Prescription for experimental determination of the dynamics of a quantum black box

Isaac L. Chuang; M. A. Nielsen

REVIEW

Prescription for experimental determination of the dynamics of a quantum black box

Isaac L. Chuang, M. A. Nielsen

Quantum Physics · Submitted 1 Oct 1996 · 20 references

Pith's verdict: The founding paper of quantum process tomography: the χ-matrix and N²-probe protocol are right and became standard, but the inversion is never constrained to be physical and the paper doesn't say so.

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arXiv IDquant-ph/9610001 (version 1)

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CitationIsaac L. Chuang, M. A. Nielsen. "Prescription for experimental determination of the dynamics of a quantum black box." arXiv:quant-ph/9610001 (1996). https://pith.science/paper/FINO244B

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Editor's note

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This is the paper that started quantum process tomography, and it deserves to be read as such. The core content is real: the χ-matrix parameterization of a quantum operation in a fixed operator basis, the observation that N² linearly independent inputs fix E everywhere by linearity, the β/κ generalized-inverse construction for extracting χ from measured λ, and the explicit one- and two-qubit reconstructions. The parameter count N⁴−N² is correct, the 12-parameter single-qubit count checks out, and the Appendix A argument that χ = κλ satisfies βχ = λ is valid. The Bloch-vector affine map with the polar decomposition of M is a genuinely nice piece of pedagogy that I still see quoted. Nothing in the central identifiability claim is circular — it rests on linearity of E and on measured outputs, not on a fitted parameter or a prior result of the authors' own. The reader's ACCEPT is right, and the stress-test note holds up on reading: I'd sharpen it rather than soften it. The paper says χ is 'positive Hermitian by definition' right after Eq. (3.2). That is true of the χ built from an operator-sum representation. It is not true of χ = κλ computed from finite-sample data through a generalized inverse, and the paper never distinguishes the two. Eqs. (3.9)–(3.10) then take √d_i of the eigenvalues of that reconstructed χ; a negative eigenvalue — routine on noisy data — gives an imaginary √d_i and the 'operator-sum representation' is not one. There is no projection onto the completely-positive trace-preserving set, no goodness-of-fit test, no error propagation. The paper states no limitation anywhere, so this is not an acknowledged gap being re-litigated. It is the gap the maximum-likelihood process tomography literature later filled, which is the fairest thing you can say about it. Two smaller things. The abstract's 'necessary and sufficient conditions' promises more than Eq. (3.6) delivers: that equation is necessary and sufficient for reproducing λ, not for χ being a valid quantum operation. And the two-qubit formula (4.12)–(4.13) is asserted, not derived — it's a Kronecker extension, so minor. The Choi–Jamiołkowski isomorphism is not cited, which in 1996 is forgivable but worth knowing. Eq. (2.1) also has a dropped division sign. Who should read it: anyone building or benchmarking a gate, and anyone teaching tomography who wants the clean linear-algebra version before the MLE version. Send it to referees — the math is sound and the framing gap is exactly what a referee should press on.

Referee report

Major objections (2)

[§III, Eqs. (3.6)–(3.10)] The estimator χ = κλ is not constrained to be positive semidefinite or trace-preserving. The paper states that χ is 'positive Hermitian by definition' immediately after Eq. (3.2), which is true of the true χ but not of the estimator. On noisy data, the inversion can return a χ with negative eigenvalues, and then Eq. (3.10) takes √d_i of a negative number, producing imaginary Kraus operators. The paper calls Eq. (3.6) 'a necessary and sufficient condition for the matrix χ to give the correct quantum operation E,' but it is only necessary and sufficient for χ to reproduce the measured λ; it does not enforce χ ≥ 0 or Eq. (2.3). Since the title and abstract promise an experimental prescription, this gap should be addressed: either state explicitly that the inversion is exact-data only and that noisy data require constrained estimation (e.g., maximum likelihood with a CP/TP constraint), or add a projection/goodness-of-fit step.
[§IV, Eq. (4.13)] The definition P = I ⊗ [(ρ00 + ρ12 + ρ21 + ρ33) ⊗ I] is dimensionally inconsistent with a permutation matrix. The ρ_nm are 4×4 density matrices, so the object inside the bracket is 16×16, and the outer tensor products make it larger still. As written, the two-qubit construction cannot be reproduced. Please correct the definition of P and clarify the index convention for the transpose P^T.

