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Comparison of Quantum Channels by Superchannels

Gilad Gour

arXiv:1808.02607v3quant-phmath-phmath.MP

TL;DR

The paper asks how to decide whether one set of quantum channels can be transformed into another by a single superchannel. It extends conditional min-entropy to bipartite channels and uses it to characterize this comparison, including through semidefinite programming. The characterization also extends quantum majorization from states to channels.

Abstract

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We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of "quantum majorization" from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels, and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.

I. INTRODUCTION

The paper extends comparison of quantum states to dynamical processes: channels are compared by whether one superchannel can transform one set into another. It proposes the extended conditional min-entropy as a complete family of distinguishability measures for this comparison.

I. INTRODUCTION

  • A superchannel maps quantum channels to quantum channels and is physically realized by pre- and post-processing the channel on which it acts.The paper motivates this framework by treating channels as dynamical processes and superchannels as their most general evolution.Source: §I · p. 1
  • Channel distinguishability must be monotone under superchannels, just as state distinguishability is monotone under quantum channels.The proposed comparison generalizes the state-conversion question, previously characterized for finite dimensions using semidefinite programming and conditional min-entropy.Source: §I · pp. 1–2
  • The central problem is whether a single superchannel can map each channel in one set to its paired channel in another set.The paper states necessary and sufficient conditions for this conversion and connects them to extended conditional min-entropy.Source: §I · p. 2

A. The space of linear maps

  • The paper sets up linear maps and supermaps as vector spaces, with associated inner products and Choi-matrix representations.Multipartite channels are described by assigning each party input and output systems, including a trivial one-dimensional input when a party holds only an output subsystem.Source: A. The space of linear maps · p. 2, B. The space of supermaps · p. 3

C. Completely Positive Preserving (CPP) Maps and Superchannels

This section defines the preservation properties that distinguish superchannels from general maps on channels. It gives equivalent characterizations that connect these properties to positivity, trace preservation, and physical realization.

C. Completely Positive Preserving (CPP) Maps and Superchannels

  • A superchannel is a map that is completely CP preserving and TP preserving; complete CP preservation requires positivity even when extended by any auxiliary system.CP preserving means mapping every CP map to a CP map, while TP preserving means mapping every TP map to a TP map.Source: C. Completely Positive Preserving… · p. 4
  • The theorem characterizes a superchannel through equivalent conditions, including positivity and specified marginal constraints on its Choi matrix.The proof also establishes an equivalent physical realization using channels with pre- and post-processing.Source: C. Completely Positive Preserving… · pp. 4–5
  • A superchannel corresponds to a CPTP map with the input and output systems rearranged, subject to the stated marginal condition.The associated map takes A1 and B0 as inputs and A0 and B1 as outputs.Source: C. Completely Positive Preserving… · p. 6

D. Entropies

The paper extends state entropy ideas to channels by requiring entropy functions to reflect noise under specified transformations. It develops the conditional min-entropy and gives its operational interpretation when one subsystem is classical.

D. Entropies

  • Entropy functions for states are required to be monotone under random unitary channels, additive under tensor products, and normalized on maximally mixed and pure states.The paper notes that other properties, such as concavity and subadditivity, are not among these three baseline requirements.Source: D. Entropies · pp. 6–7

1) The conditional min-entropy:

  • The conditional min-entropy can be computed by semidefinite programming, with a dual formulation that optimizes over CPTP maps.The paper defines it for a bipartite state and notes that its value reduces to the min-entropy when the conditioning system is one-dimensional.Source: 1) The conditional min-entropy · p. 7
  • When A1 is classical, the conditional min-entropy expression gives the optimal probability of guessing A1 after measuring A0.Source: 1) The conditional min-entropy · p. 7

2) Entropies and support functions of convex sets:

The paper connects entropy functions to convex-set support functions and introduces noise models for superchannels. It distinguishes random unitary superchannels from the broader doubly stochastic class and gives the latter a physical interpretation.

