Ancilla-Assisted Quantum Process Tomography

• AAQPT is a quantum process tomography method that converts channel reconstruction into state tomography using bipartite probe states.
• It employs the Choi–Jamiołkowski isomorphism and faithfulness criteria to enable flexible probe states beyond just maximally entangled ones.
• Adaptive estimators and operator Schmidt rank analysis in AAQPT optimize resource scaling and sample complexity for practical implementations.

Ancilla-assisted quantum process tomography (AAQPT), often called ancilla-assisted process tomography (AAPT), is a process-identification framework in which an unknown quantum channel acts on only one subsystem of a bipartite probe state, after which ordinary state tomography is performed on the joint output. In the ideal Choi–Jamiołkowski setting, the probe is a maximally entangled state and the output is the Choi state of the channel; more generally, AAQPT allows non-maximally entangled, mixed, and even separable probe states, provided the input state is sufficiently faithful or useful for reconstruction (Lie et al., 2022, Levy et al., 2021, Singh, 18 May 2026).

1. Operational formulation and Choi-state representation

In the basic AAQPT setup, one prepares a bipartite state $\rho_{AB}$, lets the unknown channel $\mathcal E$ act only on system $A$, and obtains

$$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$

If different channels always produce different bipartite outputs, then tomography of $\mathcal E_A(\rho_{AB})$ determines $\mathcal E$ (Lie et al., 2022).

The standard state-channel representation used in AAQPT is the Choi–Jamiołkowski isomorphism. For a channel $E$ on an $n$-qubit system, the Choi matrix is written as

$\Lambda_{E} = (\mathds{I} \otimes E)\big(\ket{\phi^{+}}\bra{\phi^{+}}^{\otimes n}\big),$

with normalized Choi state

$\rho_{\Lambda} = \frac{1}{2^n}\Lambda,$

and channel action

$\mathcal E$0

This turns process tomography into state tomography on a doubled Hilbert space (Levy et al., 2021).

A closely related formulation represents the process by a process matrix $\mathcal E$1 satisfying

$\mathcal E$2

For trace-preserving channels, $\mathcal E$3. In the maximally entangled-input case, the Choi state obeys

$\mathcal E$4

so tomography of the bipartite output directly reconstructs the process matrix (Xiao et al., 7 Sep 2025).

AAQPT is therefore distinguished from standard quantum process tomography by replacing many input preparations with one correlated system–ancilla preparation and a joint-output tomography stage. The literature summarized here also emphasizes that a maximally entangled probe is not mandatory in practice; a known input state plus full joint measurement data can suffice, provided the state satisfies the relevant faithfulness criterion (Rahman et al., 2024).

2. Faithfulness, invertibility, and sensitivity

The central structural notion in AAQPT is faithfulness. A bipartite state $\mathcal E$5 is faithful on $\mathcal E$6 if the map

$\mathcal E$7

is injective on the class of channels under consideration (Lie et al., 2022).

The key object is the Jamiołkowski map associated with $\mathcal E$8,

$\mathcal E$9

or equivalently

$A$0

A rigorous characterization states that a bipartite state is faithful for process tomography on $A$1 iff its Jamiołkowski map $A$2 is left invertible; equivalently, $A$3 is surjective (Lie et al., 2022).

An equivalent criterion is expressed through realignment. For

$A$4

with swap operator $A$5, usefulness for AAQPT is equivalent to the existence of $A$6. The realignment map

$A$7

has the same singular values as $A$8, so the criterion can be stated as full-rank realignment: a bipartite input is useful for AAQPT iff $A$9 exists (Lu et al., 2022).

The same work also introduces a weaker notion, sensitivity, defined by

$$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$0

Faithfulness asks whether the channel can be recovered; sensitivity asks only whether any nontrivial channel can be detected. Faithfulness implies sensitivity in general, but the converse need not hold. For classes of maps that form a group, however, the two notions coincide; unitary operations are the canonical example (Lie et al., 2022).

For restricted channel classes, several nontrivial equivalences emerge. In particular,

$$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$1

whereas faithfulness to unitary operations alone is strictly weaker (Lie et al., 2022). For sensitivity to unital, random-unitary, and unitary channels, the relevant state-side criterion is the absence of any nontrivial local classical observable on the system side. In the paper’s terminology, a PC-Q state satisfies

$$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$2

and the characterization is that only bipartite states that has no local classical observable at all can be used to sense the effect of unital channels (Lie et al., 2022).