Minor objections (9)

[§II, Eq. (2.1)] There is a typo: 'ρ → E(ρ)tr(E(ρ))' should read 'ρ → E(ρ)/tr(E(ρ))'.
[§IV, Eqs. (4.7)–(4.8)] There is a labeling inconsistency. Working through the algebra with ρ_1 = |0⟩⟨0|, ρ_2 = |0⟩⟨1|, ρ_3 = |1⟩⟨0|, ρ_4 = |1⟩⟨1|, Eq. (4.7) evaluates to E(|1⟩⟨0|) = ρ'_3, not ρ'_2, and Eq. (4.8) evaluates to E(|0⟩⟨1|) = ρ'_2, not ρ'_3. The labels should be swapped or the definitions of ρ_2 and ρ_3 adjusted so the one-qubit construction is reproducible.
[§IV, line defining ρ_3] The line 'ρ2 = ρ1σx, ρ3 = σxρ2' gives ρ_3 = σ_x ρ_1 σ_x = ρ_4, not ρ_3; it should presumably read 'ρ3 = σxρ1'.
[§III, first paragraph] The statement that 'N² pure quantum states |ψ1⟩⟨ψ1|,..., |ψN²⟩⟨ψN²| are experimentally prepared' is loose: the reconstruction actually uses |n⟩, |m⟩, |n+⟩, and |n−⟩, which are not all of the form |ψj⟩⟨ψj| for a single fixed set. The count N² is right, but the phrasing could mislead a reader trying to implement the procedure.
[§III, after Eq. (3.2)] The statement that 'χ will contain N⁴ − N² independent parameters' should make explicit that this count assumes trace preservation (which the paper does state) and treats positivity as an inequality constraint that does not reduce the count.
[§IV, Eq. (4.17)] The term '( |α_l|² − Σ_p a_lp a*_lp ) δ_jk' would be clearer if the text noted that Σ_p a_lp a*_lp = |a_l|².
[Abstract] The abstract says 'We show necessary and sufficient conditions for this to be possible,' but the paper does not state a formal theorem; the condition is that the N² inputs are linearly independent (equivalently, that β has a generalized inverse). A sentence stating this explicitly would match the abstract.
[§V, Eq. (5.1)] The entanglement fidelity is attributed to ref. [16], which is listed as 'in preparation.' Since the paper uses this quantity as a headline application, a self-contained definition and a brief justification would help.
[References] References [8], [16], [17], and [18] are unpublished ('to appear' or 'in preparation'). Since the paper relies on [8] for the quantum operations formalism and [16] for entanglement fidelity, the editor may wish to note this.

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The paper

The paper claims

An arbitrary quantum black box can be fully characterised by feeding it a complete set of input states and measuring the outputs, yielding a finite matrix of numbers that encodes its entire dynamics.

The reading

The paper asks a practical question: if you are handed a quantum device you do not understand — a noisy logic gate, a communication channel that may contain an eavesdropper, a decohering quantum computer — can you determine its complete behaviour from the outside, by experiment alone? The authors answer yes, and give an explicit recipe. Prepare $N^2$ linearly independent input states of an $N$-dimensional system, measure each output by state tomography, and invert a fixed linear relation to recover a complex matrix $chi$ that specifies the device's quantum operation $E$ completely. The payoff is a finite, measurable fingerprint — $N^4 - N^2$ real parameters — for a system that may be coupled to an unobserved environment, and from that fingerprint one can compute fidelity, entanglement fidelity and channel capacity for the device.

What carries the argument

The load-bearing object is the $chi$ matrix — the 'classical' error-correlation matrix that, once a basis of operators $ ilde{A}_i$ is fixed, replaces the abstract quantum operation with a concrete array of complex numbers. The bridge from experiment to $chi$ is the matrix $eta$, defined by expanding each basis sandwich $ ilde{A}_m ho_j ilde{A}_n^dagger$ in the input basis $ho_k$; its generalized inverse converts measured output coefficients into $chi$. The second piece of machinery is the diagonalisation $chi = U d U^dagger$, which turns the numerical answer back into physical operators $A_i = sqrt{d_i} sum_j U_{ij} ilde{A}_j$, so that the experiment ends in the same operator-sum language theorists use. For a single qubit the authors add a geometric reading: the map becomes an affine map of the Bloch sphere, $vec{lambda} ightarrow M vec{lambda} + vec{c}$, whose polar decomposition $M = OS$ separates deformation from rotation, with relaxation-like rates read off from $M$ and $vec{c}$.

