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Communication through coherent control of quantum channels
Alastair A. Abbott, Julian Wechs, Dominic Horsman, Mehdi Mhalla, Cyril Branciard
TL;DR
The paper asks how coherent control can transmit information through fully depolarising channels and what determines the action of a controlled channel. It studies coherent control of channel choice and reports that implementation details affect the joint output, so a CPTP map alone does not fully specify the controlled action.
Problem
Method
Results
Takeaways & Limitations
A complete description of coherently controlled channels in the studied setups must include transformation matrices alongside the CPTP maps.
Source: §6 · p. 6For certain depolarising-channel implementations, including cases where T0 = 0 or T1 = 0, the global channel cannot transmit information.
Source: §4 · p. 5
Abstract
from arXiv · showhide
A completely depolarising quantum channel always outputs a fully mixed state and thus cannot transmit any information. In a recent Letter [D. Ebler et al., Phys. Rev. Lett. 120, 120502 (2018)], it was however shown that if a quantum state passes through two such channels in a quantum superposition of different orders - a setup known as the "quantum switch" - then information can nevertheless be transmitted through the channels. Here, we show that a similar effect can be obtained when one coherently controls between sending a target system through one of two identical depolarising channels. Whereas it is tempting to attribute this effect in the quantum switch to the indefinite causal order between the channels, causal indefiniteness plays no role in this new scenario. This raises questions about its role in the corresponding effect in the quantum switch. We study this new scenario in detail and we see that, when quantum channels are controlled coherently, information about their specific implementation is accessible in the output state of the joint control-target system. This allows two different implementations of what is usually considered to be the same channel to therefore be differentiated. More generally, we find that to completely describe the action of a coherently controlled quantum channel, one needs to specify not only a description of the channel (e.g., in terms of Kraus operators), but an additional "transformation matrix" depending on its implementation.
1 Introduction
The introduction motivates coherent control as a way to transmit information through fully depolarising channels, even without indefinite causal order. It also argues that a coherently controlled channel cannot be fully specified by its CPTP map alone.
1 Introduction
- The quantum switch had shown that fully depolarising channels can transmit information when their order is coherently controlled [3].The introduction presents this result as the motivation for studying coherent control of channel choice.Source: §1 · p. 1
- Coherently multiplexing fully depolarising channels allows information transmission, despite the absence of indefinite causal order.The control selects which channel is used, rather than the order in which two channels are applied.Source: §1 · p. 1
- The action of a coherently multiplexed channel depends on the channels’ specific realisations, not only on their CPTP maps.The paper says an additional description is needed to specify the controlled channel’s action.Source: §1 · p. 1
2 Communication through the “depolarising quantum switch”
The quantum switch coherently controls the order of two channels, producing indefinite causal order. With fully depolarising channels, the switched process can transmit information even though the individual channels and their fixed-order compositions cannot.
2 Communication through the “depolarising quantum switch”
- The quantum switch uses a control qubit to select the order of two operations on a target; a superposed control produces a superposition of orders.The paper identifies this as indefinite causal order.Source: §2 · p. 2
- The switched channel transmits information with fully depolarising channels, although each channel and either fixed-order composition cannot.The switch’s output depends only on the CPTP maps implemented by the operations.Source: §2 · p. 2
- In the photonic setup illustrated in Figure 1, the control is encoded in photon polarisation and the target in another internal degree of freedom.The inset depicts the quantum switch; the main figure shows the first half of a related setup.Source: Figure 1 · p. 2
3 Communication through coherently-controlled depolarising channels
The section examines coherent control between fully depolarising channels, where a target passes through one channel in a superposition of alternatives. This setup transmits information without indefinite causal order, and its capacity and output depend on features of the implementation and the joint control-target state.
