PaperlayerView exact revision on arXiv

Source-linked AI summary

Tensor networks and graphical calculus for open quantum systems

Christopher J. Wood, Jacob D. Biamonte, David G. Cory

arXiv:1111.6950v3quant-ph

TL;DR

The paper addresses the difficulty of relating multiple representations of completely positive maps in open quantum systems. It develops a tensor-network graphical calculus for representing and transforming these maps, then demonstrates the approach through channel constructions, process tomography, and fidelity derivations. The resulting calculus unifies common representations and supplies concise graphical transformations and proofs.

  • Problem

    Transformations among the numerous representations of completely positive maps are often cumbersome, motivating an intuitive framework for their unification and interoperability.

  • Method

    The paper casts open-systems theory into Penrose-style tensor networks expressed in Dirac notation, using wire manipulations to represent and transform map representations.

  • Results

    The calculus unifies common CPTP-map representations and supports graphical derivations for composite channels, ancilla-assisted process tomography, and channel fidelity quantities.

  • Takeaways & Limitations

    Diagrammatic manipulation provides concise transformations and proofs across the Kraus, system-environment, superoperator, Choi, and related representations.

Abstract

from arXiv · show

We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate fidelity and entanglement fidelity of a channel in terms of each of the different representations of the channel.

I. INTRODUCTION

The paper presents a tensor-network graphical calculus to unify representations of completely positive maps and make their transformations more intuitive. It introduces the notation and demonstrates applications to channel composition, process tomography, and fidelity quantities.

  • Motivation: Open-system evolution is modeled by completely positive, trace-preserving maps rather than necessarily unitary dynamics.A quantum channel acts linearly on density operators, preserves positivity under extensions, and preserves trace.
  • Motivation: The paper addresses the cumbersome transformations among several mathematically equivalent representations of completely positive maps.Its graphical calculus aims to provide an intuitive unification and interoperability among these representations.
  • Graphical calculus: The approach combines Penrose-style tensor diagrams with Dirac notation to manipulate, visualize, and contract quantum circuits and open-system equations.States, effects, operators, and scalars are represented by triangles, boxes, and diamonds, while wire orientation identifies tensor type.
  • Applications: Applications include composite and reduced channels, a simpler condition and reconstruction method for ancilla-assisted process tomography, and graphical derivations of channel fidelity formulas.The paper explicitly presents these as demonstrations of the calculus rather than new results.
  • Graphical calculus: Tensor-network diagrams represent linear equations involving tensor composition and contraction, and diagram manipulations yield equivalent equations.The notation uses open wires for tensor indices and adopts a right-to-left orientation convention.

A. Bipartite Matrix Operations

The paper treats bipartite matrices as fourth-order tensors and expresses partial traces, transposition, swaps, and reshuffling as tensor-index operations. These operations provide the graphical primitives needed for manipulating representations of completely positive maps.

  • Bipartite representation: Bipartite matrices on X ⊗ Y can be represented as fourth-order tensors with indices for each subsystem input and output.The paper also gives a two-index representation using a standard basis for X ⊗ Y.
  • Bipartite representation: The equivalence between composite indices and subsystem indices is fixed by α = d_y m + μ and β = d_y n + ν.Here d_y is the dimension of Y.
  • Bipartite operations: The principal bipartite operations are partial traces, transposition, bipartite-SWAP, and row- and column-reshuffling.Their tensor actions are specified by summing or permuting subsystem indices.
  • Bipartite operations: The notation treats reshuffling as a reindexing operation and distinguishes row-reshuffling from the col-reshuffling convention used generally for R.Partial transpose and partial-SWAP can be represented by acting only on the wires associated with one subsystem.

B. Vectorization of Matrices

Vectorization reshapes matrices into vectors so they can be used in the superoperator formalism, with column-stacking, row-stacking, and arbitrary operator-basis conventions. The paper connects these conventions to wire manipulations and to broader representations of CPTP maps.

  • Vectorization conventions: Vectorization reshapes an m × n matrix into a vector and is required for describing open-system dynamics in the superoperator formalism.The paper distinguishes column-stacking and row-stacking conventions.
  • Vectorization conventions: Col-vec and row-vec representations are related by exchanging wires through the SWAP operation.The graphical definitions arise by bending a wire in opposite directions.
  • Vectorization conventions: The unnormalized Bell state is the vectorized identity operator, linking vectorization to bipartite-state notation.This identification is used in graphical definitions of vectorization.
  • Basis changes: Vectorization in an arbitrary orthonormal operator basis extracts the coefficients of the basis elements into a standard-coordinate vector.Basis-change operators transform between vectorized representations in different orthonormal operator bases.
  • Basis changes: The paper uses col-vec by default and treats changes to arbitrary operator bases or row-vec as explicit transformations.Roth’s lemma supplies the vectorization rule for matrix products, while the CPTP-map framework includes Kraus, Stinespring, superoperator, and Choi representations.
  • CPTP-map representations: The Kraus representation expresses a CPTP map through operators satisfying a completeness relation, but the Kraus operators are not unique because of unitary freedom.The canonical Kraus representation selects an orthogonal operator set for transformations between representations.