3. Correlations, operator Schmidt rank, and the role of entanglement

AAQPT does not require entanglement per se. A decisive quantity is the operator Schmidt rank (OSR), defined from the operator Schmidt decomposition

$$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$3

where $$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$4 and $$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$5 are orthonormal operator bases and $$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$6. In the framework of correlation-assisted process tomography, standard AAQPT is possible when

$$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$7

Thus, even a separable but sufficiently correlated state can be enough for AAPT (Caiaffa et al., 2018).

This operator-space viewpoint yields a quantitative interpolation between standard QPT and AAQPT. If $$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$8, then one can generate a faithful set using

$$\mathcal E_A(\rho_{AB}) := (\mathcal E \otimes id_B)(\rho_{AB}).$$9

local operations on the probe. The number of required local input preparations therefore scales inversely proportional to the operator Schmidt rank. For pure probe–ancilla states of Schmidt rank $\mathcal E_A(\rho_{AB})$0, a faithful set can be obtained with

$\mathcal E_A(\rho_{AB})$1

local unitaries, matching the inverse-OSR law because $\mathcal E_A(\rho_{AB})$2 for pure states (Caiaffa et al., 2018).

Several misconceptions are corrected by later work. Entanglement is not necessary for AAQPT usefulness, but it is also not sufficient. There exist entangled states $\mathcal E_A(\rho_{AB})$3 such that $\mathcal E_A(\rho_{AB})$4 is singular, hence they are useless for AAQPT. Explicit examples include a two-qutrit entangled state and a two-qutrit bound entangled state whose realigned matrices have zero singular values (Lu et al., 2022).

Conversely, certain PPT entangled and bound entangled states can be faithful. A family

$\mathcal E_A(\rho_{AB})$5

with

$\mathcal E_A(\rho_{AB})$6

can be PPT yet entangled for suitable parameters, while its realignment remains invertible by continuity. The same work gives an explicit $\mathcal E_A(\rho_{AB})$7 PPT entangled example $\mathcal E_A(\rho_{AB})$8 with all realigned singular values strictly positive, so it is faithful for AAQPT (Singh, 18 May 2026).

This literature also separates entanglement detection from tomography usefulness. The CCNR criterion tests separability through

$\mathcal E_A(\rho_{AB})$9

but AAQPT requires all singular values of $\mathcal E$0 to be nonzero. Local filtering operations may improve the trace norm of the realignment criterion, yet rank-reducing subspace filters can destroy faithfulness by creating zero singular values. The resulting state may then be unusable for AAQPT despite improved CCNR value (Singh, 18 May 2026).

4. Reconstruction schemes and estimator families

Because AAQPT maps process tomography to state tomography on the joint output, it supports several estimator families. In projected least-squares QPT, one first computes the least-squares estimator of the Choi matrix and then projects it onto the convex set of Choi matrices,

$\mathcal E$1

with

$\mathcal E$2

For AAQPT with MUB measurements, the least-squares estimator is

$\mathcal E$3

and the projection stage can be implemented numerically by the hyperplane intersection projection (HIP) algorithm (Surawy-Stepney et al., 2021).

A distinct line of work extends a two-stage solution from standard QPT to AAPT. Starting from an operator-Schmidt decomposition

$\mathcal E$4

the joint output

$\mathcal E$5

yields

$\mathcal E$6

The method reconstructs a raw process estimate and then projects it onto the physically admissible set. In this framework, the maximally entangled state is the optimal input state for AAPT (Xiao et al., 2023).

Classical-shadow methods provide another AAQPT realization. In ancilla-assisted ShadowQPT, one prepares a maximally entangled state, applies the channel to one half, and performs randomized measurements on the resulting Choi state. The snapshot estimator is

$\mathcal E$7

and averaging reconstructs the Choi matrix. This formulation permits arbitrary a posteriori evaluation of input-output quantities through

$\mathcal E$8

with favorable scaling for reduced-process tasks (Levy et al., 2021).

AAQPT can also be combined with ancilla-assisted quantum state tomography to produce single-shot process tomography (SSPT). In that construction, AAPT encodes the process into one enlarged state, AAQST reconstructs that state in a single collective measurement of commuting observables, and the process matrix $\mathcal E$9 is then obtained from the linear system

$E$0

The method was demonstrated for several single-qubit processes and a twirling process in a three-qubit NMR register (Shukla et al., 2014).

5. Resource scaling, sample complexity, and optimality

AAQPT does not automatically improve all resource measures. For non-adaptive incoherent measurements, the sample complexity of channel tomography in diamond norm is

$E$1

A lower bound applies even for ancilla-assisted strategies:

$E$2

while a matching ancilla-free upper bound holds up to logarithmic factors. In this regime, ancillas are allowed, but under non-adaptive incoherent measurements they do not improve the sample complexity (Oufkir, 2023).

