"In classical physics, the notion of the “state” of a physical system is quite intuitive...there exists a one-to-one correspondencebetween the physical properties of the object (and thus the entities of the physical world) and their formal and mathematical representation in the theory...

With the advent of quantum theory in the early twentieth century, this straightforward bijectivism between the physical world and its mathematical representation in the theory came to a sudden end. Instead of describing the state of a physical system by means of intuitive symbols that corresponded directly to the “objectively existing” physical properties of our experience, in quantum mechanics we have at our disposal only an abstract quantum state that is defined as a vector (or, more generally, as a ray) in a similarly abstract Hilbert vector space." (emphasis added; italics in original)"

Schlosshauer, M. A. (2007). Decoherence and the Quantum-To-Classical Transition. Springer.

I think that's the most concise description of the ontological nature of the formalisms of quantum physics (and in particular quantum mechanics) that is both relatively uncontentious and not overly simplistic. Shankar's classical text has side-by-side columns comparing the postulates of QM and their closest classical analogues (when they exist) that is perhaps even clearer, but this is due to the "graphical" (organizational formatting)  effect, not the description.

The "Copenhagen interpretation" (CI) and the standard/orthodox interpretation of QM (such as either exist and what their relationship may be; see attached) holds that this question is meaningless. Bohr's approach to the deeply troubling nature of the quantum realm was to ignore it as any reality. Mark P. Silverman (in Quantum Superposition) promotes this view baldly, forcefully, and unambiguously: "What quantum mechanics ‘adds up to’ is that it is an irreducibly statistical theory". One of the most problematic aspects of modern physics is the extent to which we rely on mathematical frameworks & formalisms for the development (even creation) of theory. With QM, are at least fairly clear as to how the correspondence between the mathematical representation of the system fails to correspond to any property or state of a physical system. This is in part because of what QM is (or one way of conceiving of what it is).

QM, from one perspective, is a mathematical structure with a corresponding procedure for experiments. It is a way to prepare systems in a specified way so that they can be transcribed into a wave-function. QM provides the way to take the system in question and its manner of preparation and apply to it a transformation function to yield a new wave-function (the final wave-function). In a very real way, the prepared system and the measured system (or the initial and final wave-functions) are two different systems. The mathematical structure of QM gives us a method for obtaining probabilities that prepared systems of specified types will yield particular outcomes given specified measurements.

For Bohr (it seems) and for others, this is where QM ends. For those who think a physical theory should be more than a procedure, the interpretation of QM is of great importance. Whatever one's position, however, there is absolutely no 1-to-1 correspondence between the wave-function or state-vector and a physical system (or its properties). After all, "observables" aren't values but functions.

     Quantum states are no real objects and only their square do it as a density of probability. The best way to see that there is not a one-to-one correspondence is that you can multiply then by one continuous exponential belonging to U(1). Therefore you have at least a correspondence to infinity values with one given state.

To Andrew Messing:

Dear Andrew,

I agree with you when you point up that the ontological character of the state vector depends on the particular interpretation of quantum mechanics. Yet I disagree when you state that “Whatever one's position, however, there is absolutely no 1-to-1 correspondence between the wave-function or state-vector and a physical system (or its properties). After all, "observables" aren't values but functions”.

For example, in the pilot wave interpretation (PWI), the wave function has an ontological meaning independently of the existence or nonexistence of the observer. The wave carries the particle into regions where the field is non-vanishing and omits regions where the quantum field cancels, therefore in PWI there is a bijection relation between the wave function (or the state vector) and the element of the reality – the physical state of the system.

In the many worlds interpretation (MWI) the ψ field is an ontological entity like for the PWI (however here there is not particle at all only a wave). Consequently, in MWI the square modulus of the ψ field is a measure of existence or a degree of reality. Again, in MWI there is a one-to-one correspondence between the state vectors and the elements of the reality: According to Everett, ψ represents an ontological field and the difference between ψ1 and ψ2 is completely physical in an objective sense (for example, those states are orthogonal in the Hilbert space).