B. System-Environment / Stinespring Representation

The system-environment representation models open-system evolution through a system coupled to an environment, typically initialized in a pure state and evolved jointly before reducing to the principal system. It is physically intuitive but non-unique and can be cumbersome when environmental dynamics are irrelevant; the paper therefore presents superoperator and Choi-matrix alternatives.

  • System-environment model: The system-environment model represents open-system evolution by coupling a principal system to an environment and evolving their joint state.The model is closely related to the Stinespring representation.
  • System-environment model: The environment may be assumed to start in a pure state, with dimension at most d^2 when the system has dimension d.
  • System-environment model: Different system-environment interactions can produce the same reduced dynamics, so the representation is not unique.The freedom includes choosing the environment’s initial state and adjusting the joint unitary accordingly.
  • System-environment model: The system-environment description can be cumbersome for calculations that do not require explicit environmental dynamics.The paper identifies other descriptions as more convenient in such contexts.
  • Scope and assumption: Complete positivity of the principal-system evolution is guaranteed when the initial system-environment state is separable, while initially correlated states can yield non-completely-positive reduced dynamics.The correlated case is outside the paper’s scope.
  • Alternative representations: The paper introduces superoperator and Choi-matrix descriptions as alternative representations, with the Choi matrix supporting complete-positivity tests through its eigenvalues.The superoperator is unique for a chosen vectorization basis, whereas the Choi matrix is unique for a chosen isomorphism convention.

E. Process Matrix Representation

The process matrix χ is a basis-dependent representation of a CPTP map obtained by expressing the map in an orthonormal operator basis. It is related to the Choi matrix by a unitary basis change and inherits its hermitian-preservation and complete-positivity structure.

  • Definition: A CPTP map can be expressed as a process matrix χ relative to an orthonormal operator basis.The process matrix is unique once the operator basis is chosen.
  • Relation to Choi matrix: The process matrix is related to the Choi matrix by a vectorization basis-change transformation.
  • Relation to superoperator: Using the col-vec operator basis makes the process matrix equal to the superoperator, χ = Λ.
  • Structural properties: Because the process matrix is a unitary transformation of the Choi matrix, it shares the Choi matrix’s structural conditions for hermitian preservation and complete positivity.Trace-preservation conditions depend on the matrix elements and chosen basis.
  • Basis changes: Changing from one orthonormal operator basis to another uses the corresponding superoperator-style basis transformation.

IV. TRANSFORMING BETWEEN REPRESENTATIONS

The graphical calculus organizes transformations among CP-map representations using wire-bending dualities and diagrammatic constructions. Reshuffling gives a unique reversible equivalence between Choi matrices and superoperators, while mappings involving Kraus or system-environment forms are generally one-way because those representations are non-unique.

  • Transformation framework: Transformations among CP-map representations use wire bending for reshuffling, vectorization, and the Choi-Jamiołkowski isomorphism.Other transformations are nonlinear decompositions or constructions represented diagrammatically.
  • Choi and superoperator: The Choi matrix and superoperator are related by bipartite reshuffling in the corresponding col- or row-vectorization convention.
  • Choi and superoperator: Reshuffling is self-inverse, making the Choi–superoperator transformation linear, bijective, and reversible.It is the only transformation among the considered representations with all three properties.
  • Kraus and system-environment mappings: Kraus operators and system-environment representations can be mapped to superoperators by vectorization, but the mappings are only surjective.Injectivity fails because Kraus and system-environment representations are non-unique.
  • Construction of superoperators: A Kraus representation produces a superoperator through vectorization, while a system-environment representation produces one from the joint unitary and initial environment state.The graphical proofs use tensor contractions and Roth’s lemma.

C. Transformations to the Choi-matrix representation

The paper describes transformations from Kraus and system-environment representations to the Choi matrix, then explains how Kraus and Stinespring forms are obtained through nonlinear decompositions. These transformations are governed by surjectivity, non-uniqueness, basis choices, and fixed-environment restrictions.

