A quantum state, via the Born Rule, assigns probabilities to the different possible values of observables of a quantum system. It is only in virtue of these probabilities — or, alternatively, expectation values — that quantum mechanics is empirically testable. Since assigning a state to a system means assigning probabilities to the values of observables, it may seem natural to read Peierls’ claim that the state ‘represents our knowledge of the system’ (Peierls [1991], p. 19) as a shorthand for saying that the state represents our knowledge of these probabilities. The latter view is sometimes even straightforwardly identified with the epistemic conception of states, for example by Marchildon, who claims that ‘[i]n the epistemic view [of quantum states], the state vector (or wave function or density matrix) does not represent the objective state of a microscopic system […], but rather our knowledge of the probabilities of outcomes of future measurements’ (Marchildon [2004], p. 1454). However, according to an argument due to Fuchs and endorsed by Timpson, 31 the idea that quantum states represent our knowledge about quantum probabilities should not be regarded as the core claim of the epistemic conception of quantum states, simply because, as they argue, this idea is in fact incompatible with the latter. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The main ambition of this work has been to explore the prospects for dissolving the foundational problems of quantum theory by adopting the epistemic conception of quantum states and reflecting on how it should be spelled out in detail. This differs (at least in emphasis, if not more profoundly) from the main ambition of those — ‘realistically minded’ — philosophers whose declared aim is to determine what reality is like if it is as quantum theory says. If the Rule Perspective is correct, quantum theory does not really say anything about reality at all: features of reality are described not by quantum states, but by NQMCs, and while quantum theory helps us ascribe probabilities to the NQMCs, it does not typically make any statements as to which NQMCs are true and which are false. However, even though quantum theory does not in that sense say anything about reality, we can learn a lot about reality by reflecting on what features it must have for quantum theory’s undisputed empirical success to be possible. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The previous chapter addressed the challenge of making sense of the notion of a quantum state assignment being performed correctly without thereby acknowledging the notion of a quantum state a quantum system ‘is in’. The present chapter discusses what interpretation of quantum probabilities goes best with the account that I have called the ‘Rule Perspective’ and started to develop in the previous chapter. The first section considers what quantum probabilities should be taken to be probabilities of in the Rule Perspective; the second investigates what philosophical interpretation of probabilities applies best to quantum probabilities thus conceived. The third section considers two objections to the resulting account of quantum probabilities. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We derive the Hilbert space formalism of quantum mechanics from epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. An epistemic formalism is developed, where concepts like individual and collective knowledge are used, and knowledge may be actual or potential. The physical state
S corresponds to the collective potential knowledge. The state
S is a subset of a state space
S={Z}, such that
S always contains several elements
Z, which correspond to unattainable states of complete potential knowledge of the world. The evolution of
S cannot be determined in terms of the individual evolution of the elements
Z, unlike the evolution of an ensemble in classical phase space. The evolution of
S is described in terms of sequential time
n∈N, which is updated according to
n→n+1 each time potential knowledge changes. In certain experimental contexts
C, there is initial knowledge at time
n that a given series of properties
P,P′,… will be observed within a given time frame, meaning that a series of values
p,p′,… of these properties will become known. At time
n, it is just known that these values belong to predefined, finite sets
{p},{p′},…. In such a context
C, it is possible to define a complex Hilbert space
HC on top of
S, in which the elements are contextual state vectors
SˉC. Born's rule to calculate the probabilities to find the values
p,p′,… is derived as the only generally applicable such rule. Also, we can associate a self-adjoint operator
Pˉ with eigenvalues
{p} to each property
P observed within
C. These operators obey
[Pˉ,Pˉ′]=0 if and only if the precise values of
P and
P′ are simultaneoulsy knowable.
If a quantum state is prescriptive then what state should an agent assign, what expectations does this justify, and what are the grounds for those expectations? I address these questions and introduce a third important idea—decoherence. A subsystem of a system assigned an entangled state may be assigned a mixed state represented by a density operator. Quantum state assignment is an objective matter, but the correct assignment must be relativized to the physical situation of an actual or hypothetical agent for whom its prescription offers good advice, since differently situated agents have access to different information. However this situation is described, it is true, empirically significant magnitude claims that make the description correct, while others provide the objective grounds for the agent’s expectations. Quantum models of environmental decoherence certify the empirical significance of these magnitude claims while also licensing application of the Born rule to others without mentioning measurement.
Applications of quantum mechanics have led to many successful predictions and explanations of puzzling phenomena, and we now apply quantum mechanics to gain, process, and communicate information in novel ways. We can understand quantum mechanics by understanding how we have applied it. We should not seek agreement on the nature of the world it represents, because this theory does not itself represent the physical world (though its applications do help us to represent it better). When applied to a quantum state, quantum mechanics yields probabilities for physical events: both state and probability are objective—not because they represent elements of physical reality, but because each exerts normative authority over the beliefs of anyone who accepts quantum mechanics and applies it relative to a physical situation they may (but need not) occupy. These events may be described by statements (about values of magnitudes) that are meaningful in an appropriate environmental context, and quantum mechanics can help one to say when that is. Measurement creates an appropriate context, so here the Born rule indirectly yields probabilities of measurement outcomes. The quantum state of a system does not “collapse” on measurement: a new state must be assigned relative to a physical situation in which information about the outcome is accessible. Understood this way, there is no measurement problem, and violation of Bell inequalities does not demonstrate “spooky” non-local action. Quantum field theories have no physical ontology of their own: a quantum field is a mathematical object in a model whose application helps to improve and extend our descriptions of the world in other terms. We cannot realise the scenario of Wigner’s friend and its recent extensions: but the data that provide overwhelming evidence for quantum mechanics are objective in the same sense as the relative measurement outcomes described in those scenarios.
The concept of a quantum state represents one of the most fundamental pillars of the paradigm of quantum theory.'3 Contrary to its mathematical elegance and convenience in calculations, the physical interpretation of a quantum state is not so transparent. The problem is that the quantum state (described either by a state vector, or density operator or a phase-space probability density distribution) does not have a well defined objective status, i.e. a state vector is not an objective property of a particle. According to Peres (see,1 p. 374): "There is no physical evidence whatsoever that every physical system has at every instant a well defined state. .. In strict interpretation of quantum theory these mathematical symbols [i.e., state vectorsi represent statistical information enabling us to compute the probabilities of occurrence of specific events." Once this point of view is adopted then it becomes clear that any "measurement" or a reconstruction of a density operator (or its mathematical equivalent) can be understood exclusively as an expression of our knowledge about the quantum mechanical state based on a certain set of measured data. To be more specific, any quantum-mechanical reconstruction scheme is nothing more than an a posteriori estimation of the density operator of a quantum-mechanical (microscopic) system based on data obtained with the help of a macroscopic measurement apparatus.3 The quality of the reconstruction depends on the "quality" of the measured data and the efficiency of the reconstruction procedure with the help of which the data analysis is performed. In particular, we can specify three different situations. Firstly, when all system observables are precisely measured. In this case the complete reconstruction of an initially unknown state can be performed (we will call this the reconstruction on the complete observation level) . Secondly, when just part of the system observables is precisely measured then one cannot perform a complete reconstruction of the measured state. Nevertheless, the reconstructed density operator still uniquely determines mean values of the measured observables (we will denote this scheme as reconstruction on incomplete observation levels) . Finally, when measurement does not provide us with sufficient information to specify the exact mean values (or probability distributions) but only the frequencies of appearances of eigenstates of the measured observables, then one can perform an estimation (e.g. reconstruction based on quantum Bayesian inference) which is the "best" with respect to the given measured data and the a priori knowledge about the state of the measured system.