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On the Possibility of Quantum Circuits Part I: the Epistemic Level

Olivier Brunet

arXiv:1605.00796v1quant-ph

TL;DR

The paper asks how to characterize whether a quantum circuit corresponds to a feasible physical experiment. It develops logical rules for epistemic verification statements, then argues that the formalism offers a Lorentz-invariant account of circuits and addresses Hardy’s paradox. Its conclusion treats quantum vector states as particular kinds of verification statements, while leaving an ontic interpretation as a possible direction for further work.

  • Problem

    The paper seeks a general way to determine whether a quantum circuit corresponds to an actually feasible physical experiment.

    Source: §1 · p. 1

  • Method

    The authors define verification statements as epistemic predictions based on prior measurement outcomes and experimental setup, then develop logical rules for them.

    Source: §1 · p. 2, §7 · p. 27

  • Results

    The authors argue that their rules provide a Lorentz-invariant realistic formulation of quantum circuits and address Hardy’s paradox by restricting conjunctions to compatible slices.

    Source: §1 · p. 2, §6 · p. 27

  • Takeaways & Limitations

    The authors propose viewing quantum vector states, when they resemble verification statements, as particular types of epistemic statements.

    Source: §7 · p. 27

    The formalism concerns a fragment of quantum mechanics and makes assumptions about obtaining measurement outcomes without invoking quantum states.

    Source: §3 · p. 5, §3.1 · p. 5

Abstract

from arXiv · show

We present a formulation of quantum circuits where the focus is set on whether a given circuit (made of unitary operators and projective measurements with definite outcomes) does reflect an actually realizable physical experiment. In order to do this, we introduce verifications statements which are purely epistemic assertions indicating whether a outcome is possible at some point and develop our formalism which, in the end, consists in a set of logical rules about verification statements. Finally, we argue that our formalism provides a Lorentz-invariant realistic formulation of quantum circuits and illustrate this by considering a circuit corresponding to Hardy's paradox and showing how our formalism prevents making contradictory assertions regarding our knowledge about the circuit.

1 Introduction

The paper proposes a possibilistic rather than probabilistic way to characterize whether quantum circuits correspond to feasible physical experiments. It introduces epistemic verification statements and applies its logical rules to argue for a Lorentz-invariant formulation, including an analysis of Hardy’s paradox.

1 Introduction

  • The paper seeks a general characterization of whether a quantum circuit represents an actually feasible physical experiment, using a possibilistic rather than probabilistic approach.Its scope is a fragment of quantum mechanics corresponding to quantum circuits.Source: §1 · p. 1
  • Two consecutive projective measurements of one particle with orthogonal outcomes make the depicted circuit impossible.After the first outcome [0], the Born rule rules out an orthogonal outcome at the second measurement.Source: §1 · p. 1
  • The proposed logical rules use verification statements to analyze Hardy’s paradox and argue that some objections to a Lorentz-invariant interpretation do not apply.The authors present the formalism as a Lorentz-invariant realistic formulation of quantum circuits.Source: §1 · p. 2

2 Quantum Circuits

Quantum circuits are represented as acyclic graphs of system nodes and operation nodes, with an induced order among nodes. The section introduces slices and circuit cuts for describing events and examining a circuit up to a chosen point.

2 Quantum Circuits

  • A quantum circuit is an acyclic graph with system nodes and operation nodes for projective measurements or unitary transformations.The graphical representation distinguishes incoming and outgoing edges; circuits are bipartite, so arrows connect nodes of different types.Source: §2 · pp. 2–3
  • The formalism considers projective measurements and unitary transformations; composite systems can also simulate POVMs by Naymark’s theorem [CN00, Per02].Each operation node is specified by ordered lists of incoming and outgoing nodes and their dimensions.Source: §2 · pp. 2–3
  • A circuit’s arrows induce a partial order, and the full-subgraph relation preserves the circuit’s time structure and is itself a partial order.The paper denotes this relation by C ⊆f C′.Source: §2 · pp. 3–4
  • A slice is an ordered set of mutually incomparable system nodes, representing events that can be considered simultaneously.In the circuit’s order relation, spacelike-separated events are not comparable.Source: §2 · p. 4
  • Cutting a circuit along a slice removes its strict future and produces a full subgraph of the original circuit.The paper illustrates this operation by cutting circuit T along {B3, C2}.Source: §2 · p. 4

3 Possible and Impossible Circuits

This section sets outcome-based assumptions for deciding whether a quantum circuit is possible, then expresses those assumptions and further measurement rules as logical deductions about verification statements.

3 Possible and Impossible Circuits

  • The assumptions concern measurement outcomes rather than quantum states and are required to remain compatible with standard quantum mechanics.They are intended to rule out situations that cannot represent an actual physical experiment.Source: §3 · p. 5

3.1 Non Contradiction

  • Two consecutive projective measurements with orthogonal outcomes are impossible; more generally, measuring q after outcome p is impossible when q ≤ p⊥.This is the Non Contradiction rule for projective measurements.Source: §3.1 · p. 5

3.2 Considering Observables

  • An observable is a finite set of orthogonal outcomes, and a possible circuit must allow at least one outcome of any applicable observable.The section also treats an observable containing the bottom element as invalid and defines verification statements as part of the logical formalism.Source: §3.2 · pp. 6–7

3.3 Possibilistic Non-Contextuality

  • Possibilistic non-contextuality makes outcome certainty independent of which observable is measured, supporting Order and Compatible Meet deductions about impossible outcomes.For compatible p and q, the rules derive the impossibility of q from that of p when q ≤ p, and of p ∨ q when both are impossible.Source: §3.3 · p. 8

3.4 A Few More Rules

  • Further rules preserve impossibility through unitary operations and compatible measurements, and extend verification across larger slices or permutations of slice blocks.The first version of the formalism is summarized in Figure 7.Source: §3.4 · pp. 9–10

4 Logical Variations

Section 4 revises the logic’s rules and generalizes the meet rule. It also illustrates how verification statements capture the transfer of properties in a teleportation circuit.