I agree with the Baldomir Professor, but i believe that your questions is related with the "reality" of the quantum state.  The quantum state is only a mathematical representation of the probability of to have a state of the reality, but the measurement and subsequent collapse of this represent the reality. An example of this is the paper "Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark State" where the authors report the imagen of the wave function of one electron in a hydrogen atom.

I hope that my answer help you, abrazo.

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.213001

To Daniel Baldomir:

Dear Daniel.

Sure, quantum states are not real objects; they are just mathematical symbols. The question is what do they represent? For example, phase points aren’t real, however they represent physical states of a mechanical system. Similarly, do state vectors correspond to states of the physical reality or only to states of knowledge?

To Thomas Cantu:

Dear Thomas.

It’s the exactly my point that “there are certain aspects of vector space mathematics that directly corresponds to real world features of the waves structure”. From a category-theoretical perspective, such a correspondence stands for an isomorphism between mathematical formalism of a physical theory and elements of the physical reality.

State vector is a thing we know how to manipulate. Easy and understandable. Reality is neither. It is much too complex and poorly defined. For example, is your question a reality. Well, it is, though it takes on just different matter on itself on my IPad than on yours, can take the form of the sound wave or an electromagnetic signal. 

You can't define simple object in terms of a complex and fuzzy one. 

      Dear Arkady,

     The reality in Quantum Mechanics is only associated to the experiments which takes into account of the possible results of a measurement by means of its the eigenvalues representing the observable (the choice of Hermitian operators is for obtaining real eigenvalues). The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator. And always the states ( U(1) vectors) are transformed under a phase indeterminacy (or gauge transformation). Therefore the states are not real physical objects at all.

Dear Arkady Bolotin:

Regarding your disagreement: "Yet I disagree when you state that '...there is absolutely no 1-to-1 correspondence between the wave-function or state-vector and a physical system (or its properties).' For example, in the pilot wave interpretation (PWI), the wave function has an ontological meaning independently of the existence or nonexistence of the observer."

I see two potential issues here. The one that is paradoxically (as is the way with quantum physics) both simpler and more subtle relates to your counter-example. It is an interpretation of QM. Were there a 1-to-1 correspondence between the physical system and the mathematical representation, we wouldn't require any such interpretation. Additionally, Everett's original formulation of what was later called the many-worlds interpretation (MWI) was an attempt not merely to provide an ontological interpretation but the simplest possible interpretation (namely, simply take as given that everything was physically realized and thus the reason we only ever see a single outcome is because of branching universes and so forth). Perhaps I should have said "everyone agrees that there is no straightforward or known 1-to-1 correspondence between the wave-function (i.e., the mathematical representation) and any physical system" or something similar. Consistent quantum theories, Bohmian mechanics, and a number of other proposals offer either interpretations of QM or an alternative theory, and it is not always clear when one falls into the former or latter category. Bohmian mechanics, for example, is often thought of and treated as an interpretation of quantum mechanics:

"Bohmian mechanics is often said to be an interpretation of quantum mechanics, which should indicate redundancy, i.e., that it is merely an interpretation. But Bohmian mechanics is not an interpretation of anything...It is a complete theory where nothing is left open, and above all, it does not need an interpretation. It is a theory of nature" (italics in original)

Dürr, D., & Teufel, S. (2009). Bohmian Mechanics: The Physics and Mathematics of Quantum Theory (Fundamental Theories of Physics). Springer.

In a very real sense, many if not all "interpretations" of QM are actually different theories, for what we call "interpretations" entail radically different theories about the nature of reality.

That said, in keeping with the tradition of referring to approaches as to the nature of QM as "interpretations" of one and the same theory, then it cannot be that there exists any 1-to-1 correspondence between the formalism and the physics (between the mathematical representation of the system's state/properties) and some physical system. For one thing (and at least part of the motivation behind ensemble interpretations) in order to prepare a system we must fundamentally disturb it. This is, in fact, what preparation is: disturbing a system in particular ways dictated by theory so that we may transcribe the result into mathematical notation. Recall also that a postulate of QM is that the state-vector/wave-function contains "all the information needed" (or something similar to this), which is prima facie odd (to put it mildly), as the uncertainty principle tells us that such complete information about any quantum system is impossible.