Extended reading notes

Core claim

The central claim is that any quantum operation $E$ on an $N$-dimensional state space is completely determined by its action on $N^2$ linearly independent inputs, and that the coefficients of $E$ in a fixed operator basis can be recovered from measured data by a single linear inversion. Writing $E( ho) = sum_{mn} ilde{A}_m ho ilde{A}_n^dagger chi_{mn}$ for a chosen basis of operators $ ilde{A}_i$, the authors show that the measured expansion coefficients $lambda_{jk}$ of the outputs satisfy $sum_{mn} eta_{mnjk} chi_{mn} = lambda_{jk}$, where $eta$ is computable from the basis choice alone, and that $chi = kappa lambda$ for any generalized inverse $kappa$ of $eta$ solves this relation. They then show how to diagonalise $chi$ and read off an operator-sum representation $E( ho) = sum_i A_i ho A_i^dagger$. For one qubit this means twelve parameters, obtainable from four input states; for two qubits, the same construction goes through with a Kronecker-product matrix $Lambda_2 = Lambda otimes Lambda$ acting on sixteen measured density matrices, and in general a $k$-qubit black box is fixed by $16^k - 4^k$ independent parameters.

Load-bearing premise

The procedure assumes that the $chi$ recovered by inverting noisy, finite-sample measurements is itself a physical quantum operation — positive and trace-preserving — whereas the inversion imposes no such constraint and can return maps with negative eigenvalues.

Editorial extensions

If this is right

Any experimental quantum gate, channel or memory can be assigned a complete numerical description by a fixed, finite set of state-preparation and tomography runs, rather than by fitting a theorist's model.
Because entanglement fidelity depends linearly on $chi$, errors in state preparation and measurement propagate linearly into errors in the reported gate quality, making the characterisation robust enough to compare devices.
The same procedure extends to measurements: by repeating each measurement and recording outcome frequencies, the per-outcome quantum operations $E_i$ of a generalised measurement, including a quantum-nondemolition measurement, can be reconstructed.
Since a Lindblad generator is the logarithm of a quantum operation, the measured $E$ can in principle be converted into an experimental estimate of the master-equation generator itself.
For $k$ qubits the cost is explicit: $16^k - 4^k$ parameters, which quantifies how quickly full characterisation becomes impractical as devices grow.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The recipe is an inverse problem with no regularisation: on real, finite-sample data the recovered $chi$ need not be positive semidefinite or trace-preserving, so practical use would require a constrained fit or a projection back onto the physical set, which the paper does not supply.
The stated scaling $16^k - 4^k$ makes the method a benchmarking tool for one- and two-qubit devices rather than a scalable diagnostic; a natural extension is to combine it with the assumption that noise is local, so that only few-qubit marginals need reconstructing.
Because the inversion is linear in the measured density matrices, the whole pipeline could be recast as a single linear map from raw tomography counts to $chi$, which would let error bars on gate fidelity be computed by ordinary propagation rather than by resampling.
The Bloch-sphere picture suggests a compact experimental summary: reporting the polar factors $O$ and $S$ plus the displacement $vec{c}$ would separate coherent control errors from decoherence in a way a single fidelity number cannot.

Original abstract

We give an explicit prescription for experimentally determining the evolution operators which completely describe the dynamics of a quantum mechanical black box -- an arbitrary open quantum system. We show necessary and sufficient conditions for this to be possible, and illustrate the general theory by considering specifically one and two quantum bit systems. These procedures may be useful in the comparative evaluation of experimental quantum measurement, communication, and computation systems.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,894 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.46 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We give an explicit prescription for experimentally determining the evolution operators which completely describe the dynamics of a quantum mechanical black box -- an arbitrary open quantum system. We show necessary and sufficient conditions for this to be possible, and illustrate the general theory by considering specifically one and two quantum bit systems. These procedures may be useful in the comparative evaluation of experimental quantum measurement, communication, and computation systems.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We give an explicit prescription for experimentally determining the evolution operators which completely describe the dynamics of a quantum mechanical black box -- an arbitrary open quantum system.	contribution 0.98	0.91	0.73	asserted 1.00
s1	We show necessary and sufficient conditions for this to be possible, and illustrate the general theory by considering specifically one and two quantum bit systems.	contribution 0.75	0.93	0.25	asserted 1.00
s2	These procedures may be useful in the comparative evaluation of experimental quantum measurement, communication, and computation systems.	interpretation 0.85	0.15	0.10	hedged possibility 1.00