2) Entropies and support functions of convex sets:

A. Random Unitary Superchannels

  • A random unitary superchannel is a convex combination of reversible superchannels, corresponding to applying a reversible transformation and forgetting which one was used.The definition is given for matching input and output dimensions.Source: A. Random Unitary Superchannels · p. 8

B. Doubly Stochastic Superchannels

  • A superchannel is doubly stochastic when both it and its dual are superchannels; this class is larger than the random unitary superchannels.The paper identifies these maps with operations that do not decrease noise.Source: B. Doubly Stochastic Superchannels · pp. 8–9
  • Doubly stochastic superchannels have a physical characterization through their realization and the condition on the post-processing channels.The theorem gives equivalent Choi-matrix and realization conditions; the class need not have the random unitary form.Source: B. Doubly Stochastic Superchannels · pp. 8–9

C. Completely Uniformity Preserving Superchannel

For equal input and output dimensions, the section characterizes superchannels that preserve marginal uniformity, including when an ancillary system is present, and relates this property to doubly stochastic superchannels.

C. Completely Uniformity Preserving Superchannel

  • Marginally uniform means the output on A1 is maximally mixed for every input, while correlations with an ancillary system may remain.The marginal on the ancillary system is not required to be maximally mixed.Source: C. Completely Uniformity Preserving… · p. 9
  • For equal input and output dimensions, a superchannel that does not decrease noise is completely uniformity preserving if it maps channels marginally uniform on A to channels marginally uniform on B, even with an ancillary system.The figure depicts this action; its curly arrows indicate maximally mixed states.Source: C. Completely Uniformity Preserving… · pp. 9–10, Figure 4 · p. 10
  • Theorem 3 gives equivalent characterizations through a Choi-matrix condition and a realization involving a channel Γ with an additional condition.The characterization applies when the input and output dimensions agree.Source: C. Completely Uniformity Preserving… · p. 10
  • Doubly stochastic superchannels are completely uniformity preserving, but the converse fails because that property does not require the additional Choi condition associated with a unital superchannel.The section describes uniformity preservation as analogous to preserving the maximally mixed state; it also notes that the complete condition is stronger than the preceding noise-preservation condition.Source: C. Completely Uniformity Preserving… · pp. 9–11

D. Completely Unital-Channel Preserving Superchannels

This section studies superchannels that preserve unital channels, including under extension by an ancillary system. It gives equivalent structural characterizations and relates preservation of random-unitary channels to preservation of all unital channels.

D. Completely Unital-Channel Preserving Superchannels

  • The motivation is that a noise-increasing superchannel should not turn a random-unitary channel into a non-unital channel.The section focuses on the stronger property of completely preserving unital channels.Source: D. Completely Unital-Channel… · p. 11
  • A superchannel that maps random-unitary channels to unital channels also maps every unital channel to a unital channel.The argument uses that unital channels can be expressed as affine linear combinations of unitary channels.Source: D. Completely Unital-Channel… · p. 11
  • For a unital-channel-preserving superchannel, the Choi matrix has a specified two-term form, and complete preservation forces the β term to vanish.The two terms are of forms I_A0 ⊗ α_A1B1 and I_A1 ⊗ β_A0B1.Source: D. Completely Unital-Channel… · p. 11
  • Complete unital-channel preservation requires the superchannel to preserve unitality even when extended by an arbitrary ancillary system.The section considers equal input and output dimensions for both channel systems.Source: D. Completely Unital-Channel… · p. 11
  • Theorem 4 gives equivalent characterizations using Choi-matrix marginals and a realization through a unital CPTP map Γ subject to an additional condition.The theorem states equivalence between complete preservation, the Choi conditions, and the realization condition.Source: D. Completely Unital-Channel… · p. 11

IV. THE ENTROPY OF A QUANTUM CHANNEL

The section extends entropy functions from quantum states to channels, defining the extended conditional min-entropy for bipartite channels. It establishes its entropy-function properties and introduces the extended conditional min-entropy as a tool for studying uncertainty in channel outputs.

