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What are quantum states? Gábor Hofer-Szabó Research Centre for the Humanities, Budapest – p. 1 Quantum interviews – p. 2 Question 4: What are quantum states? Jeffrey Bub: “the state is simply a credence function, a bookkeeping device for keeping track of probabilities” Christopher Fuchs: “a quantum state is a set of numbers an agent uses to guide the gambles he might take on the consequences of his potential interactions with a quantum system” GianCarlo Ghirardi: “quantum states are the mathematical entities that characterize, in the most accurate way possible in principle, the situation of an individual physical system” Tim Maudlin: “ultimately, there should be but one quantum state: that of the whole universe” – p. 3 Question 4: What are quantum states? Lucien Hardy: “a state corresponds to a list of probabilities that can be used to calculate the probability for any outcome of any measurement that may follow the preparation with which the state is associated” David Mermin: “Quantum states are mathematical symbols” Lee Smolin: “quantum states are to be regarded as statistical states, which result from averaging over more fundamental nonlocal degrees of freedom” Anton Zeilinger: “individual systems can be described by quantum states” – p. 4 Statistical interpretation Ballentine, 1970: “A quantum state represents an ensemble of similarly prepared systems, but does not provide a complete description of an individual system.” “Quantum mechanics predicts nothing which is relevant to a single measurement.” – p. 5 Statistical interpretation Mermin, 1993: “This heresy [the statistical interpretation] takes the state vector to describe an ensemble of systems and maintains that in each individual member of that ensemble every observable does indeed have a definite value, which the measurement merely reveals when carried out on that particular individual system.” – p. 6 Statistical interpretation Claims: 1. The rejection of statistical interpretation is the result of mixing it with a hidden variable interpretation. Suggestion: use the term “minimal interpretation.” 2. Interpreting the quantum state statistically many of the quantum mechanical measurement problems disappear. – p. 7 Two problem classes in quantum mechanics 1. Measurement problems: Superposition — Schrödinger’s cat — collapse — Wigner’s friend — decoherence — many-worlds, many-minds — ontic-epistemic debate (PBR theorem) 2. Local realism problems: EPR scenario — Bell inequalities — local causality — common cause — hidden variables — no-go theorems (Neumann, Jauch-Piron, Kochen-Specker) – p. 8 Quantum mechanics Representation: System State: s Measurement: a Outcomes: Ai −→ H: Hilbert space −→ Ws : density operator −→ Oa : self-adjoint operator −→ Pai : spectral projections Born rule: Tr(Ws Pai ) provides the probabilities – p. 9 Interpretations of quantum mechanics A11 a1 A62 a2 A21 a1 – p. 10 Interpretations of quantum mechanics A11 a1 Minimal Interpretation (MI): State: ensemble preparation Measurement settings: am Measurement outcomes: Aim A62 a2 Probability = long-run relative frequency: i ∧a ) #(A m m i ps (Am |am ) = #(am ) A21 Born rule: a1 Tr(Ws Paim ) = ps (Aim |am ) – p. 11 Interpretations of quantum mechanics Property Interpretation (PI): A11 α 11 a1 i Properties: αm j ) = δij ps (Aim |am ∧ αm ps (am ∧ αni ) = ps (am ) ps (αni ) A62 a2 α 62 Consequently: i ) Tr(Ws Paim ) = ps (Aim |am ) = ps (αm A12 a1 α 12 – p. 12 Interpretations of quantum mechanics Propensity Interpretation (PrI): A11 q α1 a1 q Propensities: αm i q ) = qm ps (Aim |am ∧ αm ps (am ∧ αnq ) = ps (am ) ps (αnq ) A62 q a2 α2 Consequently: X Tr(Ws Paim ) = ps (Aim |am ) = q ) q i ps (αm q A12 q a1 α1 – p. 13 Interpretations of quantum mechanics Copenhagen Interpretation (CI): A51 ω5 α 1 a1 αω Wave function: αω ps (Aim |am ∧ αω ) = Tr(Ws Paim ) that is Ws and αω is associated A26 ω6 ω a2 α α 2 ✭ ✭ ✭ ✭ ω ✭✭✭✭ ps (am✭ ∧✭α✭ )✭= ps (am ) ps (αω ) ✭✭✭ Projection postulate: obtaining i j ω ω n An α changes into α A14 ω a1 α α ω 41 ps (Aim |am where ωni i ∧ α ) = Tr(Wsn Paim ) i Wsn = Pnj Ws Pnj Tr(Pnj Ws Pnj ) – p. 14 Interpretations of quantum mechanics The ontology of the different interpretations: outcomes Aim & settings am PI: outcomes Aim & settings am i & properties αm PrI: outcomes Aim & settings am q & propensities αm CI: Aim MI: outcomes & settings am ω & wave function α , α i ωn – p. 15 Remarks 1. The minimal interpretation is not a property (hidden variable) interpretation. 2. But it is open towards supplementing it by properties, propensities, wave functions, etc. 3. On certain conditions (locality, non-contextuality) there is no property interpretation of quantum mechanics. 4. But this does not invalidate the minimal interpretation. 5. The minimal interpretation is not a subjective (Bayesian) interpretation. 6. The Copenhagen interpretation is a hidden variable (propensity) interpretation where the hidden variable is the wave function itself. – p. 16 Individual systems Question: Do we have any empirical or theoretical reason to refer quantum states to individual systems? – p. 17 Individual systems Zeilinger, 2011: “I should emphasize that in my opinion, individual systems can be described by quantum states. We can certainly prepare an individual system to be in an eigenstate of a specific apparatus.” – p. 18 Individual systems – p. 19 Individual systems Alter and Yamamoto, 2001: “The quantum wavefunction contains all relevant information about the single physical system. However, in order to obtain this information, and determine the wavefunction experimentally, one needs to consider the statistics of the results of a series of measurements, where each measurement is performed on a single system in an ensemble of identical systems.” – p. 20 Individual systems Question: Do we have any empirical or theoretical reason to refer quantum states to individual systems? – p. 21