4.1 The Mes rule, revisited

  • For mutually compatible propositions, Ord and Compatible Meet yield an equivalent formulation that permits replacing the Mes rule.Source: §4.1 · p. 10

4.2 The Sasaki Rule

  • The Sas rule replaces both Non Contradiction and Compatible Preservation in the revised logic.The paper presents the Sas rule as valid and shows that it suffices, with Top and Ord, to recover those rules.Source: §4.2 · pp. 12–13

4.3 Generalizing the Compatible Meet Rule

  • If distinct atoms are both verified on a slice of dimension at least 3, the circuit is impossible.Source: §4.3 · p. 13
  • For d(Γ) ≥ 3, SC,Γ cannot have two distinct minimal elements; for a minimal e, every p in SC,Γ satisfies e ≤ p, so SC,Γ = e↑, and kC(Γ) is called the epistemic state.Source: §4.3 · pp. 13–14
  • Assuming the circuit transformation is always possible and yields d(Γ :: [γ]) ≥ 3, Theorem 6 applies, allowing Compatible Meet’s compatibility requirement to be dropped in favor of the more general Meet rule.Source: §4.3 · pp. 13–14

4.4 Quantum Teleportation

  • In the teleportation circuit, any property verified by A1, R1 is transferred to C3, R1 after the measurements and corresponding operation.The depicted circuit represents one of four possibilities, determined by the measurement outcomes and corresponding operation.Source: §4.4 · pp. 14, 17

5 Some More Properties

The formalism establishes that verification can be proved using only a circuit’s strong past and gives verification statements an operational meaning. It also characterizes how knowledge changes when systems are entangled and other systems are acted upon.

5.1 Verification and Full Subgraphes

  • If a circuit is possible, each of its full subgraphs is also possible.Verification is monotone under full-subgraph extension.Source: §5.1 · p. 17
  • Any verification statement about a slice can be proved using only its strong past.The strong past consists of nodes every path from which crosses that slice.Source: §5.1 · pp. 17–18

5.2 The Meaning of Verification

  • The logic treats verification as an atomic statement rather than syntactic sugar, refining its intended meaning.The presentation says the logic no longer needs references to Imp.Source: §5.2 · pp. 18, 20

5.3 Knowledge and Entanglement

  • If one slice is in the strong past of a later slice, verification at the earlier slice is preserved in the later one.The authors describe “@” as meaning “after (inclusive).”Source: §5.3 · p. 21
  • When an entangled partner is left untouched, acting on the other system can only increase knowledge about it.The result includes measurements on the other system.Source: §5.3 · p. 21
  • Opening box C narrows the knowledge about A1 and B1 from [00] + [|01⟩+|10⟩] to [|01⟩+|10⟩].The authors interpret this as increased knowledge after learning the particle is not in C.Source: §5.3 · p. 22

6 Realism and Lorentz Invariance

The paper applies its verification formalism to Hardy’s paradox, where reasoning across Lorentz frames appears to rule out detector coincidences that quantum mechanics and experiments allow. It argues that restricting assertions to compatible slices avoids the contradiction and supports a Lorentz-invariant account of reality in quantum circuits.

Hardy’s paradox

  • Two interferometers with overlapping arms form Hardy’s setup; a positron and electron taking both overlapping arms annihilate.The circuit is depicted as Hardy’s paradox circuit H.Source: §6 · p. 23, Figure 9 · p. 23
  • Frame-dependent inferences imply both particles cannot reach the d detectors, although quantum mechanics and experiments say that coincidence is possible.The paper identifies the possible term in the epistemic state and cites experimental confirmation.Source: §6 · p. 24
  • The formalism avoids the contradiction because the separate verification statements cannot be combined on a slice containing both required sets of events.The stated obstacle is that no such slice exists.Source: §6 · p. 25

Lorentz-invariant realism

  • The authors propose attaching elements of reality to spacetime regions represented by slices, rather than to a time or single spacetime event.They state that this makes the elements Lorentz invariant by specifying which hypersurfaces may be considered.Source: §6 · pp. 22, 26, 27
  • The Meet rule is compatible with the analysis because conjunction applies only to verification statements on compatible whole slices.The paper says the rule has been shown correct in this setting.Source: §6 · p. 27

7 Conclusion

The conclusion presents verification statements as epistemic predictions about possible outcomes, defined from accessible experimental information. The resulting circuit formalism provides a Lorentz-invariant realistic formulation in this domain, while treating quantum vector states as particular epistemic cases.

Conclusion

  • Verification statements concern possible outcomes and are defined using accessible information about prior measurements and the experimental setup.The paper associates each slice with subspaces, including a minimal one called its epistemic state.Source: §7 · p. 27
  • The authors say their logical formalism provides a Lorentz-invariant realistic formulation of quantum mechanics in the context of quantum circuits.They place this approach among formulations that associate states with spacelike hypersurfaces.Source: §7 · p. 27
  • Quantum vector states are presented as particular verification statements and therefore as epistemic in nature, especially when a slice’s epistemic state is one-dimensional.The authors identify ontic interpretations via models of the theory as a possible direction for further consideration.Source: §7 · p. 27
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