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Quantum Shannon theory with superpositions of trajectories

Giulio Chiribella, Hlér Kristjánsson

arXiv:1812.05292v3quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

Quantum Shannon theory typically treats message states as quantum but their trajectories as classical, leaving coherent path propagation outside its usual scope. This paper proposes a framework for coherently superposed trajectories, with examples including noiseless communication through noisy channels under specified conditions.

  • Problem

    Quantum Shannon theory allows quantum messages but traditionally assumes their trajectories are classical, while a broader model must separate message encoding from path propagation.

    Source: §I · p. 2

  • Method

    The framework models vacuum-extended devices and defines communication resources including the number of coherently superposed paths and transmitted particles.

    Source: §I · p. 2

  • Results

    25% probability of noiseless quantum communication is possible through the complete depolarising channel as N→∞ with access to its vacuum extension; other examples show perfect classical erasure and perfect qubit dephasing communication at large N.

    Source: §I · p. 3, §3 · p. 13

  • Takeaways & Limitations

    The framework opens the study of new quantum channel capacities and shows that correlations between internal and external degrees of freedom can be exploited by a receiver to enhance decoding.

    Source: §I · p. 3, §V · p. 17

    The basic model assumes that external degrees of freedom are not subject to noise.

    Source: §V · p. 17

Abstract

from arXiv · show

Shannon's theory of information was built on the assumption that the information carriers were classical systems. Its quantum counterpart, quantum Shannon theory, explores the new possibilities arising when the information carriers are quantum systems. Traditionally, quantum Shannon theory has focussed on scenarios where the internal state of the information carriers is quantum, while their trajectory is classical. Here we propose a second level of quantisation where both the information and its propagation in spacetime is treated quantum mechanically. The framework is illustrated with a number of examples, showcasing some of the counterintuitive phenomena taking place when information travels simultaneously through multiple transmission lines.

V. Conclusions 17

The paper extends quantum Shannon theory by treating communication paths as coherent quantum degrees of freedom, while separating path from message encoding. It defines operational resources for superposed channels and illustrates how independent or correlated processes can alter communication.

I. INTRODUCTION

  • A communication model must distinguish information encoded in internal degrees of freedom from propagation along external degrees of freedom, while preventing the path from carrying message information.The proposed model prohibits message encoding in the path and forbids intermediate repeaters from correlating the message with it.Source: §I · p. 2

II. FRAMEWORK

  • The framework builds channel superpositions by coherently routing devices to receive either the message or vacuum, without specifying their environmental interactions.It identifies vacuum extensions, superposed paths, and transmitted-particle counts as resources available in the communication model.Source: §I · p. 2

I. INTRODUCTION

  • The framework models vacuum-extended channels, coherently superposed paths, and transmitted-particle counts as resources, defining a sequence of capacities that increases with path number N.At N = 1, the capacities are the standard quantum Shannon theory capacities.Source: §I · pp. 2–3
  • Superposed paths let a qubit erasure channel carry a perfect classical bit and a qubit dephasing channel become a perfect qubit channel at large N; correlated memory can also support noiseless quantum transmission.The latter phenomenon is not possible with a superposition of independent channels.Source: §I · p. 3

A. Systems and sectors

  • A sector is a subspace of a larger system’s Hilbert space; an effective channel on it requires that the larger channel map the sector into itself.Under the No Leakage Condition, the effective channel is the restriction of the larger channel to that sector.Source: A. Systems and sectors · pp. 3–4

B. Superposition of channels

  • A superposition of channels has Kraus operators A_i ⊕ B_i, but its form can depend on the chosen Kraus representations and the channels’ environmental realizations.The characterized physical realization jointly routes system and environment to distinct regions; routing the environment can be problematic when an adversary controls it.Source: B. Superposition of channels · pp. 4–5

C. The vacuum extension of a quantum channel

The paper models a communication device as acting on both a message sector and an orthogonal vacuum sector. It explains how vacuum extensions relate to device physics and characterizes features of their extreme points.

