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arXiv:2208.05132 · Annalen der Physik

Not All Entangled States are Useful for Ancilla‐Assisted Quantum Process Tomography

Guodong Lu, Zhou Zhang, Yue Dai, Yuli Dong, Chengjie Zhang

Published Feb 21, 2022• 2 citations
arXivHTMLPDF

Abstract

Abstract It is well known that one can extract all the information of an unknown quantum channel by means of quantum process tomography, such as standard quantum‐process tomography and ancilla‐assisted quantum process tomography (AAQPT). Furthermore, it is shown that entanglement is not necessary for AAQPT; there exist separable states that are also useful for it. Surprisingly, in this work we find that not all entangled states are useful for AAQPT; there also exist some entangled states that are useless. The realignment operation used in entanglement detection can be related to the question whether a bipartite state is useful for AAQPT. The relationship between them is derived and the process of extracting the complete information of an unknown channel by the realignment operation shown. Based on this relationship, examples of a two‐qutrit entangled state and a two‐qutrit bound entangled state are presented. Both of these two examples are entangled but they cannot be used for AAQPT. Last but not the least, experimental verification is been performed on the IBM platform.

DOI
10.1002/andp.202100550

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Abstract It is well known that one can extract all the information of an unknown quantum channel by means of quantum process tomography, such as standard quantum‐process tomography and ancilla‐assisted quantum process tomography (AAQPT). Furthermore, it is shown that entanglement is not necessary for AAQPT; there exist separable states that are also useful for it. Surprisingly, in this work we find that not all entangled states are useful for AAQPT; there also exist some entangled states that are useless. The realignment operation used in entanglement detection can be related to the question whether a bipartite state is useful for AAQPT. The relationship between them is derived and the process of extracting the complete information of an unknown channel by the realignment operation shown. Based on this relationship, examples of a two‐qutrit entangled state and a two‐qutrit bound entangled state are presented. Both of these two examples are entangled but they cannot be used for AAQPT. Last but not the least, experimental verification is been performed on the IBM platform.
Lu \textit{et al.} [Ann. Phys. (Berlin) \textbf{534}, 2100550 (2022)] posed the question of whether bound entangled states can be used for ancilla-assisted quantum process tomography (AAQPT). In this work, we answer this question in the affirmative by explicitly demonstrating that certain bound entangled states can be used for AAQPT. We further show that, although local filtering operations may improve the trace norm of the realignment criterion, they are essentially ineffective for AAQPT, as they render the resulting states \emph{unfaithful} and therefore unsuitable for reliable process reconstruction. Further, we investigate the efficiency of bound entangled states for AAQPT and establish quantitative bounds by comparing their performance with Werner and isotropic states. Our results therefore provide a new application of bound entangled states in the context of ancilla-assisted quantum process tomography.
Complete and precise characterization of a quantum dynamical process can be achieved via the method of quantum process tomography. Using a source of correlated photons, we have implemented several methods, each investigating a wide range of processes, e.g., unitary, decohering, and polarizing. One of these methods, ancilla-assisted process tomography (AAPT), makes use of an additional "ancilla system," and we have theoretically determined the conditions when AAPT is possible. Surprisingly, entanglement is not required. We present data obtained using both separable and entangled input states. The use of entanglement yields superior results, however.
Standard quantum process tomography on a ddd-dimensional input is performed by preparing several states of an input probe that then evolve under the action of the quantum channel corresponding to the progress. The final states of the probe are reconstructed by means of state tomography. An alternative is offered by ancilla-assisted process tomography: a single probe-ancilla state is used, and the correlations existing between probe and ancilla are exploited to fully reconstruct the information on the channel. In order for ancilla-assisted process tomography to be possible, the probe-ancilla input state does not need to be entangled, but still needs to have maximal operator Schmidt rank. Here we establish and analyze a framework for process tomography that interpolates between these two methods, aiming at exploiting any correlations that may exist between probe and ancilla to allow process tomography with as few input preparations as possible, when the probe-ancilla state may be operator-Schmidt-rank deficient. The main object of our investigation is the minimal number of initial local operations on the input probe for a given starting probe-ancilla state that are needed to allow process tomography. We prove that such a number scales inversely proportional to the operator Schmidt rank of the input probe-ancilla state for arbitrary local input processing. We also provide results for the case where this initial local processing is restricted to be a unitary rotation, in particular showing that the mentioned scaling is satisfied for pure input entangled states, and that in the case of mixed states there might be extreme cases where the fixed input probe-ancilla state provides information on a ≈d2/2\approx d^2/2≈d2/2-dimensional subspace of the operator space and a single additional input unitary rotation allows process tomography.
The standard procedure for quantum process tomography (QPT) requires a series of experiments. Each experiment involves initialization of the system to a particular basis state, applying the quantum process $\ensuremath{\varepsilon}$ on the system, and finally characterizing the output state by quantum state tomography (QST). The output states collected for a complete set of basis states enable us to calculate the $\ensuremath{\chi}$ matrix characterizing the process $\ensuremath{\varepsilon}$. The standard procedure for QST itself requires independent experiments, each involving measurement of a set of commuting observables. Thus QPT procedure demands a number of independent measurements and, moreover, this number increases rapidly with the size of the system. However, in ensemble systems, the total number of independent measurements can be greatly reduced with the availability of ancilla qubits. Ancilla-assisted process tomography (AAPT) has earlier been shown to require a single QST of system-ancilla space. Ancilla-assisted quantum state tomography (AAQST) has also been shown to perform QST in a single-scan measurement of an ensemble system. Here we combine AAPT with AAQST to realize a single-scan QPT (SSPT), a procedure to characterize a general quantum process in a single ensemble measurement. We demonstrate experimental SSPT by characterizing several single-qubit processes using a three-qubit NMR quantum register. Furthermore, using the SSPT procedure, we experimentally characterize the twirling process and compare the results with theory.
We present an efficient experimental procedure that certifies nonvanishing quantum capacities for qubit noisy channels. Our method is based on the use of a fixed bipartite entangled state, where the system qubit is sent to the channel input. A particular set of local measurements is performed at the channel output and the ancilla qubit mode, obtaining lower bounds to the quantum capacities for any unknown channel with no need of quantum process tomography. The entangled qubits have a Bell state configuration and are encoded in photon polarization. The lower bounds are found by estimating the Shannon and von Neumann entropies at the output using an optimized basis, whose statistics is obtained by measuring only the three observables ${\ensuremath{\sigma}}_{x}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{x}$, ${\ensuremath{\sigma}}_{y}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}\ensuremath{\bigotimes}{\ensuremath{\sigma}}_{z}$.