dorsal/arxiv
View SchemaRetrodiction of Generalised Measurement Outcomes
| Authors | Anthony Chefles, Masahide Sasaki |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0205091 |
| URL | https://arxiv.org/abs/quant-ph/0205091 |
| DOI | 10.1103/PhysRevA.67.032112 |
Abstract
If a generalised measurement is performed on a quantum system and we do not know the outcome, are we able to retrodict it with a second measurement? We obtain a necessary and sufficient condition for perfect retrodiction of the outcome of a known generalised measurement, given the final state, for an arbitrary initial state. From this, we deduce that, when the input and output Hilbert spaces have equal (finite) dimension, it is impossible to perfectly retrodict the outcome of any fine-grained measurement (where each POVM element corresponds to a single Kraus operator) for all initial states unless the measurement is unitarily equivalent to a projective measurement. It also enables us to show that every POVM can be realised in such a way that perfect outcome retrodiction is possible for an arbitrary initial state when the number of outcomes does not exceed the output Hilbert space dimension. We then consider the situation where the initial state is not arbitrary, though it may be entangled, and describe the conditions under which unambiguous outcome retrodiction is possible for a fine-grained generalised measurement. We find that this is possible for some state if the Kraus operators are linearly independent. This condition is also necessary when the Kraus operators are non-singular. From this, we deduce that every trace-preserving quantum operation is associated with a generalised measurement whose outcome is unambiguously retrodictable for some initial state, and also that a set of unitary operators can be unambiguously discriminated iff they are linearly independent. We then examine the issue of unambiguous outcome retrodiction without entanglement. This has important connections with the theory of locally linearly dependent and locally linearly independent operators.
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"abstract": "If a generalised measurement is performed on a quantum system and we do not\nknow the outcome, are we able to retrodict it with a second measurement? We\nobtain a necessary and sufficient condition for perfect retrodiction of the\noutcome of a known generalised measurement, given the final state, for an\narbitrary initial state. From this, we deduce that, when the input and output\nHilbert spaces have equal (finite) dimension, it is impossible to perfectly\nretrodict the outcome of any fine-grained measurement (where each POVM element\ncorresponds to a single Kraus operator) for all initial states unless the\nmeasurement is unitarily equivalent to a projective measurement. It also\nenables us to show that every POVM can be realised in such a way that perfect\noutcome retrodiction is possible for an arbitrary initial state when the number\nof outcomes does not exceed the output Hilbert space dimension. We then\nconsider the situation where the initial state is not arbitrary, though it may\nbe entangled, and describe the conditions under which unambiguous outcome\nretrodiction is possible for a fine-grained generalised measurement. We find\nthat this is possible for some state if the Kraus operators are linearly\nindependent. This condition is also necessary when the Kraus operators are\nnon-singular. From this, we deduce that every trace-preserving quantum\noperation is associated with a generalised measurement whose outcome is\nunambiguously retrodictable for some initial state, and also that a set of\nunitary operators can be unambiguously discriminated iff they are linearly\nindependent. We then examine the issue of unambiguous outcome retrodiction\nwithout entanglement. This has important connections with the theory of locally\nlinearly dependent and locally linearly independent operators.",
"arxiv_id": "quant-ph/0205091",
"authors": [
"Anthony Chefles",
"Masahide Sasaki"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.67.032112",
"title": "Retrodiction of Generalised Measurement Outcomes",
"url": "https://arxiv.org/abs/quant-ph/0205091"
},
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