IV. THE ENTROPY OF A QUANTUM CHANNEL

  • The section defines channel entropy through monotonicity, additivity, and normalization, while noting that minimum output entropy fails the additivity requirement.Minimum output entropy would qualify if additivity were weakened to apply only to tensor products of replacement maps.Source: §IV · p. 12
  • The extended min-entropy satisfies monotonicity, additivity, and normalization, and is monotonic under the larger class of doubly stochastic superchannels.The section contrasts this with the minimum output entropy, which fails the full additivity required by its entropy-function definition.Source: §IV · pp. 12–13

A. The extended conditional min-entropy

  • The extended conditional min-entropy extends conditional min-entropy from states to bipartite channels and is developed as a measure of uncertainty about one dynamical system conditioned on another.Its definition is expressed using the normalized Choi matrix of the bipartite channel and an optimization problem that can be solved as an SDP.Source: A. The extended conditional min-entropy · p. 13
  • The channel version preserves properties of conditional min-entropy and is defined as a function of bipartite channels, not general comb operators.The section relates its expression to superchannels and notes that the normalization factor is consequential.Source: A. The extended conditional min-entropy · p. 13

B. Properties of the extended conditional min-entropy

The extended conditional min-entropy retains several properties of conditional min-entropy, including a replacement-channel case, additivity, and conditioning behavior. Its properties also include a relation under superchannel processing.

B. Properties of the extended conditional min-entropy

  • For replacement channels, the extended conditional min-entropy reduces to the conditional min-entropy of the channel’s normalized Choi matrix.This is presented as its generalization of conditional min-entropy.Source: B. Properties of the extended… · p. 14
  • The conditioning property applies to the marginal channel formed by fixing an input state on C and tracing out its output.The theorem states the resulting inequality holds for every input density matrix on C.Source: B. Properties of the extended… · p. 14
  • The extended conditional min-entropy is additive for product channels.The proof treats the normalized Choi matrix of the product channel as a product of the component Choi matrices.Source: B. Properties of the extended… · p. 14
  • The theorem includes a data-processing property involving any superchannel applied to the channel.The section also describes the extended conditional min-entropy as quantifying uncertainty about one dynamical system conditioned on another.Source: B. Properties of the extended… · p. 15

C. Operational interpretation as a guessing probability

When system B is classical, the extended conditional min-entropy has an operational interpretation as a guessing probability. In particular, it captures the maximum probability of guessing a classical output, including a channel-identification setting with a uniformly chosen input.

C. Operational interpretation as a guessing probability

  • When B is classical, the extended conditional min-entropy has an operational interpretation as a guessing probability.In a channel-identification case, 2^-Hext_min(B|A) is the maximum probability of guessing which channel Alice holds when Bob’s input is chosen uniformly.Source: C. Operational interpretation as a… · pp. 15–16
  • For a fixed Bob input, Alice can send a possibly entangled state through her share of the channel and measure her output together with the retained system to guess Bob’s outcome.The guessing probability optimizes over input states and POVMs; averaging over a uniformly chosen Bob input gives the guessing probability for the quantum-classical channel.Source: C. Operational interpretation as a… · pp. 15–16
  • The interpretation also applies when only B1 is classical: the extended conditional min-entropy is the maximum probability of correctly guessing B1’s output.In this setting B0 may be entangled with Alice’s systems, and the strategy optimizes over isometries and POVMs.Source: C. Operational interpretation as a… · pp. 16–17, Figure 6 · p. 17

V. COMPARISON OF QUANTUM CHANNELS

The section extends channel comparison to bipartite channels and characterizes when a superchannel transforms one into another. It relates this preorder to channel divergences and the extended conditional min-entropy.