C. The vacuum extension of a quantum channel

  • A vacuum sector represents the device input when no signal is sent and is assumed orthogonal to the message sector.A device is modeled as a channel on the direct-sum system of message and vacuum sectors.Source: C. The vacuum extension of a quantum… · p. 5
  • The framework assumes exact orthogonality between message and vacuum, while noting that some weak-coherent-state and quantum-field scenarios may require approximate orthogonality.The paper retains exact orthogonality for a simpler presentation.Source: C. The vacuum extension of a quantum… · p. 5
  • A vacuum extension preserves the message sector’s channel action, but its mathematical form is non-unique and may or may not preserve coherence with vacuum.The extension depends on the Kraus representation and vacuum amplitudes; some constructions turn message-vacuum superpositions into classical mixtures.Source: C. The vacuum extension of a quantum… · pp. 5–6
  • The physical device determines its vacuum extension, which can be specified through input-output tomography without specifying the channel’s full unitary realization.This preserves the separation between the communication system and its environment, which may be controlled by an adversary.Source: C. The vacuum extension of a quantum… · p. 6
  • Extreme vacuum extensions have nonzero vacuum amplitudes wherever their corresponding Kraus operators are nonzero.The vacuum extensions of a given channel form a convex set, and its extreme points are free from classical randomness.Source: C. The vacuum extension of a quantum… · p. 6

D. The superposition of two independent channels

The framework defines channel superposition by restricting independent vacuum-extended devices to the one-particle sector, then generalizes the construction to N channels. Operationally, a particle is coherently routed through one device while the others receive vacuum.

D. The superposition of two independent channels

  • The two-channel superposition restricts the product of vacuum-extended channels to the one-particle sector, and its construction from physical devices is a legitimate quantum supermap.The one-particle sector consists of a particle in either channel, with vacuum in the other.Source: D. The superposition of two… · p. 6

E. Superposition of multiple independent channels

III. COMMUNICATION THROUGH A SUPERPOSITION OF INDEPENDENT CHANNELS

  • The model separates information encoded in internal degrees of freedom from propagation controlled by external degrees of freedom.Source: §III · p. 7

A. Communication model with coherent control over independent transmission lines

1. Single-particle communication

The single-particle model encodes a quantum message in an internal system and routes it through independent vacuum-extended channels using a path degree of freedom. To keep the path a routing resource rather than part of the message, encodings must be product-form, excluding signalling schemes that bypass channel noise.

1. Single-particle communication

  • A single particle carries the message in internal system M and uses path qubit P to traverse independent vacuum-extended channels through their one-particle sector.The unitary U maps between M ⊗P and the one-particle sector; transmission applies the product channel and then maps back with U†.Source: §1 · pp. 7–8
  • The encoding keeps message and path uncorrelated: E(ρ) = M(ρ) ⊗ω, with a fixed path state independent of the message.Source: §1 · p. 8

2. Extension vs restriction

  • Encoding information in path would make it part of the message, so the framework restricts encodings to preserve distinct message and path roles.The paper presents this restriction as enabling a hierarchy of capacities that includes standard quantum channel capacities as its first level.Source: §2 · p. 8

3. No-signalling vs signalling encodings

  • With one input particle, controlled-SWAP routes it through either channel while the other receives vacuum, and this encoding satisfies no-signalling.Source: §3 · p. 9, Figure 4 · p. 9
  • Two-input controlled-SWAP and CNOT phase kickback can transfer a bit to the path or control and bypass noisy channels; the model excludes these signalling encodings.Source: §3 · p. 9
  • No-signalling alone can still hide a qubit in path and recover it after complete dephasing using a message-held key; product encoding excludes this route.Source: §3 · p. 10, Figure 7 · p. 10
  • Capacities are evaluated asymptotically on the effective channel; for identical lines, N-path capacity is non-decreasing in N, with N = 1 equal to standard capacity.Source: §4 · p. 10

B. Examples

The examples show that coherent path superpositions can enable communication unavailable with the channels in definite configurations, including classical communication through pure erasure channels and probabilistic quantum transmission through entanglement-breaking channels.