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (20)

[1]

C. H. Bennett, G. Brassard, and A. K. Ekert, Sci. Am. 267, 50 (1992)

View on Pith 1992
[2]

R. Hughes et al. , Contemp. Physics 36, 149 (1995)

View on Pith 1995
[3]

W. G. Unruh, Phys. Rev. A 51, 992 (1995)

View on Pith 1995
[4]

I. L. Chuang, R. Laﬂamme, P. Shor, and W. H. Zurek, Science 270, 1633 (1995)

View on Pith 1995
[5]

Q. A. Turchette et al. , Phys. Rev. Lett. 75, 4710 (1995)

View on Pith 1995
[6]

C. Monroe et al. , Phys. Rev. Lett. 75, 4714 (1995)

View on Pith 1995
[7]

K. Kraus, States, Eﬀects, and Operations (Springer- Verlag, Berlin, 1983)

View on Pith 1983
[8]

B. W. Schumacher, LANL e-print quant-ph/9604023, to appear in Phys. Rev. A (1996)

View on Pith Pith review arXiv 1996
[9]

W. H. Louisell, Quantum Statistical Properties of Radia- tion (Wiley, New York, 1973)

View on Pith 1973
[10]

C. W. Gardiner, Quantum Noise (Springer-Verlag, New York, 1991)

View on Pith 1991
[11]

H. Mabuchi, quant-ph/9608020 (1996)

View on Pith Pith review arXiv 1996
[12]

M. Raymer, M. Beck, and D. McAlister, Phys. Rev. Lett. 72, 1137 (1994)

View on Pith 1994
[13]

U. Leonhardt, Phys. Rev. A 53, 2998 (1996). 5

View on Pith 1996
[14]

D. Leibfried et al. , unpublished (1996)

View on Pith 1996
[15]

R. A. Horn and C. R. Johnson, Topics in matrix analysis (Cambridge University Press, Cambridge, 1991)

View on Pith 1991
[16]

M. A. Nielsen, B. W. Schumacher, C. M. Caves, and H. Barnum, in preparation (1996)

View on Pith 1996
[17]

S. Lloyd, LANL e-print quant-ph/9604015, submitted to pra (1996)

View on Pith Pith review arXiv 1996
[18]

B. W. Schumacher and M. A. Nielsen, LANL e-print quant-ph/9604022, to appear in Phys. Rev. A (1996)

View on Pith Pith review arXiv 1996
[19]

V. B. Braginsky and F. Y. Khalili, Quantum Measure- ment (Cambridge Unviersity Press, Cambridge, England, 1992)

View on Pith 1992
[20]

L. Ljung, System Identiﬁcation: Theory for the User (Prentice Hall PTR, Upper Saddle River, 1987). 6

View on Pith 1987

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Optimal learning of covariant quantum states and channels
quant-ph 2026-09 conditional novelty 7.0 of 10

Known symmetries reduce the copy and query cost of quantum tomography to sharp bounds, and the Hayashi optimal pure-state measurement gains an efficient circuit implementation.
Efficient estimation of Pauli channels
quant-ph 2019-07 conditional novelty 7.0 of 10

Pauli channels on n qubits can be estimated to relative precision epsilon with O(epsilon^-2 n 2^n) measurements, and structured Pauli channels with O(epsilon^-2 n^2 log n) measurements, using a randomized benchmarking...
Pseudo-randomness and Learning in Quantum Computation
quant-ph 2010-06 accept novelty 7.0 of 10

Random quantum circuits are efficient approximate 2-designs, and mixing a classical 2k-tensor-product expander with a Fourier transform yields an efficient quantum k-design for k = O(n/log n).
Distance measures to compare real and ideal quantum processes
quant-ph 2004-08 accept novelty 6.0 of 10

Six physically motivated criteria narrow the field of quantum-process error measures to the stabilized process distance and stabilized process fidelity, which bound worst-case failure probabilities in quantum computation.
Connecting Quantum Tomography and Quantum Retrodiction
quant-ph 2026-06 unverdicted novelty 5.0 of 10

The Petz recovery map equals the gradient of the log-likelihood in maximum-likelihood tomography, unifying retrodiction and state reconstruction via a shared iterative procedure.

Show 1 more citing papers

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hep-th 2026-06 unverdicted novelty 2.0 of 10

The paper organizes important open questions in quantum gravity and quantum information into four themes without presenting new results or derivations.