Finally, there is the method whereby we actually obtain values from measurement that tell us the relationship between the physical system and our formalisms. In classical mechanics, position and momentum were enough to completely characterize any dynamical system (as all other properties of interest could be derived from (x, p). In QM, "obervables" are matrices that "act on" the system in accordance with the statistical structure of QM. Thus even given an interpretation which posits acorrespondence between the mathematical notation and the physical system, this correspondence cannot be 1-to-1 as there are no values to correspond to observable properties. In classical physics, I can refer to values such as work, energy, momentum, etc., and never worry about correspondence because these values directly (1-to1) correspond to the notions I measure. In QM, such "observables" are Hermitian matrices that are derived theoretically (for the most part).

To Daniel Baldomir:

Dear Daniel.

What you are saying in your response represents a pragmatic and instrumentalist interpretation of QM, in which only macroscopic apparatuses and detectors possess a clear definition.

But I am sure you’re already know this. Such a way of thinking does not try to solve the problem of the status of a wave function but only uses it as a practical tool for defining a probability as obtained by a ‘classical observer’.

Of course, there is nothing wrong about the instrumentalist interpretation of QM. However, questions such as about the meaning of the wave function for the Universe, or about the Schrodinger cat, or about the Heisenberg shifty split, and so on cannot be answered within the frame of such a minimalist approach to QM.

To Andrew Messing:

Dear Andrew,

Allow me again to disagree with you. You said “Were there a 1-to-1 correspondence between the physical system and the mathematical representation, we wouldn't require any such interpretation”.

First, there is a bijection relation between elements of the theory and elements of the reality – it depends whom you ask.

Recently several ontological models of quantum mechanics have been introduced, that is, theories which are predictively equivalent to quantum mechanics, but providing a possibly richer description of the microscopic reality through the so-called ontic state, (i.e., the most accurate specification of the physical state of the system, at least in principle). For example, R. Colbeck and R. Renner [Nature Comm. 2, 411 (2011)] derive completeness of quantum mechanics and the one-to-one correspondence between quantum state vectors and elements of reality from the assumptions of the validity of the predictions of quantum mechanics.

Second, I do not insist on any isomorphism existing between state vectors and the ontic states: Let me remind you that this is my question, not a statement.

To Igor Goliney:

Dear Igor,

Frankly, I did not get what you’re trying to say in your response except that everything is gloom and doom. However, even though I see some rationale in such a pessimistic point of view, I cannot share it completely.

Dear Arkady Bolotin:

A bijection, or 1-to1 correspondence, between "elements of the theory and elements of reality" isn't the same as a 1-to1 correspondence between the physical and the formalisms. You are providing me with examples of metatheoretical approaches, and I freely grant that many of these (most, actually) posit a certain relationship between the formalism or "elements of the theory" and "elements of reality." The only reason there are such interpretations/approaches is because such a relationship isn't intrinsic to the theory. Were we to survey the literature from Galileo to the emergence of quantum theory, we would find a vast amount of literature written on the interpretation of probability theory, logic, language, etc., but not mechanics. Alternatively, we need only examine the use of terms from classical physics and their quantum mechanical equivalents. Again, the measured values from classical physics corresponded directly to what was measured, making terms like "observable" at best redundant and at worst simply confusing. According to the OED (the real OED, the one that currently exists only via electronic database, not the various Oxford dictionaries) the first example of the sense of "observable" to refer to physics is Dirac's 1930 Principles of Quantum Mechanics.

