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   system and apparatus prior to the measurement, that |qₖ⟩ represents the   
   system as having the property qₖ, and that |Aₖ⟩ represents the apparatus   
   as indicating this outcome. For it is self-contradictory to interpret   
   the sum Σₖ cₖ |Aₖ⟩⊗|qₖ⟩ as the combined physical state of   
   system and   
   apparatus and also to interpret the respective terms |qₖ⟩ and |Aₖ⟩ as   
   representing the system in possession of the property qₖ and the   
   apparatus as indicating this outcome. This is exactly the befuddled   
   thinking which leads to the notorious Schrödinger-cat state   
      
   |S-cat⟩ = c₁ |A₁⟩⊗|cat(alive)⟩ + c₂ |A₂⟩⊗|cat(dead)⟩,   
      
   where |cat(alive)⟩ is supposed to represents the cat as being alive and   
   |A₁⟩ is supposed to represents the apparatus as signaling this fact —   
   and ditto for |cat(dead)⟩ and |A₂⟩ — even as the sum of the two terms   
   then represents the cat as being neither dead nor alive (or both dead   
   and alive).   
      
   Here is the scenario as originally thought up by Schrödinger4:   
      
        One can even set up quite ridiculous cases. A cat is penned up in a   
   steel chamber, along with the following diabolical device (which must be   
   secured against direct interference by the cat): in a Geiger counter   
   there is a tiny bit of radioactive substance, so small, that perhaps in   
   the course of one hour one of the atoms decays, but also, with equal   
   probability, perhaps none; if it happens, the counter tube discharges   
   and through a relay releases a hammer which shatters a small flask of   
   hydrocyanic acid. If one has left this entire system to itself for an   
   hour, one would say that the cat still lives if meanwhile no atom has   
   decayed. The first atomic decay would have poisoned it. The ψ-function   
   of the entire system would express this by having in it the living and   
   the dead cat (pardon the expression) mixed or smeared out in equal   
   parts. It is typical of these cases that an indeterminacy originally   
   restricted to the atomic domain becomes transformed into macroscopic   
   indeterminacy, which can then be resolved by direct observation.   
      
   The fact of the matter is that the symbol |x⟩ denotes a vector in a   
   certain mathematical space. While by itself it bears no relation to   
   either physical reality or human experience, it is often used (and   
   should in fact only be used) as a convenient shorthand for another   
   mathematical animal, which is designated by the symbol |x⟩⟨x|. This   
   represents a measurement outcome — either that to which a probability is   
   assigned or that on the basis of which probabilities are assigned.   
      
   Probabilities are calculated as outputs of a mathematical machine T,   
   which has two input slots. The first slot is for the outcome on the   
   basis of which probabilities are assigned, the second slot is for the   
   outcome to which a probability is assigned. Thus if |A₁⟩⟨A₁| is   
   inserted   
   into the first slot, T serves to assign probabilities to the possible   
   outcomes of another measurement B, based on the information provided by   
   |A₁⟩⟨A₁|, which is that the cat is alive. If |A₁⟩⟨A₁| is   
   inserted into   
   the second slot, T serves to assign a probability to finding (by means   
   of another measurement B) that measurement A indicates that the cat is   
   alive, conditional on whatever measurement outcome is fed into the first   
   slot.   
      
   Share Aurocafe   
      
    From quantum theory’s early days, the goal of making physical sense of   
   the theory’s mathematical formalism has been pursued along two   
   apparently divergent lines, one fundamentally philosophical, the other   
   essentially mathematical; one spearheaded by Niels Bohr, the other set   
   in motion by the mathematician, physicist, and computer scientist John   
   von Neumann.   
      
   When Bohr wrote5 that “the physical content of quantum mechanics is   
   exhausted by its power to formulate statistical laws governing   
   observations obtained under conditions specified in plain language,”   
   most people took him to advocate a naïve realistic view of measuring   
   instruments and other macroscopic objects. What he was actually trying   
   to defend was an essentially Kantian stance. To him, the events which   
   quantum mechanics correlates statistically were experiences capable of   
   objectivation,6 which requires communication in terms that everybody can   
   understand. (His insistence on the use of plain language would make no   
   sense if he were merely advocating a metaphysically sterile   
   instrumentalism.)   
      
   As regards von Neumann, it is well known that the mathematical formalism   
   of quantum mechanics was worked out by him in a systematic and   
   mathematically precise way and summed up in his celebrated 1932 book.7   
   Many published discussions of interpretive issues in quantum mechanics   
   present von Neumann as viewing the “quantum state” |Ψ⟩ as a   
   representation of a physical state that is capable of changing   
   (“evolving”) in two distinct ways: continuously between measurements (as   
   also during the so-called pre-measurement stage); and discontinuously at   
   the objectification stage, when a measurement is completed and the   
   system’s state is said to “collapse.”   
      
   It is much less well known that, soon after the publication of his book,   
   von Neumann rejected in favor of |Ψ⟩⟨Ψ| the central role he had assigned   
   to |Ψ⟩. While, mathematically speaking, |Ψ⟩ is a vector in some vector   
   space V, |Ψ⟩⟨Ψ| is an operator that projects vectors into a subspace of   
   V. Von Neumann thus abandoned the notion of a physical state with two   
   distinct modes of change, and instead espoused as the physically   
   relevant core of quantum mechanics the conditional probabilities defined   
   by the “trace operator” T with its two input slots for projection   
   operators.8   
      
   While both Bohr and von Neumann (after the publication of his book) thus   
   were on convergent tracks, too many physicists and philosophers of   
   science today see the issue of interpreting quantum mechanics as a   
   choice between a view that Bohr never held (instrumentalism) and a view   
   that von Neumann soon abandoned (quantum state realism).   
      
   There are actually two measurement problems. One, sometimes called the   
   “big” measurement problem, is the problem of explaining how measurement   
   outcomes come about “dynamically,” i.e., as a result of a single   
   continuous mode of change, without invoking a second, discontinuous mode   
   of change. This problem arises from the false premise that |Ψ⟩   
   represents a physical state, and that its dependence on time is the   
   continuous time dependence of a physical state.   
      
   While the passage of time between the first measurement (on the basis of   
   whose outcome probabilities are assigned) and the second measurement (to   
   the possible outcomes of which probabilities are assigned) is taken care   
   of by an operator that depends on the respective times of the two   
   measurements, this operator does not represent a physical process that   
   brings about a physical change. Any story purporting to relate what   
   happens between the two measurements is (in Wolfgang Pauli’s felicitous   
   phrase) “not even wrong,” inasmuch as it can be neither proved nor   
   disproved.   
      
   “Observations,” as Schrödinger wrote, “are to be regarded as discrete,   
   disconnected events. Between them there are gaps which we cannot fill   
   in.” The reason we cannot fill in these gaps is that the concepts at our   
   disposal — in particular: position and momentum, time and energy,   
   causality and interaction — owe their meanings in large part to the   
      
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