How later papers used it

239 later papers cite it

Evidence: substantial. 239 of the citing papers share no author with it, and 42 of those carry a confident label from Pith's citation classifier.

Pith read 38 of the 341 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

15 later papers by other authors are listed below: 15 used its method or tool.

Uses its method or tool 15 of 40 papers with this label

The measurements are then used to reconstruct the quantum process matrix [ref] in the Pauli basis $σ_0, σ_x, σ_y, σ_z$ with a maximum likelihood algorithm.

Full Bell-basis measurement of an atom-photon 2-qubit state and its application for quantum networks (2023) · Step 2: Bell-state projection · 1.00
Used quantum process tomography from this paper to seek more explicit information on whether errors result from miscalibration, decoherence, or parasitic Hamiltonian terms.

Extensive characterization of a family of efficient three-qubit gates at the coherence limit (2022) · 1.00 · paraphrased: the licence does not allow quoting
Computed the process matrix from four input states and simulated output states using this paper's method.

Adiabatic quantum state transfer in a semiconductor quantum-dot spin chain (2020) · 1.00 · paraphrased: the licence does not allow quoting
Determined the T-gate process fidelity using quantum process tomography from this paper with readout error mitigation.

Blind Quantum Computation on a Modular Superconducting Processor (2026) · 1.00 · paraphrased: the licence does not allow quoting
Performed full quantum process tomography from this paper to characterize qubit state transfer from node A to node B.

Deterministic Quantum State Transfer and Generation of Remote Entanglement using Microwave Photons (2017) · 1.00 · paraphrased: the licence does not allow quoting
Extracted the process maps using this paper's method, assuming ideal input states, and calculated process fidelities with respect to ideal process maps.

Realization of a Universal Quantum Gate Set for Itinerant Microwave Photons (2021) · 1.00 · paraphrased: the licence does not allow quoting
Computed the process matrix from the density matrices following the procedure in this paper.

Realizing Quantum Convolutional Neural Networks on a Superconducting Quantum Processor to Recognize Quantum Phases (2021) · 1.00 · paraphrased: the licence does not allow quoting
Performed quantum process tomography from this paper to characterize the STIRAP and saSTIRAP protocols for the open-system qutrit case.

Majorana representation of adiabatic and superadiabatic processes in three-level systems (2020) · 1.00 · paraphrased: the licence does not allow quoting
Performed quantum process tomography from this paper by preparing input states at the sender and measuring output states at the receiver.

Deterministic Quantum Communication Between Fixed-Frequency Superconducting Qubits via Broadband Resonators (2025) · 0.99 · paraphrased: the licence does not allow quoting
Extracted the process matrix describing the transformation from input to output density matrix in terms of the Pauli operators, following this paper.

Enhanced Quantum Interface with Collective Ion-Cavity Coupling (2014) · 0.98 · paraphrased: the licence does not allow quoting
Applied the standard quantum process tomography technique from this paper to characterize processes in any dimension d.

Characterizing $d-$dimensional quantum channels by means of quantum process tomography (2018) · 0.97 · paraphrased: the licence does not allow quoting
The single-ququart process tomography (QPT) [ref] is performed by sandwiching the unknown quantum channel $ℰ$ with different combinations of pre-rotation and post-rotation gates, where the pre-rotation set $𝒮$ is shown in Table [ref].

Exploring Ququart Computation on a Transmon using Optimal Control (2023) · Tomography and readout histogram · 0.97
We finally demonstrate applications in full characterization of the quantum channel through quantum process tomography [ref] .

Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons (2018) · Introduction · 0.95
Used this paper's PT algorithm, which assumes IID, to get the matrix describing the distribution of measurement results between Alice and Bob in its QKD protocol.

Efficient Device-Independent Quantum Key Distribution (2023) · 0.92 · paraphrased: the licence does not allow quoting
The process matrix describes how the protocol maps the prepared memory qubit onto the photonic qubit at the HTW in terms of the Pauli operators ${σ_0, σ_x, σ_y, σ_z}$ [ref] .

Demonstration of quantum network protocols over a 14-km urban fiber link (2024) · Ion-photon entanglement and ion-to-photon quantum state teleportation · 0.89

How the literature uses it

33 arXiv papers cite this in 53 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 35 sentences without a confident label

Uses its method or tool 3 papers

For a fixed value of $t$, Alice can use process tomography [ref] to determine the unitary operator $Û(t)$.