It is this that I (and the source I quoted initially) refer to when I say that QM lacks a 1-to-1 correspondence. The observable properties of classical systems, like momentum, have no values in QM. There cannot exist a 1-to-1 relationship between the momentum of a system and a theoretically derived matrix operating on a abstract vector also theoretically derived in an infinite dimensional space. There can exist a relationship between the formalism and reality, but the fact hat various such relationships are posited and none agreed upon is a testament to the lack of a direct correspondence of the type we find in classical physics. Additionally, the fact that there exist physicists who argue that there does not exist any such relationship and they cannot be shown to be false, despite the various interpretations and theories since Bohr's original and "orthodox" statistical interpretation, again is a testament to a lack of any 1-to-1 correspondence. I'm not disagreeing that there can exist a correspondence; indeed, I feel strongly that one exists and that certain types of experiments (particularly those that demonstrate clearly quantum phenomena well-beyond the subatomic or even atomic scale) make an irreducibly statistical interpretation untenable. This does not change the fact that whatever else may be, we don't know what it is, and therefore it isn't 1-to-1 (i.e., the "observables" aren't actually observed values, the prepared system is said to be in a state that quantum theory dictates we can't know yet mathematically we claim to know exactly, and measurements are merely a way to connect via the mathematical apparatus of QM the manner or preparation to experimental outcome).

To Andrew Messing:

Dear Andrew,

I think there’s some misunderstanding from you of what I asked in this thread. The key issue here is the status of the quantum state vector or the wave function. Does the wave function correspond directly to some kind of physical reality (e.g. a wave)? If it does then it implies that the wave function is in one-to-one correspondence with elements of reality and may therefore be itself considered an element of reality of the system.

Or, is the quantum state something less than real? In particular, it is often argued that the quantum state does not correspond directly to reality, but represents an experimenter's knowledge or information about some aspect of reality. In such a case, different state vectors could correspond to the same ontic state of the system and so any bijection relation between them and elements of the reality wouldn’t be possible.

The problem is that if we assume that the quantum state is a real physical state, then collapse would be a mysterious physical process, whose precise time of occurrence is not well defined. On the other hand, if the quantum state merely represents information about the real physical state of a system, then experimental predictions are obtained which contradict those of quantum theory (see, for example, Pusey M., Barrett J., Rudolph T.: “On the reality of the quantum state”, Nature Phys. 8, 476, 2012).

Dear Arkady Bolotin:

I understood the question. If you look at my initial response, it addresses the idea of correspondence quite directly and notes that there exists no 1-to-1 correspondence. The state-vector cannot be "created" (i.e., we cannot transcribe according to quantum mechanical theory the specifications on the prepared system) without necessarily ensuring that there is no direct correspondence between the state of the system as represented in the mathematical representations and whatever there may be in reality.

Does the wave-function or state-vector correspond to some sort of reality? Of course. There is no theory or interpretation that holds otherwise, as this holds true even of statistical mechanics which we are well-aware is wrong (or a simplification).

The "quantum state" as per the formalism of QM obviously doesn't correspond to any state of any known physical system. If it did, we would not have the very interpretations of QM that you have proffered. Moreover, the state of the system is prepared by fundamentally and repeatedly disturbing the would-be system of interest according to the statistical nature of QM, such that when we let the system "run" and then "measure" it, we can use the theoretically derived system and the theoretically derived matrix operator to relate experimental outcomes to the initial system.

As the transcribed quantum state is always a "ensemble" system of repeated disturbances to obtain the "state-vector" which contains all the information relevant to the system in direct contradistinction to the uncertainty principle were it not for the face that this "state" has no direct., known correspondence to anything physical, QM would be useless.

Procedurally, interpreting QM in terms of probabilistic outcomes neither yields mysterious processes nor contradictory results. It is simply insufficient and undesirable. This doesn't mean, however, that we should seek to contradict either QM theory or experimental outcomes in order to force an interpretation so fill gaps in our knowledge.

Dear Arkady and Andrew,

It seems that both share the idea of an alternative existence to the quantum mechanics. Let me to summarize my answer in some points:

1. There is only one Quantum Mechanics which is working everyday, in all that I know. You can called it instrumentalist as if it could be another different one.

2. The square of the wave function is a physical observable, i.e. the density of probability as has interpreted Max Born. Therefore there is a relationship between states and physical observables (density of probability) but not 1-to-1 in absolute. One simple form to see it is that you transform a complex magnitude in one real one by cutting the pure complex axes.

3. One practical example is the theory of density functional theory (DFT) where a density of probability is found for a fundamental state and after that the states are calculated with the Kohn-Sham equations. These states are so good as the ones that we could calculate with the Schrödinger equations in spite to have a very different form.