V. COMPARISON OF QUANTUM CHANNELS

  • The relation Ψ_RB ≺q Φ_RA holds when a superchannel acting on subsystem A transforms Φ_RA into Ψ_RB.The paper calls this relation quantum majorization and extends the state preorder to bipartite channels.Source: §V · p. 17

A. Channel Divergences

  • Channel divergences provide necessary conditions for superchannel transformations and are contractive under superchannels.The section also notes invariance under unitary superchannels.Source: A. Channel Divergences · p. 18

B. Characterization of Quantum Majorization for Channels

  • The characterization reduces without loss of generality to a classical reference system R with one-dimensional input.The resulting theorem characterizes the preorder using the extended conditional min-entropy and provides an SDP formulation.Source: §V · p. 17, B. Characterization of Quantum… · pp. 18–19

CXY B

The proof reformulates the characterization using channel-valued tests and relates those tests to Choi-matrix constraints.

CXY B

  • The relevant Choi matrices are required to be positive semidefinite and to satisfy a marginal constraint.The supplied passage does not specify the marginal value.Source: CXY B · p. 19

JXY B

  • The proof shows that the conditions can be tested using CPTP maps instead of arbitrary linear maps.This replaces a general test map with a channel-valued test while preserving the stated equivalence.Source: JXY B · p. 19
  • The final step removes the tilde from the test map, completing the proof of the characterization.The text identifies this as the completion of the proof.Source: JXY B · p. 20

C. Comparison of channels with the extended conditional min-entropy

The extended conditional min-entropy supplies a complete characterization of channel comparison and supports an SDP solution. The section also applies these results to Gibbs-preserving superchannels and summarizes their broader channel-theoretic scope.

C. Comparison of channels with the extended conditional min-entropy

  • For comparing channel pairs, the extended conditional min-entropy yields a complete family of divergences that is necessary and sufficient for a superchannel transformation.The corollary is the special case of the general characterization with dY = 1.Source: C. Comparison of channels with the… · p. 20

D. Characterization with semidefinite programming

  • The characterization involves infinitely many conditions, but the comparison problems can be solved with semidefinite programming.The resulting SDP has polynomially many constraints in the dimensions and can be used with standard packages such as CVX.Source: D. Characterization with semidefinite… · p. 20, JXY B · p. 21

JXY B

  • The proof expresses the SDP using direct-sum real vector spaces and a linear map encoding its constraints.It replaces a linear equality constraint with equivalent inner-product conditions.Source: JXY B · pp. 20–21

E. An application to thermodynamics

  • For Gibbs-preserving superchannels, the characterization gives necessary and sufficient conditions for simulating one channel from another.The extended conditional min-entropy supplies a complete family of athermality monotones, and the SDP provides an efficient solution; extensions to other thermodynamic models are left for future work.Source: E. An application to thermodynamics · p. 21

VI. SUMMARY AND CONCLUSIONS

  • Doubly stochastic superchannels are completely uniformity preserving, and the extended min-entropy is monotonic under them.The paper also describes the extended min-entropy as monotonic under a larger set of operations than random-unitary superchannels.Source: §VI · p. 21
  • The extended conditional min-entropy has properties resembling its state version, including a guessing-probability interpretation when one subsystem is classical.The paper uses it to extend quantum majorization to bipartite channels and characterizes that preorder with necessary-and-sufficient tests and an efficient SDP.Source: §VI · p. 21

APPENDIX A STRONG DUALITY IN CONIC LINEAR PROGRAMMING

The appendix states strong-duality conditions for conic linear programs, applies them to establish an optimization equivalence, and outlines an operational optimization for the extended conditional min-entropy.

APPENDIX A STRONG DUALITY IN CONIC LINEAR PROGRAMMING

  • Under the stated closedness and feasibility conditions, the primal and dual optimal values coincide.The appendix also gives a Slater-condition route to no duality gap when a strictly feasible primal point and a primal optimum exist.Source: Appendix A · p. 22

A. Proof of the Equivalence of (62) and (63)

  • The text identifies the problem as the primal of the stated conic program, says it is identical to (104), derives its dual as (105), and states the resulting expression is equivalent to (63).Source: A. Proof of the Equivalence of (62)… · p. 22

APPENDIX B OPERATIONAL INTERPRETATION OF Hext

  • The operational optimization uses a quantum instrument and an isometry, with positive semidefinite constraints on its operators.The text says this optimization is illustrated in Fig. 6.Source: Appendix B · pp. 22–23
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