  • Transformations to the Choi matrix: Kraus and system-environment representations map to the Choi matrix through the Choi–Jamiołkowski wire-bending duality.The transformations are surjective but not injective because the source representations are non-unique.
  • Choi to Kraus: The graphical calculus then turns the Choi matrix into Kraus operators through spectral decomposition, with the number of Kraus operators equal to the Choi matrix rank.
  • System-environment to Kraus: Kraus representations can also be obtained from system-environment representations by decomposing the partial trace in a chosen environment basis.
  • Kraus and Stinespring: For a fixed environment basis and initial state, the partial-trace transformation between Kraus and Stinespring representations is invertible and therefore bijective.The inverse is Stinespring dilation, while the transformation remains nonlinear because it depends on a basis choice.
  • Kraus to system-environment: Constructing a system-environment representation from a superoperator requires reshuffling to the Choi matrix, spectral decomposition to canonical Kraus operators, and then Stinespring dilation.Completing the full joint unitary beyond the initial-environment subspace is unnecessary for describing the CP-map evolution and can be cumbersome.

V. APPLICATIONS

The applications section develops tensor-network tools for composite-system vectorization, including unravelling, subsystem wire permutations, and Bell-state constructions.

  • Composite-system vectorization: Composite vectors can be represented with either one concatenated index or separate subsystem indices, connected by index permutations.The tensor-network representation makes these permutations explicit through SWAP operations.
  • Bell-state construction: The unnormalized Bell state is defined as the vectorization of the identity operator on a composite system.Its tensor network provides a graphical primitive for representing vectorized composite operators.
  • Bell-state construction: Vectorizing a composite operator bends all system wires upward while preserving the order of the subsystems.This gives an equivalent graphical representation of the column-vectorized operator.
  • Composite-system vectorization: The graphical calculus uses vectorization and unravelling to relate joint-system tensors to tensors on individual subsystems.The unravelling operation separates vectorized composite operators into subsystem factors, with an inverse that restores the joint representation.
  • Graphical examples: For N = 2, 3, and 4, the paper illustrates explicit tensor networks and a graphical proof of the unravelling construction.These examples demonstrate how the general operation is realized diagrammatically.

B. Composing superopators

This section shows how tensor-network unravelling constructs composite superoperators and represents common matrix operations, including trace, transpose, and partial transformations.

  • Composing superoperators: The tensor product S1 ⊗ S2 acts on separately vectorized inputs, so unravelling is required for the vectorization of a composite input.The correct joint superoperator is built by combining subsystem superoperators with the unravelling operation and its inverse.
  • Composing superoperators: A joint superoperator is constructed from subsystem superoperators {Sk} by applying the corresponding tensor-network rearrangement.The construction is illustrated explicitly for N = 2 and N = 3.
  • Basis transformations: The same composition framework supports Pauli-basis transformations for N-qubit systems and operations on selected subsystems.The paper notes that col-vec and row-vec calculations may be computationally efficient, while Pauli-basis forms can be convenient for analysis.
  • Matrix operations: The trace of a vectorized matrix is implemented by a trace superoperator represented graphically through the adjoint of the unnormalized Bell state.For composite systems, subsystem traces and other operations are inserted alongside identity superoperators.
  • Matrix operations: The transpose superoperator is a swap superoperator, and bipartite operations include partial traces, transposition, and col-reshuffling.These operations receive direct tensor-network representations for composite matrices.

D. Reduced superoperators

The reduced-superoperator application constructs an effective map for one subsystem by initializing and post-selecting an ancilla, while the tomography example gives a recovery condition for unknown channels.

  • Reduced superoperators: A larger superoperator on X ⊗ Y can be reduced to an effective map on X when Y is initialized in τ0 and post-selected in τ1.The construction applies the composite superoperator between the ancilla preparation and post-selection operations.
  • Ancilla-assisted process tomography: Quantum process tomography reconstructs an unknown channel from measurement statistics, while AAPT obtains the channel through a bipartite input and measured output state.EAPT is the maximally entangled special case, where the output is the rescaled Choi matrix.
  • Ancilla-assisted process tomography: AAPT succeeds exactly when the reshuffled input density matrix ρAS is invertible, equivalently when its Schmidt number equals d2.The measured output can then be post-processed with an inverse recovery map to reconstruct the Choi matrix.
  • Ancilla-assisted process tomography: The graphical proof treats the bipartite input state as the Choi matrix of an effective channel and composes superoperators to derive recovery.For maximally entangled inputs, the state has the form ρAS = |V⟩⟩⟨⟨V| for a unitary V.
  • Ancilla-assisted process tomography: When the input is not maximally entangled, increasing closeness to a singular matrix increases the condition number and error amplification during inversion.This identifies numerical conditioning as a practical boundary for channel reconstruction.