B. Examples

  • The examples illustrate protocols unavailable in standard quantum Shannon theory, including striking effects as the number of paths grows.The model’s sender uses product encodings, with the path state fixed independently of the message.Source: B. Examples · p. 10, §1 · p. 11

1. Classical communication through pure erasure channels

  • Two complete-erasure channels can communicate classically in superposition when their vacuum extensions preserve coherence between the message and vacuum.The effective channel is a measure-and-reprepare channel equivalent to a Z channel.Source: §1 · p. 11
  • log2(5/4) ≈ .32 is the classical capacity of the resulting Z channel, achievable using polar codes.The sender can encode 0 as |α⟩ and 1 as an orthogonal state |α⊥⟩, then use an optimal classical code.Source: §1 · p. 11
  • Without coherence-preserving vacuum extensions, the output is independent of the input and communication is impossible.The effective channel is equivalent to measuring the path followed by erasing the message.Source: §1 · p. 11

2. Quantum communication through entanglement-breaking

  • A complete-decoherence channel permits noiseless transmission of one qubit with probability 1/4 through the destructive-interference branch.That branch is proportional to Z, which the receiver can undo; the remaining constructive branch is the channel 2/3 ZρZ, with quantum capacity 1 − h(1/3) ≈ .08.Source: §2 · p. 12
  • Superpositions of a finite number of independent noisy channels cannot produce a noiseless channel unless the original channel is unitary.The same conclusion holds for superpositions of two different channels and, more generally, any finite number of independent channels.Source: §2 · p. 12

3. Perfect communication through asymptotically many

In the large-path limit, examples achieve perfect classical or quantum communication, or noiseless quantum communication probabilistically. The paper also extends its model to correlated channels, showing that spatially correlated environments can realize some, but not all, product channels.

3. Perfect communication through asymptotically many

  • As N grows, the channel tends toward a mixture of the operations F · F† and A − F · F†, corresponding to constructive and destructive interference.The path state is diagonal in the Fourier basis.Source: §3 · p. 12
  • Perfect classical communication of one bit is achieved as N → ∞ for the pure erasure channel with the specified vacuum extension.The effective channel measures {|α⟩, |α⊥⟩} and prepares orthogonal path states, allowing perfect transmission of classical bits.Source: §3 · p. 12
  • Perfect quantum communication is achieved as N → ∞ for the complete dephasing channel with the specified vacuum extension.The supplied passage states this result but does not specify further conditions or a probability.Source: §3 · p. 12
  • Noiseless quantum communication with probability 25% becomes possible as N → ∞ for the complete depolarising channel with the specified vacuum extension.The example uses eA0 = (I ⊕ 1)/2.Source: §3 · p. 13
  • When a path measurement heralds the quantum operation F · F†, noiseless transmission of a qubit occurs; in this case, the vacuum interference term is proportional to a unitary gate.Source: §3 · p. 13

IV. COMMUNICATION THROUGH A SUPERPOSITION OF CORRELATED CHANNELS

  • The model is extended to channels correlated across alternative paths, considering spatial and temporal correlations.Spatial interactions are described by unitary channels coupling the particle to local environments, which may have previously interacted.Source: §IV · p. 13, A. Spatially correlated channels · p. 13

A. Spatially correlated channels

  • Preexisting correlations between environments on alternative paths can realize every random-unitary channel on the internal degree of freedom while leaving the path untouched.With correlated channels, the effective channel acts nontrivially only on the internal degree of freedom; uncorrelated channels generally correlate the path and internal degrees of freedom.Source: A. Spatially correlated channels · p. 14
  • A pure erasure channel tensored with the path identity cannot be realized by the spatially correlated-channel construction.The paper’s intuition is that complete information transfer to two spatially separated environments is impossible because of the no-cloning theorem.Source: A. Spatially correlated channels · p. 14

B. Channels with correlations in time: realising the output of the quantum SWITCH

The section constructs a physical circuit whose output is the quantum SWITCH of two channels, using a particle in a superposition of paths through regions with time-correlated environments. The realization has restricted message-to-path signalling and depends on local vacuum extensions and environmental memories, so it simulates rather than implements the abstract SWITCH.