4. The theory of Quantum Mechanics (QM) is not a "logic theory" coming from the common sense or being interpreted directly from the physical phenomena as Classical Mechanics does. This was made by Einstein or Bohm looking for hidden variables to substitute it, but all that they have found are effects as the Aharonov-Bohm, which paradoxically are explained perfectly in QM.

5. I think that it would be very interesting if you could show the new problems or difficulties solved by the "alternative" theories that you said.

Therefore, trying to answer your question, the relation between measurable real phenomenon and wave function is infinity-to-1 and no 1-to-1

Dear Daniel Baldomir :

1) There is only one QM if it is treated as a procedure and mathematical structure. Historically, as well as today, physical theories have consisted of more than recipes.

2) There is no such thing as a "physical observable" in QM. Observables are mathematical functions that yield results, but are never, ever observed.

3) Your example is of the type I have referred to: fundamentally non-classical and bearing no readily comparable relationship to measurable/observable values in classical physics. Admittedly, it doesn't take long when working with quantum formalisms and systems to forget the significance of this divide, yet it exists nonetheless.

4) There is no mathematical system that is not also a logical theory, and von Neurmann's formulation of quantum logical was but the first and perhaps most fundamental. There is no such thing as a non-logical formal systems. That would be a contradiction in terms.

5) I have maintained that no new solutions can be offered other than by speculation. QM is what it is; interpretations are another thing altogether.

Thanks for your insight! 

To Andrew Messing:

Dear Andrew,

I am very sorry but I do not understand: According to you, does the wave function correspond directly to some kind of physical reality, or does not?

If you put “Does the wave-function or state-vector correspond to some sort of reality? Of course”, then how come you state just in a sentence later that “The "quantum state" as per the formalism of QM obviously doesn't correspond to any state of any known physical system”?

It seems to me that you have your own interpretation of QM, so it would be very helpful if you highlight the key points of it before we would argue further.

Classical properties are "isomorph" to classical reality, but classical reality is not real.  For example, a momentum can't be assigned to a real particle, it has only a meaning inside the classical theory, and some experiments show that the measured value isn't deterministic.  That is to say, theories are not reality, mathematics isn't Nature, and quantum mechanics, quantum field theory, however precisely they decribe some phenomena, are no exception.

Dear Arkady Bolotin:

The orthodox interpretation of the wave-function is that it does not in any direct way whatsoever correspond to any physical system's state or properties. It is irreducibly statistical. In a reply above I cited Silverman saying as much, and this dates back to Bohr's interpretation.

I am not as skeptical as those who would sever ties to a statistical mechanics that is intended to describe the fundamental constituents and processes of all that is real. I therefore find myself as most do: stuck between a formalism that cannot be 1-to1 and yet is related to reality.

Apart from my belief that the orthodox interpretation is wrong, and that there is a correspondence between the formalism and the physics, I cannot say anything more, and any who do are merely speculating.

Dear Daniel,

I trust you are aware that although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations.

I’m sure you also know that there exist a number of contending schools of thought (at least 9), differing over whether quantum mechanics can be understood to be deterministic, which elements of quantum mechanics can be considered "real", and other matters.

So, what’s the matter with your response? Do you really think that despite the fact that many physicists continue to show a strong interest in this subject (just look for example a few issues of Foundation of Physics journal), they all are mistaken and spend their valuable time for trifles?

To Claude Pierre Massé:

Dear Claude,

Please recall that isomorphism is not equality. A point in the multidimensional phase space is not real (it’s a mathematical abstraction) but it is isomorphic to a real physical state of a system that can be measured. Being isomorphic, the theoretical elements of classical mechanics (e.g., phase points) are in a one-to-one correspondence with the elements of the reality – the physical states of the mechanical system.

Now the question is, are the elements of QM – such as quantum state vectors – also isomorphic to the element of the reality – the physical states of the quantum system?

1) There is only one QM if it is treated as a procedure and mathematical structure. Historically, as well as today, physical theories have consisted of more than recipes.

    --I fully agree.

