Formalize Open Quantum Problem #43: Are all extensibly causal processes purifiable?

[MarioKrenn6240]

What is the conjecture

This is problem #43 in the Open Quantum Problems master list, introduced there by M. Araújo and C. Brukner.

Problem (Open Quantum Problem #43: “Are all extensibly causal processes purifiable?”).
In the process-matrix framework, a process is extensibly causal if it remains causal under extension with arbitrary ancillary input systems prepared in an arbitrary joint quantum state.
A process is purifiable if it can be obtained from a pure process by feeding a fixed pure state into an auxiliary past system and tracing out an auxiliary future system.
The question is whether every extensibly causal process is purifiable.

The OQP page phrases causality via causal inequalities; a standard formalization is in the language of process matrices:

• Fix N local laboratories with input/output Hilbert spaces A_I^1, A_O^1, ..., A_I^N, A_O^N.
• A local operation of party i is a completely positive (CP) map 𝓜_{a_i|x_i}, with Choi matrix M_{a_i|x_i}.
• A process matrix is an operator
  W ∈ A_I^1 ⊗ A_O^1 ⊗ ... ⊗ A_I^N ⊗ A_O^N
  (or more generally with additional global past/future systems)
  such that
  p(a_1,...,a_N|x_1,...,x_N) = Tr[(M_{a_1|x_1} ⊗ ... ⊗ M_{a_N|x_N}) W]
  defines a valid probability distribution for every choice of local quantum instruments.
  Equivalently, W ≥ 0 and W satisfies the usual linear normalization constraints of the process-matrix formalism.

Causality is then an operational property of the correlations generated by W:

• In the bipartite case, a correlation p(a,b|x,y) is causal iff it can be written as
  p_causal = λ p^{A<B} + (1-λ) p^{B<A},
  where p^{A<B} allows signaling from Alice to Bob but not from Bob to Alice, and p^{B<A} vice versa.
• For arbitrary numbers of parties, there is a standard recursive generalization allowing dynamical causal order.
• A process matrix W is causal iff every distribution generated from W by local instruments is causal in that sense.

The notion relevant to the open problem is stronger:

• W is extensibly causal iff for every ancillary quantum state ρ supplied jointly to the parties on extra input systems, the extended process W ⊗ ρ is still causal.
• Equivalently: no entangled ancilla extension can activate noncausal correlations.

Purifiability is the proposed “physicality” criterion:

• Allow global past and future systems P and F (possibly trivial).
• A process
  S ∈ P ⊗ A_I^1 ⊗ A_O^1 ⊗ ... ⊗ A_I^N ⊗ A_O^N ⊗ F
  is pure iff, whenever unitary transformations are applied in all local laboratories, the induced transformation from the global past P to the global future F is unitary.
• A process W is purifiable iff there exist auxiliary systems P' and F' and a pure process
  S ∈ P ⊗ P' ⊗ A_I^1 ⊗ A_O^1 ⊗ ... ⊗ A_I^N ⊗ A_O^N ⊗ F ⊗ F'
  such that
  W = Tr_{P'F'} [ S^T (|0⟩⟨0|^{P'} ⊗ I^{F'}) ],
  where the transpose is the one coming from the chosen Choi/Jamiołkowski convention.
  Informally: W admits a pure reversible dilation, from which it is recovered by fixing an ancilla input and discarding an ancilla output.

So the problem can be phrased as:

• Main question / conjectural implication: for every finite multipartite process matrix W,
  extensibly causal(W) ⇒ purifiable(W)?
• Equivalently, if EC_N denotes the set of N-party extensibly causal processes and Pur_N the set of N-party purifiable processes, is
  EC_N ⊆ Pur_N
  for all N and all finite local dimensions?

Important related distinctions:

• There are causal processes that are not causally separable.
• There are causally separable (hence causal) processes that become noncausal after extension with entangled ancillas.
• Purifiability is not known to coincide with causality: the purification paper gives a tripartite purifiable process that can still violate causal inequalities, so the open problem is specifically about the one-way implication
  extensibly causal ⇒ purifiable,
  not a general equivalence between causal and purifiable processes.

A natural sub-problem / useful boundary case is:

• Bipartite case: determine whether every bipartite extensibly causal process is purifiable.
  A later result shows the converse direction there, namely that all purifiable bipartite processes are extensibly causal, so the remaining open direction is exactly
  extensibly causal ⇒ purifiable
  in the bipartite setting.

Physical significance (as emphasized on the OQP page):

• Extensibly causal processes look like the natural “physically reasonable” subset of all process matrices, because they remain compatible with causal behavior even after arbitrary entangled ancilla extensions.
• Purifiability is motivated by reversibility: if all physically realizable higher-order processes should admit a pure reversible dilation, then the problem asks whether extensible causality already forces that structure.
• A negative answer would mean either that some extensibly causal processes are not physically realizable, or that purifiability is not a necessary condition for physical realizability.

Where to find the details / references

Primary sources:

• Open Quantum Problems site (Problem #43)
• Open Quantum Problems master list (for numbering/metadata)
• M. Araújo, A. Feix, M. Navascués, and Č. Brukner, “A purification postulate for quantum mechanics with indefinite causal order”:
  https://quantum-journal.org/papers/q-2017-04-26-10/
  https://arxiv.org/abs/1611.08535

Key related references (as listed on the OQP page):

• O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order” (Nat. Commun. 3, 1092 (2012); arXiv:1105.4464)
  (introduces the process-matrix formalism and causal inequalities)
• M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, “Witnessing causal nonseparability” (New J. Phys. 17, 102001 (2015); arXiv:1506.03776)
  (causal witnesses; the quantum switch as a causally nonseparable process that does not violate causal inequalities)
• O. Oreshkov and C. Giarmatzi, “Causal and causally separable processes” (New J. Phys. 18, 093020 (2016); arXiv:1506.05449)
  (defines causal, causally separable, extensibly causal, and extensibly causally separable processes; shows activation of noncausality by entangled ancillas)
• C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, “The simplest causal inequalities and their violation” (New J. Phys. 18, 013008 (2016); arXiv:1508.01704)
  (explicit causal inequalities and violations in the process-matrix framework)

(Recent additional pointer, directly relevant to the converse direction in the bipartite case:)

• W. Yokojima, M. T. Quintino, A. Soeda, and M. Murao, “Consequences of preserving reversibility in quantum superchannels” (Quantum 5, 441 (2021); arXiv:2003.05682)
  (shows that all purifiable bipartite processes are extensibly causal)

Prerequisites needed

• Finite-dimensional quantum mechanics: density operators, quantum channels, instruments, unitary evolution
• Completely positive maps and the Choi–Jamiołkowski representation
• Process matrices / higher-order quantum maps and the generalized Born rule
• Signaling, no-signaling, causal order, and causal inequalities
• Linear algebra / operator theory: positive semidefinite operators, tensor products, partial trace, purification / dilation
• (Optional) causal separability, the quantum switch, and classically controlled circuits as canonical examples

AMS categories

• ams-81 (Quantum theory)
• ams-47 (Operator theory)
• ams-15 (Linear and multilinear algebra; matrix theory)
• ams-94 (Information and communication theory)

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• I plan on adding this conjecture to the repository
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