B. Channels with correlations in time: realising the output of the quantum SWITCH

  • The circuit’s output depends only on channels A and B, not on their local vacuum extensions.The identity action on the vacuum sectors and the circuit’s sequential interactions with both environments erase dependence on vacuum amplitudes.Source: B. Channels with correlations in time:… · p. 15
  • The effective channel is the quantum SWITCH S(A, B), a correlated superposition of A ◦B and B ◦A.The two orders arise from alternative paths through regions RA and RB with ideal local quantum memories.Source: B. Channels with correlations in time:… · pp. 14–15
  • Encoding initializes the path independently of the message, and the intermediate SWAP is a product operation that flips only the path.Thus neither operation induces signalling from the message to the path or mixes the message and path degrees of freedom.Source: B. Channels with correlations in time:… · p. 15
  • The circuit supplies a vacuum extension as well as the switched channel, making superposition of orders composable with superposition of paths.Its circuital realizations can be implemented with existing photonic technologies [18] [19].Source: B. Channels with correlations in time:… · pp. 15–16
  • The Figure 11 circuit simulates the abstract quantum SWITCH using local vacuum extensions and two perfect environmental memories; it is not a genuine implementation of the abstract map.Other physical realizations of the SWITCH exist, including the circuit proposed in Ref. [42].Source: B. Channels with correlations in time:… · p. 15

C. Communication model with time-correlated channels

The communication model extends path-superposition protocols to channels correlated across time, using message-independent encoding and product repeaters. Examples include self-switching effects and perfect quantum communication for certain noisy channels; the conclusion assumes noiseless external degrees of freedom and identifies path noise as future work.

C. Communication model with time-correlated channels

  • Time-correlated channel superpositions can be realized without access to the environment, and quantum combs describe such correlations using only the parties’ local input/output systems.The protocol encodes Q into the one-particle subspace, alternates T channel transmissions with repeaters, and decodes the output back to Q.Source: C. Communication model with… · p. 16

1. Self-switching

  • A quantum channel switched with itself acts trivially when its Kraus operators commute; noncommuting operators permit nontrivial self-switching effects.For the pure erasure channel described in the passage, the resulting communication channel has classical capacity log2(5/4) ≈.32.Source: §1 · p. 17

2. Perfect communication through a coherent superposition

  • Perfect quantum communication is possible for the random-unitary channel A = 1/2(X + Y) when X² = Y² and {X, Y} = 0.The path can be measured and a conditional correction applied to the message; the passage contrasts this complete noise removal with irreducible noise for superpositions of noisy channels on independent paths.Source: §2 · p. 17

V. CONCLUSIONS

  • The framework separates internal message information from external path propagation, whose correlations with the message can be used by a receiver to improve decoding.The examples include classical communication with pure erasure channels and quantum communication with entanglement-breaking channels.Source: §V · p. 17
  • The basic model assumes noiseless external degrees of freedom; dephasing and particle loss are proposed as extensions, with tolerable noise left for future study.The open question is how much path noise preserves an advantage over conventional communication along a single well-defined path.Source: §V · pp. 17–18

Appendix A: Proof of Theorem 1

The appendix proves the equivalence between the No Leakage Condition and the stated channel structure by decomposing Kraus operators across sectors, constructing a Stinespring dilation, and recovering the channel from block-diagonal unitary dynamics.

Appendix A: Proof of Theorem 1

  • The No Leakage Condition is characterized in terms of Kraus operators, whose restrictions to sectors A and B form Kraus representations of the respective channels.The proof uses the condition on both sectors to show each Kraus operator has the direct-sum form Ai ⊕Bi.Source: Appendix A · p. 20
  • The proof constructs a Stinespring representation with a direct-sum isometry VA ⊕VB from those sector-wise Kraus operators.Each isometry is extended to a unitary, and the initial environment states can be chosen equal.Source: Appendix A · p. 21
  • Conversely, block-diagonal Hamiltonian evolution followed by tracing out the environment yields a channel that is a superposition of A and B.The constructed evolution is U = exp[−i(HAE ⊕HBE)T/ℏ].Source: Appendix A · pp. 21–22

Appendix B: Vacuum extensions with non-trivial vacuum dynamics

This appendix defines vacuum extensions that may have nontrivial vacuum dynamics and uses them to construct operational superpositions of channels. A vacuum-discarding map recovers the one-particle sector and returns a failure state outside it; the appendix also states a converse construction for suitable Kraus operators.