2)There is no such thing as a "physical observable" in QM. Observables are mathematical functions that yield results, but are never, ever observed.

    ..I do not know if I understand you, perhaps it is a question of language. A physical observable is a measurable operator, where by a sequence of physical operations it is possible an eigenvalue (number) acting on a system state. For instance, these operations could be to apply one external magnetic field to fix the phase of the state or put an array of magnetic vortices for calculating the Berry phase. Obviously this is measurable and also this corresponds to the idea of observable. Another thing is that the measurement process avoid to know exactly the value because it interferes with the experiment.

3)Your example is of the type I have referred to: fundamentally non-classical and bearing no readily comparable relationship to measurable/observable values in classical physics. Admittedly, it doesn't take long when working with quantum formalisms and systems to forget the significance of this divide, yet it exists nonetheless.

   -My example is DFT and it is well known that provides eigenvalues of energies (Fermi energy, bands, density of states and so on).

4) There is no mathematical system that is not also a logical theory, and von Neurmann's formulation of quantum logical was but the first and perhaps most fundamental. There is no such thing as a non-logical formal systems. That would be a contradiction in terms.

    – Let me to put you one example for fixing the language. Is it logic that we have only two quantum statistical behaviours: fermions or bosons? For me this is just a fact and no a logical deduction. The same could happen with their commutation rules, etc...

5) I have maintained that no new solutions can be offered other than by speculation. QM is what it is; interpretations are another thing altogether.

  – I fully agree.

Dear Claude,

Just another thing I forgot to mention. The isomorphic relation between state vectors and elements of the reality has nothing to do with Born’s rule, i.e., the statistical interpretation of Psi vectors.

Indeed, suppose that Psi vectors are isomorphic to the physical states of the systems. Then it would imply that two different state vectors Psi1 and Psi2 would correspond to two different sets Y1 and Y2 of values of the system’s observables such that Y1 and Y2 would have the disjoint probability distributions for measured values.

Alternatively, if state vectors Psi1 and Psi2 aren’t isomorphic to the physical states 1 and 2 then two sets Y1 and Y2 of the values may have an overlapping distribution.

Dear Arkady,

I am very ignorant because I don't know most of the things that you assume that I know. Sorry!

Could you tell me one experiment in QM which several interpretations?

Could you name any of these schools of thought devoted to QM? Are they physicists? Do you think that there are yet physicists that they think that QM can be formulated as a deterministic theory? Thank you in advance.

Arkadi, the question is, which reality?  In the prevaling philosophy (positivism,) reality isn't what is modelled, it is what is confronted to the prediction of the theories, and is able to falsify them.  There is no object in Nature that is isomorph to a point in phase space.  There is only under a particular approximation, that is, under particular experimental conditions.  In the case of quantum mechanics, the condition is a weak gravitational field, since it is incompatible with general relativity.

    Dear Claude Pierre,

     Quantum Mechanics is compatible with Gravitation everyday, another thing is that we don't know the way to formulate Quantum Gravity Theory. And from my humble point of view in Physics reality is what we can measure in different conditions.

Dear Daniel,

Glad to oblige.

Except the different QM formulations (e.g., the matrix formulation, the wave function formulation, Feynman’s path integral formulation, Wigner’s phase space formulation, and, of course, the density matrix formulation), which have different problems but also provide different insights, it is noteworthy to mention (among many others) those interpretations of QM, which add new physics (absent in the listed formulations of QM).

Such interpretations are

• the pilot wave interpretation (de Broglie–Bohm) introducing some real quantum field that guides the particles,
• the many-worlds interpretation (Everett) introducing the (physically real) wave function of the Universe,
• the transactional interpretation (Cramer) adding to physics retarded waves (moving forward in time) and advanced waves (moving backward in time),
• and the objective collapse theories (including the Ghirardi-Rimini-Weber theory and the Penrose interpretation) that take the process of collapse of the wave function as ontologically objective.

Examples of interpretations that formulate QM as deterministic theory are

• the pilot wave interpretation, which postulates an actual configuration that exists even when unobserved,
• and the interpretation proposed by Gerard ’t Hooft, which states that beneath QM there may be a deterministic theory with (local) information loss (like superstrings; this may lead to a sufficiently complex vacuum state, and to an apparent non-locality in the relation between the deterministic “ontological” states and the quantum states, of the kind needed to explain away the Bell inequalities).