3 Communication through coherently-controlled depolarising channels
- With the control prepared in |+⟩c, the target passes once through a superposition of two fully depolarising channel implementations.The considered implementation randomises over orthogonal unitaries; the calculations also generalise to other initial control states.Source: §3 · pp. 2–3
- In interferometer-type settings, localised operations provide structural information about the joint Hilbert space that makes coherent control possible.The paper notes that coherent control of completely unknown operations is generally impossible.Source: §3 · p. 2
- The coherently controlled channel transmits some information, whereas classical control or control decohered in the σz basis transmits none.The paper attributes the advantage over classical control to the global channel arising from coherent control.Source: §3 · p. 3
- The communication advantage requires no indefinite causal order: the two channels are used in parallel rather than one after the other.The authors say this result raises questions about the role of causal indefiniteness in the quantum-switch result.Source: §3 · p. 3
- ≈0.16 bits for the coherently controlled setup versus ≈0.05 bits for the full depolarising quantum switch, for a qubit target system.These are lower-bound Holevo-information comparisons reported by the paper.Source: §3 · p. 3
- The output distributes input information across the control-target system; the control alone does not contain all of the transmitted information.Tracing out the control leaves the target maximally mixed, while some information is carried in correlations.Source: §3 · p. 3
4 Dependence on channel implementation
In coherent control, the output can depend on how a channel is implemented, not just on its CPTP map or Kraus representation. Transformation matrices capture this extra implementation information and help specify the joint control-target output.
4 Dependence on channel implementation
- Independent Stinespring dilations purify the channels, with the control governing each unitary extension on both the target and its corresponding environment.This construction applies generally because any channel can be purified in a form equivalent to a Stinespring dilation.Source: §4 · pp. 4–5
- Different orthogonal-unitary implementations of the same depolarising channels can yield different control-target outputs, while a particular choice recovers the depolarising quantum-switch state.Adding relative phases to the unitary operators can change the output even when the local unitary channels remain the same.Source: §4 · p. 4
- The controlled circuit’s output depends on the implementations of C0 and C1, so their channel maps or Kraus representations alone do not fully specify it.Even a global phase on an operation can become a physically significant relative phase between interferometer arms.Source: §4 · p. 4
- Transformation matrices T0 and T1 encode how Kraus operators are combined using environment-state coefficients, completing the specification of the controlled output.The circuit figure supplements the channel operations with T0 and T1 to specify the joint state.Source: §4 · p. 4, Figure 2 · p. 5
- For a d-dimensional fully depolarising channel, an implementation’s transformation matrix must satisfy Tr[T†T] ≤ 1/d, and the resulting constraint characterizes the attainable outputs.The constraint is applied separately to T0 and T1.Source: §4 · p. 5
- If either transformation matrix is zero, the resulting global channel cannot transmit information.This is a limitation for particular implementations of depolarising channels, not for every implementation.Source: §4 · p. 5
5 Distinguishing different implementations of coherently-controlled channels
Coherent control makes distinct implementations of the same channel distinguishable through the joint control-target output. The section gives both an optimal discrimination strategy and examples, including identity and depolarising channels.
5 Distinguishing different implementations of coherently-controlled channels
- For fixed C0, T0, and C1, differing transformation matrices T1 and T′1 generally produce different global channels, enabling discrimination of C1's implementations.For equally probable alternatives, the maximum success probability is 1/2(1 + D(MT1, MT′1)).Source: §5 · p. 5
- With C0 = I and T0 = 1, the upper bound on implementation distinguishability is attainable using the specified input state and optimal output-state discrimination.The input state is chosen to maximize separation associated with T1 − T′1.Source: §5 · p. 5
- The interferometric setup can perfectly distinguish the operations |ψ⟩ → |ψ⟩ and |ψ⟩ → −|ψ⟩, although both implement the identity channel.These implementations have transformation matrices +1 and −1 on the relevant subspace.Source: §5 · p. 5
- Two implementations of the fully depolarising channel can also be distinguished when C0 = I, T0 = 1, and the target input is chosen as specified.The passage identifies the setting but does not provide the complete input-state expression.Source: §5 · p. 5
6 Discussion
The discussion frames coherent channel control as a communication resource and emphasizes that the resulting circuit action depends on implementation details beyond the CPTP map. It also identifies limits and open questions about the scope of the advantage and the description of controlled channels.
