, assuming a qubit->qubit quantum channel # # T is a numpy object of size (dX*dY, dX*dY) representing a # square root of a bipartite state (the current point of the # random walk). #print(T) # To compute the Choi matrix of the process encoded in T: # # Compute the bipartite state (Choi matrix weighted by input state) rho_RB = np.dot(T, T.conj().T) rho_q_RB = qutip.Qobj(rho_RB, dims=[[2,2],[2,2]]) # conjugate by inverse square root of input state to get Choi matrix invsqrtmrho_R = qutip.tensor(invsqrtm(rho_q_RB.ptrace([0])), qutip.qeye(2)) E_RB = invsqrtmrho_R * rho_q_RB * invsqrtmrho_R #print(E_RB) #print(E_RB.ptrace([0])) assert npl.norm( E_RB.ptrace([0]).data.toarray() - np.eye(2), 'nuc' ) <= 1e-6 # Maximally entangled ket between R and A: Phi_RA = (\sum_k |k>_R |k>_A) Phi_RA = qutip.Qobj(np.eye(2).reshape([4,1]), dims=[[2,2],[1]]) # <\Phi|_RA E_RB |\Phi>_RA / (2*2) : divide by (2*2) because \Phi is not normalized val = qutip.expect(E_RB, Phi_RA) / (2*2) return val # helper function: compute matrix inverse square root of a Hermitian `qutip.Qobj` matrix def invsqrtm(Aq): d, V = npl.eigh(Aq.data.toarray()) return qutip.Qobj(np.dot(np.dot(V, np.diag(np.reciprocal(np.sqrt(d)))), V.conj().T), dims=Aq.dims)
ref_channel_XY: the reference channel to which to calculate the diamond norm distance to. This argument is not used if fig_of_merit is not ‘diamond-distance’. Specify the channel by its Choi matrix on \(X\otimes Y\), as a \(d_X d_Y \times d_X d_Y\) NumPy array. The normalization should be such that the reduced state on \(X\) is the identity operator, ie., the trace of the full matrix should be equal to \(d_X\).
If dimX==dimY, you may set ref_channel_XY=None, in which case the identity process (with respect to the canonical basis) is used as reference channel.
hist_params: the parameters of the histogram to collect, i.e., range and number of bins. Must be a
tomographer.HistogramParamsobject.mhrw_params: the parameters of the random walk, i.e., the step size, the sweep size, number of thermalization sweeps and number of live run sweeps. Specify as a
tomographer.MHRWParamsobject.jumps_method: one of
"full"or"light", depending on the requested method of random walk step. This argument has the same effect as the jumps_method= argument oftomographer.tomorun.tomorun().dnorm_epsilon: the precision at which to calculate the diamond norm (which is calculated by numerically solving the corresponding semidefinite program using SCS). The default is 1e-3.
num_repeats: the total number of random walks to run. By default, this is set to the number of available cores.
binning_num_levels: number of levels in the binning analysis. By default, or if the value -1 is specified, an appropriate number of levels is determined automatically.
progress_fn, progress_interval_ms, ctrl_step_size_params, ctrl_converged_params: these parameters are treated the same as for
tomographer.tomorun.tomorun().