But you may ask, what’s the difference? Is the simplest instrumentalist interpretation not enough to explain everything? Apparently not.

For example, according to such an interpretation the time evolution of an unobserved (unperturbed) system is unitary. This leads to the quantum deterministic principle meaning that one can in principle solve the Schrödinger equation for a given physical system with the initial condition to predict the state of the system at any future time t.

However, the quantum deterministic principle brings about the black hole information loss paradox resulting from the apparent conflict between this principle and Hawking’s black hole radiation, which suggests that since the final quantum state of a physical system falling into a black hole will be mixed (thermal), the information about the initial quantum state of the system will disappear when the black hole evaporates.

In other words, the complete information about the quantum state of this system at the initial point in time cannot determine the system’s final quantum state. As it is understood now, such a paradox is to a large extent independent from a quantum treatment of the space–time degrees of freedom, i.e. a quantum theory of gravity, but depends crucially on assuming a limitless feasibility of the quantum deterministic principle.

Thus, resolving foundation issues of QM can also resolve the black hole information loss paradox.

Dear Arkady Bolotin:

You asked whether I am asserting the wave function corresponds directly to some kind of physical reality or not. I am saying it does not correspond directly, that the so-called Copenhagen interpretation (or standard/orthodox interpretation) of QM says "absolutely not", and that I and many others believe it corresponds to physical reality in some way that we do not know of, but certainly not directly (for, were there a direct correspondence between measurable properties of a system and these physical properties, we would not require "observables" as they exist in QM; we'd continue to have the direct correspondence of classical physics between measured values and the properties measured).

I do not have a personal interpretation of QM (at least not in any sense one normally refers to). I suppose my position is that the orthodox interpretation (the procedural approach to QM in which the state-vector or wave-function is purely a mathematical entity that we use to compute probabilities of particular types) isn't physics. To clarify, allow me to compare QM to classical statistical mechanics. Both are statistical in structure. Both use probabilities and other notions to describe systems without any one-to-one correspondence (albeit for entirely different reasons). However, in statistical mechanics we explicitly refer to thermodynamic potentials, that pressure is given by 1/3 rho times the average speed squared, that when we refer to distributions we really mean distributions and there is no question of interpretation. None of this is true in quantum mechanics.

In statistical mechanics, a statistical ensemble is precisely that. In quantum mechanics, it's an interpretation in the tradition of Einstein in which a quantum system is really an ensemble of systems (as in statistical mechanics). 

QM combines the language of classical mechanics with the structure (or a structure similar to that) of statistical mechanics. We describe a system's state we've derived via theoretical and statistical methods, and the only thing we do know about the nature of the state of this system is that we don't know how it corresponds to anything in the physical world other than the way experimental preparation is connected to experimental outcome. We refer to "observables" to mean those properties of a system we can measure, but we don't actually mean the properties of any system we mean mathematical functions we've replaced the classical values with. I share Einstein's deep concern with any physical theory that doesn't tell us how it relates to physical reality, although I do not subscribe to the ensemble interpretation of QM. I simply think that the use of terms from classical physics to mean something entirely different is not just misleading but makes QM a statistical theory masquerading as mechanics. The phase space of classical physics isn't like the Hilbert space of quantum mechanics, and not just because one is a complex (usually infinite-dimensional) function space and the other Euclidean, but because the ways in which we characterize systems in the phase space or phase plane (some make distinctions here but it isn't relevant enough to bother with) is not at all the way we characterize systems in QM. In classical physics, we completely characterize the system by position and momentum, and every dynamical variable we wish is theoretically derivable from these values. In QM, we represent the state of the system by a ket in Hilbert space and we represent position and momentum by Hermitian operators which yield (when acting on the mathematical "system") something conditional about the nature of measurement given the manner of preparation (and the type of system).