6 Discussion
- Coherent control increases transmission capacity without postselection, including for completely depolarising channels, and may inform error correction and mitigation.The authors suggest possible relevance to quantum communication and computation.Source: §6 · pp. 5–6
- The studied setup transmits classical information, but certain noisy channels allow perfect quantum communication with the quantum switch and not with this setup.Which resource is more powerful, and the role of indefinite causal order in switch-based advantages, remain open questions.Source: §6 · p. 6
- The circuit's global transformation depends on channel implementation, so a CPTP map or its Kraus descriptions alone do not specify coherently controlled action.The authors connect access to implementation information with possible applications in error correction, security, and channel characterisation.Source: §6 · p. 6
- Coherently controlling channels is ill-defined without further specification, although the studied setups are experimentally realizable and less demanding than full-switch implementations.The authors call for circuit descriptions that accommodate quantum-controlled operations and operations applied only on subspaces; broader generalization is left for future work.Source: §6 · p. 6
A Characterisation of the possible transformation matrices T of a channel C
This appendix characterizes transformation matrices available for implementations of a channel, using the Choi isomorphism to express channels and their properties compactly.
A Characterisation of the possible transformation matrices T of a channel C
- The appendix sets out to characterize all transformation matrices obtainable from implementations of a given channel and uses the Choi isomorphism to state and prove that characterization.Source: A Characterisation of the possible… · p. 9
A.1 Choi isomorphism
- The Choi vector represents an operator between input and output Hilbert spaces, with a fixed input basis, and can be mapped back to its operator.Source: §A.1 · p. 9
- A channel's Choi matrix represents the channel and permits its recovery; complete positivity makes the matrix positive semidefinite, while trace preservation fixes its output-space partial trace.Source: §A.1 · p. 9
- With Kraus operators, the Choi matrix is expressed through their Choi vectors, which lie in the matrix's range.Source: §A.1 · p. 9
A.2 General constraints on the transformation matrices
The appendix gives constraints characterizing transformation matrices for a fixed channel, then illustrates the available choices for several channels.
A.2 General constraints on the transformation matrices
- A valid transformation matrix T must have its Choi vector in the range of C and satisfy ⟨⟨T|C+|T⟩⟩ ≤ 1; the appendix characterizes the full set using these constraints.C+ is the Moore–Penrose pseudoinverse of the channel's Choi matrix.Source: §A.2 · p. 10
- The appendix establishes that every operator meeting those range and pseudoinverse constraints can arise from a particular channel implementation.It constructs Kraus operators and an initial environment state to obtain the desired transformation matrix.Source: §A.2 · p. 10
A.3 Examples
- Even the identity channel admits T = α1 for |α| ≤ 1; for any unitary channel U, the possible matrices are T = αU with |α| ≤ 1.The text attributes the freedom not only to global phase but also to coherent control of an operation acting nontrivially on the environment.Source: §A.3 · p. 10
- For the fully depolarising channel, the appendix applies the general constraints using its Choi matrix and pseudoinverse, then constructs an implementation for an allowed T.Source: §A.3 · p. 10
- The partially dephasing qubit examples give transformation matrices combining the identity with the relevant Pauli operator, with coefficients whose squared magnitudes sum to at most one.The examples use σz for phase flip and σx for bit flip, and describe implementations using their respective Kraus operators.Source: §A.3 · p. 11
B.1 Holevo information of the coherently controlled depolarising channels
The section evaluates classical information transmission through coherently controlled fully depolarising channels using the Holevo information. A chosen input ensemble gives a lower bound that exceeds the bound for the depolarising quantum switch.