F. Average Gate Fidelity

The paper derives average gate fidelity through tensor-network manipulations, reducing the state average to channel-representation expressions and extending the method to higher moments.

  • Gate fidelity: Gate fidelity measures the closeness of a noisy channel E to a desired channel F, often a unitary map.The comparison can be reduced to fidelity with the identity channel by composing with the target unitary’s adjoint.
  • Average gate fidelity: Average gate fidelity is obtained by averaging the gate-fidelity function over pure states under the Fubini–Study measure.The paper uses the pure-state form FE(|ψ⟩⟨ψ|) = ⟨ψ|E(|ψ⟩⟨ψ|)|ψ⟩.
  • Graphical derivation: The graphical derivation uses the symmetric-subspace projector to evaluate averages of tensor products of pure states.Permutation operators and SWAP constructions provide the tensor-network form of the symmetric projector.
  • Channel representations: The resulting average gate fidelity is expressed in terms of the channel’s Choi matrix and, by transformations, its other representations.The paper states that corresponding expressions are obtained for the superoperator, Kraus, χ-matrix, and Stinespring representations.
  • Higher moments: The tensor-network method can be extended to n > 2 state moments, where permutation operators contain n! wire permutations.For n = 3, the permutations can be decomposed into SWAP gates, supporting higher-order fidelity calculations.

G. Entanglement Fidelity

The paper uses graphical techniques to derive entanglement-fidelity expressions from the Choi representation and transform them into equivalent forms for other channel representations. These examples illustrate broader applications of the calculus to channel constructions, tomography, and fidelity proofs.

  • Entanglement fidelity: Entanglement fidelity is independent of the particular purification used for the input state.The paper presents this independence before deriving equivalent representation-specific expressions.
  • Graphical derivation: Graphical tensor manipulations provide a simple derivation in the Choi representation, with transformations to other representations supplied through the channel-conversion framework.The paper directs the transformations to Appendix C.
  • Representation changes: The resulting entanglement-fidelity formulas are given for the superoperator, Choi-matrix, Kraus, χ-matrix, and Stinespring representations.The representations are denoted by S, Λ, {K_j}, χ, and A, respectively.
  • Applications: The calculus constructs composite and effective reduced-system superoperators by vectorizing composite systems and transforming between joint and individually vectorized descriptions.These constructions support updating or modifying subsets of composite systems.
  • Applications: Reshuffling bipartite states into effective superoperators yields a succinct necessary-and-sufficient condition for ancilla-assisted process tomography and a simpler recovery construction.The paper describes the result as equivalent to a previously known condition.
  • Applications: Graphical representations also shorten the proof of average gate fidelity by expressing state averages through the symmetric-subspace projector and permutation operators.Tensor-network manipulations rearrange copies of |ψ⟩⟨ψ| before the symmetric-subspace substitution.

Appendix A: Tensor network proofs

The appendix establishes the consistency of the paper’s tensor-network primitives through algebraic and diagrammatic proofs. It covers summation, traces, snake identities, transposition, Bell-state contractions, basis changes, and fidelity transformations.

  • Basic tensor-network identities: The appendix introduces color summation as a diagrammatic convention for summing tensor indices, including Kronecker-delta contractions.The convention is used in subsequent proofs.
  • Basic tensor-network identities: Tensor-network trace diagrams correspond algebraically to Bell-state contractions such as ⟨Φ+|A ⊗ 1l|Φ+⟩ = Tr[A].The appendix uses these identities to verify the trace construction.
  • Wire bending and snake equations: Snake equations are proved from equivalences for tensor products of basis elements and from the reflected wire-bending construction.The appendix treats both the S-bend and its reflection.
  • Transposition: The appendix verifies transposition tensor networks algebraically and by counter-clockwise wire bending, including a Bell-state contraction proof.These proofs rely on the equivalence relations established for the graphical calculus.
  • Vectorization basis changes: For orthonormal operator bases, the vectorization change-of-basis operator Tσ→ω is shown to be unitary.The construction uses the computational basis of X ⊗ X.
  • Fidelity derivations: Average gate-fidelity and entanglement-fidelity expressions are derived for multiple channel representations by applying the channel transformations to the Choi-matrix formulation.The χ-matrix derivation assumes an orthonormal basis satisfying Tr[σ_j] = δ_j0.
Loading 1111.6950v3…