Why protective measurement does not establish the reality of the quantum state · Resource counting argument · 0.95
A unique artifact of this choice of quantum gates is that one can decompose multi-controlled gates using the traditional Toffoli staircase decomposition [ref] .

Primitive Quantum Gates for an SU(3) Discrete Subgroup: $\Sigma(36\times3)$ · Basic Gates · 0.92
Quantum process tomography [ref] is performed by preparing the input states at the sender and measuring the output states at the receiver.

Deterministic Quantum Communication Between Fixed-Frequency Superconducting Qubits via Broadband Resonators · Deterministic quantum state transfer and generation of remote entanglement · 0.99

Uses its data or values 1 paper

First, while the CNOT is transversal, the Toffoli gate decomposes into six CNOTs and seven T gates [ref] .

Primitive Quantum Gates for an SU(3) Discrete Subgroup: $\Sigma(36\times3)$ · Resource Costs · 0.87

Cites it as background 9 papers

The single-process application techniques include, for example, Standard QPT [ref] , Ancilla-Assisted QPT (AAPT) [ref] , Direct Characterization of Quantum Dynamics (DCQD) [ref] , Compressed sensing QPT [ref] , Gradient-Descent QPT [ref] , Self-Consistent QPT [ref] , Projected Least-Squares QPT [ref] , etc.

Multipass Quantum Process Tomography: Precision and Accuracy Enhancement · Introduction · 0.93
One convenient choice of basis operators is [ref] : [equation] where: [equation] We can now define the process matrix representation, which is expressed as: [equation] where $χ_nm = a_n a_m^†$, $χ∈ℂ^b^2 ×b^2$. $χ$ thus contains the products of the coefficients of basis operators $Ã_n$ and $Ã_m$.

Using non-convex optimization in quantum process tomography: Factored gradient descent is tough to beat · Notation and QPT-related definitions · 0.81
Quantum process tomography [ref] is the standard for the verification and characterization of quantum operations on well-controlled quantum systems.

Scalable reconstruction of unitary processes and Hamiltonians · Introduction · 1.00

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

C. H. Bennett, G. Brassard, and A. K. Ekert, Sci. Am. 267, 50 (1992)

View on Pith 1992

[2] [2]

R. Hughes et al. , Contemp. Physics 36, 149 (1995)

View on Pith 1995

[3] [3]

W. G. Unruh, Phys. Rev. A 51, 992 (1995)

View on Pith 1995

[4] [4]

I. L. Chuang, R. Laﬂamme, P. Shor, and W. H. Zurek, Science 270, 1633 (1995)

View on Pith 1995

[5] [5]

Q. A. Turchette et al. , Phys. Rev. Lett. 75, 4710 (1995)

View on Pith 1995

[6] [6]

C. Monroe et al. , Phys. Rev. Lett. 75, 4714 (1995)

View on Pith 1995

[7] [7]

K. Kraus, States, Eﬀects, and Operations (Springer- Verlag, Berlin, 1983)

View on Pith 1983

[8] [8]

B. W. Schumacher, LANL e-print quant-ph/9604023, to appear in Phys. Rev. A (1996)

View on Pith Pith review arXiv 1996

[9] [9]

W. H. Louisell, Quantum Statistical Properties of Radia- tion (Wiley, New York, 1973)

View on Pith 1973

[10] [10]

C. W. Gardiner, Quantum Noise (Springer-Verlag, New York, 1991)

View on Pith 1991

[11] [11]

H. Mabuchi, quant-ph/9608020 (1996)

View on Pith Pith review arXiv 1996

[12] [12]

M. Raymer, M. Beck, and D. McAlister, Phys. Rev. Lett. 72, 1137 (1994)

View on Pith 1994

[13] [13]

U. Leonhardt, Phys. Rev. A 53, 2998 (1996). 5

View on Pith 1996

[14] [14]

D. Leibfried et al. , unpublished (1996)

View on Pith 1996

[15] [15]

R. A. Horn and C. R. Johnson, Topics in matrix analysis (Cambridge University Press, Cambridge, 1991)

View on Pith 1991

[16] [16]

M. A. Nielsen, B. W. Schumacher, C. M. Caves, and H. Barnum, in preparation (1996)

View on Pith 1996

[17] [17]

S. Lloyd, LANL e-print quant-ph/9604015, submitted to pra (1996)

View on Pith Pith review arXiv 1996

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