B.1 Holevo information of the coherently controlled depolarising channels
- For two fully depolarising channels, the weighted ensemble with probabilities 3/5 and 2/5 on |0⟩⟨0| and |1⟩⟨1| supplies a lower bound on the Holevo information.Source: §B.1 · p. 11
- For d = 2, the chosen ensemble gives χ(M[N0, T0, N1, T1]) ≳0.16, a significant increase over the depolarising quantum switch.For larger d, this bound decreases, but less rapidly than the exact Holevo information obtained for the switch.Source: §B.1 · p. 11
- The comparison with the quantum switch is not captured by a simple degradation argument: noncommutativity of switch Kraus operators also matters, and the switch can sometimes outperform coherent control [39].The authors say that further study is needed to understand the comparison fully.Source: §B.1 · p. 12
B.2 Quantum information transfer through coherently controlled dephasing channels
For coherently controlled phase-flip and bit-flip channels, a chosen input state gives a positive lower bound on quantum capacity for every mixing parameter. At p = 1/2, the bound is positive even though the individual channels cannot transmit quantum information.
B.2 Quantum information transfer through coherently controlled dephasing channels
- The calculation takes Z(p) and X(p) with the same mixing parameter and transformation matrices TZ(p) = √p σz and TX(p) = √p σx.The coherent-information lower bound is evaluated using a maximally entangled input state.Source: §B.2 · p. 12
- The quantum-capacity lower bound is positive for all p and equals −3/4 log2(3/4) ≃0.31 at p = 1/2, when both channels are fully dephasing.The bound also exceeds the capacity of either channel individually for all p.Source: §B.2 · p. 12
- The bound exceeds that obtained for the full quantum switch for phase-flip and bit-flip channels, whose bound is positive only for p ≲0.13 and p ≳0.60.The full-switch bound exceeds either individual channel's capacity only for p ≳0.62.Source: §B.2 · p. 12
- The comparison has a scope qualification: the target passes through just one dephasing channel in this setup, which may partly explain the higher bound.The quantum switch can in some cases enable perfect quantum communication where coherent control cannot [52].Source: §B.2 · p. 12
C Channel implementation independence for the full quantum switch and other quantum processes
For the full quantum switch and more general processes in which each channel acts exactly once, tracing out the environments removes dependence on the channels’ Kraus representations and environment initial states. This implementation independence rests on the once-only action condition.
C Channel implementation independence for the full quantum switch and other quantum processes
- After the environments are traced out, the full quantum switch output depends neither on their initial states nor on the Kraus representations chosen for the channels.Each channel acts once, with the two orders coherently controlled.Source: C Channel implementation independence… · pp. 12–13
- For more general combinations of channels, if each channel acts once and only once, the output after tracing out environments is independent of the Kraus representations and environment initial states.The transformation may combine terms in different orders, as in the quantum switch.Source: C Channel implementation independence… · p. 13
- The once-only action assumption is central to the process-matrix framework, under which the process depends on the induced completely positive maps rather than their specific implementations [12].This extends the implementation-independence point beyond the quantum switch.Source: C Channel implementation independence… · p. 13
D Distinguishing different implementations of coherently-controlled channels
The section bounds how distinguishable two coherently controlled channels are when one channel’s transformation matrix changes. For the depolarising-channel example, the proposed implementations attain the largest possible distance under the bound.
D Distinguishing different implementations of coherently-controlled channels
- The diamond-norm distance compares the maximum output trace distance over joint target-reference inputs, allowing entanglement with a reference system.The definition notes that such entangled inputs can be necessary for optimal channel discrimination.Source: D Distinguishing different… · p. 13
- The upper bound on the diamond-norm distance can be saturated, so it is tight for the stated setup.Saturation is obtained with T0 = 1 and C0 = I, without a nontrivial reference system.Source: D Distinguishing different… · p. 14
- For the depolarising-channel example, the two proposed implementations are maximally distinguishable among the considered transformation matrices.Their distance reaches the bound; the passage identifies 1/d as the largest possible value.Source: D Distinguishing different… · p. 14