By contrast, adaptivity changes the asymptotic infidelity behavior. A unified formalism for state, detector, and process tomography introduces the error metric $E$3 and proves that

$E$4

iff both the mean-squared error and the total spurious weight in the estimated zero-eigenspace scale as $E$5. Guided by this criterion, a three-step adaptive AAQPT algorithm first estimates the joint output state, then measures in the estimated eigenbasis, and finally converts the output-state estimate to a process estimate. For both trace-preserving and non-trace-preserving AAQPT, this yields

$E$6

whereas static methods typically give only $E$7 worst-case scaling (Xiao et al., 7 Sep 2025).

Finite-sample bounds are also available for projected least-squares estimators. For a $E$8-qubit channel with Choi rank $E$9, the Frobenius- and trace-norm error bounds depend explicitly on the measurement design and rank, and for low ranks the projection step improves the error rates of the least-squares estimator by a factor $n$0 (Surawy-Stepney et al., 2021). In the two-stage AAPT framework, the overall computational complexity is

$n$1

and the error upper bound depends on the measurement design, sample size, ancilla dimension, and the operator-Schmidt coefficients of the input state (Xiao et al., 2023).

These results together imply a nuanced resource picture. AAQPT can reduce the number of input preparations, can support adaptive $n$2 infidelity scaling, and can be embedded into fast reconstruction pipelines, yet ancilla assistance alone does not remove the fundamental copy complexity imposed by non-adaptive incoherent measurement models (Oufkir, 2023, Xiao et al., 7 Sep 2025).

6. Experimental realizations and related reinterpretations

AAQPT has been implemented in several experimentally distinct settings. A deployed quantum-network demonstration used one photon of a polarization-entangled pair as the ancilla and the other as the system sent through a 1.6 km deployed fiber-optic link. Using 36 joint polarization projections and Bayesian inference with preconditioned Crank–Nicolson MCMC, the experiment reconstructed the input state, output state, Choi matrix, and Pauli-basis process matrix, reporting a steady process fidelity of 95.1(1)% over a 24 h period (Rahman et al., 2024).

On near-term quantum hardware, ancilla-assisted ShadowQPT was implemented on the IonQ trapped-ion quantum computer for processes up to $n$3 qubits, using both Pauli and Clifford measurements. The work emphasizes that once the Choi state has been shadow-reconstructed, many different input-output overlaps can be evaluated classically without rerunning the device (Levy et al., 2021).

Realignment-based AAQPT has also been verified experimentally on the IBM platform. In that implementation, the channel

$n$4

was reconstructed from bipartite input-output state tomography via

$n$5

with reported input and output state fidelities $n$6 and $n$7, respectively (Lu et al., 2022).

In quantum optics, adaptive ancilla-assisted process tomography has been demonstrated on a two-qubit photonic platform, where the principal qubit is photon polarization and the ancilla qubit is photon path. The reported experiments reached, for the first time, the optimal infidelity scaling in ancilla-assisted process tomography (Xiao et al., 7 Sep 2025). Earlier NMR work implemented SSPT in a three-qubit register and characterized several single-qubit gates together with a twirling process through a single collective measurement (Shukla et al., 2014).

AAQPT has also been used as an interpretive framework outside conventional tomography. In “quantum imaging with undetected photons,” the object can be modeled as an unknown process acting on an idler subsystem, while detected signal photons function as ancilla-like degrees of freedom. In the original measurement configuration, the detector probabilities are

$n$8

so only the combination $n$9 is accessible; adding a phase shifter

$\Lambda_{E} = (\mathds{I} \otimes E)\big(\ket{\phi^{+}}\bra{\phi^{+}}^{\otimes n}\big),$0

upgrades the scheme from partial AAQPT to full tomography by making $\Lambda_{E} = (\mathds{I} \otimes E)\big(\ket{\phi^{+}}\bra{\phi^{+}}^{\otimes n}\big),$1 and $\Lambda_{E} = (\mathds{I} \otimes E)\big(\ket{\phi^{+}}\bra{\phi^{+}}^{\otimes n}\big),$2 separately recoverable (Ghalaii et al., 2015).

Across these implementations, a common pattern is that AAQPT is best viewed not as a single protocol but as a family of channel-learning methods organized around one principle: correlations convert a dynamical reconstruction problem into a state-reconstruction problem. The technical content of the modern literature concerns exactly which correlations suffice, how reconstruction should be regularized, and which resource measures are or are not improved by the ancilla.