Appendix B: Vacuum extensions with non-trivial vacuum dynamics

  • A vacuum extension of C preserves the channel C on its sector, satisfies the No Leakage Condition, and may have Kraus operators Ci ⊕CVac,i.When vacuum dynamics are the identity, the operators simplify to Ci ⊕γi I.Source: Appendix B · p. 22
  • The isometry V embeds A ⊕B into the one-particle subspace of (A ⊕Vac) ⊗(B ⊕Vac), using a fixed vacuum vector.The embedded states occupy either A ⊗Vac or Vac⊗B.Source: Appendix B · p. 22
  • A vacuum-discarding map inverts V on the one-particle sector and outputs a fixed failure state on the zero- or two-particle sectors.Its success sector is specified by Psucc := PA ⊗PVac + PVac ⊗PB.Source: Appendix B · pp. 22–23
  • The operational superposition combines the two vacuum-extended channels with V and the vacuum-discarding map; its Kraus form shows that interference is mediated by the vacuum.The appendix also states a converse: suitable coefficients yield vacuum extensions whose Kraus operators define a superposition of A and B.Source: Appendix B · p. 23

Appendix C: Vacuum extensions and unitary dilations

The appendix relates superpositions defined through vacuum extensions to unitary implementations. It proves an equivalence for vacuum subspaces of arbitrary dimension and recovers Oi’s construction when the vacuum is one-dimensional.

Appendix C: Vacuum extensions and unitary dilations

  • The appendix clarifies the relation between superpositions defined through channels’ action on the vacuum and those defined through their unitary implementations.Source: Appendix C · p. 24
  • When the vacuum is one-dimensional, Oi’s unitary-implementation superpositions coincide with superpositions specified by vacuum extensions.The auxiliary system G can be omitted, with its unitaries taken as identity after redefining the environment states.Source: Appendix C · pp. 24, 26
  • For a vacuum subspace of dimension v, a standard superposition specified by vacuum extensions is equivalent to a channel with the stated unitary realization.This is the equivalence asserted in Theorem 2.Source: Appendix C · p. 24

Appendix D: Extreme vacuum extensions

The appendix characterizes extreme vacuum extensions and derives bounds on their Kraus representations. These bounds constrain how many Kraus operators can have zero system-sector components.

Appendix D: Extreme vacuum extensions

  • An extension is extreme in Vac(C, v) exactly when the operators {eC†i eCj} are linearly independent.Vac(C, v) is the convex set of vacuum extensions of C for fixed vacuum dimension v.Source: Appendix D · p. 26
  • For an extreme extension with r linearly independent Kraus operators, r² ≤ L + v², where L counts the linearly independent operators C†jCi.The bound is stated for a v-dimensional vacuum subspace.Source: Appendix D · p. 26
  • For an extreme extension, the system and vacuum evolutions must be coherent; its Kraus operators cannot separate into system-zero and vacuum-zero groups.Source: Appendix D · p. 27
  • For a one-dimensional vacuum, none of the Kraus operators Ci can be zero.More generally, the number z of zero Ci operators is bounded in terms of the vacuum dimension.Source: Appendix D · p. 27

Appendix E: Proof of Proposition 1

The appendix proves that a pure erasure channel tensored with an identity channel cannot have the specified realization. The contradiction arises because complementary channels would both be correctable, violating no-cloning.

Appendix E: Proof of Proposition 1

  • The channel E0 ⊗ IP does not admit a realization of the specified form.The proof assumes E0 is a pure erasure channel acting on the message system.Source: Appendix E · pp. 27, 29
  • The contradiction is that complementary channels M and Mc are both correctable, which would recover two copies of the same state.Source: Appendix E · pp. 28–29
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