In short, I think that simply thinking of wave-function as only something used to calculate probabilities is wrong, at least insofar at by "only" I mean that QM is considered irreducibly statistical and to ask the question "to what does the wave-function correspond to?" is like asking what the normal distribution or conditional probability corresponds to. But even though I reject this view, I do so only in the sense that it is a meaningful question that remains unanswered and perhaps will always remain so.

Dear Andrew,

Let me evaluate what you have just said (sorry for the unavoidable pedagogism – occupational hazard).

1.You said that the wave function does not correspond directly to some kind of physical reality and “…the so-called Copenhagen interpretation (or standard/orthodox interpretation) of QM says "absolutely not".

Unfortunately, that’s not correct. According to the ‘orthodox’ interpretation of QM, a pure quantum state provides a complete description of reality such that the quantum state vectors describing a system are one-to-one related to the ontic states (i.e., complete specification of the properties) of this system. In other words, pure quantum states are associated one-to-one with ontic states so that knowing the quantum state to be ψ implies having a state of complete knowledge about the ontic state of the system.

2. You said “I suppose my position is that the orthodox interpretation (the procedural approach to QM in which the state-vector or wave-function is purely a mathematical entity that we use to compute probabilities of particular types) isn't physics.”

But such a position cannot be called `orthodox’, rather it is close to the instrumentalist interpretation of QM, whose motto is “Ignorance is bliss”.

3. You said that the question "to what does the wave-function correspond to?" “remains unanswered and perhaps will always remain so”.

But this is exactly the key question, the answer to which divides physicists into rival camps: those who answer that quantum states are isomorphic to the ontic states support ψ-ontic models of QM, and those who say quantum states aren’t isomorphic to the reality support ψ-epistemic models of QM.

It’s my understanding that you (as many other practicing physicists) probably never thought hard about foundations of QM considering this subject either historical or philosophical and definitely not as a matter of a real physical theory. However, it’s not true. Wherever in the modern physics you take a look – string theory, quantum gravitation, standard model – you will see that without deep and profound understanding of the QM foundations any further progress is impossible.

And if I succeed even in the slightest convincing you of that I would be more than happy.

Dear Arkady Bolotin:

Regarding the orthodox (or standard, and of Copenhagen) interpretation, we can survey a number of sources of diverse nature and, while we find differences we will not find a view described as any of these that asserts a quantum state "provides a complete description of reality" or that there exists any known relation between the state-vector and reality (rather, the postulate that is always given as something akin to asserting that all the information about the system is encapsulated in the state-vector). For example, here are what two of the more popular quantum mechanics textbooks state:

""The orthodox position: The particle wasn't really anywhere. It was the act of measurement that forced the particle to "take a stand" (though how and why it decided on the point C we dare not ask). Jordan said it most starkly: "Observations not only disturb what is measured, the produce it...This view (the so-called Copenhagen interpretation) is associated with Bohr and his followers. Among physicists it has always been the most widely accepted position." (italics and emphasis in original)

From Griffiths' Introduction to Quantum Mechanics

"we have a single ket |ψ> representing the particle in Hilbert space, and it contains the statistical prediction for all observables...As far as the state vector |ψ> is concerned, there is just one space, the Hilbert space, in which it resides."

From Shankar's Principles of Quantum Mechanics.

Going to a different kind of literature (monograph series), we find more detailed discussions and clearer expositions:

"In classical physics, the notion of the “state” of a physical system is quite intuitive...there exists a one-to-one correspondence between the physical properties of the object (and thus the entities of the physical world) and their formal and mathematical representation in the theory...

With the advent of quantum theory in the early twentieth century, this straightforward bijectivism between the physical world and its mathematical representation in the theory came to a sudden end. Instead of describing the state of a physical system by means of intuitive symbols that corresponded directly to the “objectively existing” physical properties of our experience, in quantum mechanics we have at our disposal only an abstract quantum state that is defined as a vector (or, more generally, as a ray) in a similarly abstract Hilbert vector space." (emphasis added; italics in original)"

Schlosshauer, M. A. (2007). Decoherence and the Quantum-To-Classical Transition. Springer.

"if the ontological differences among the versions of the Collapse-Free interpretation are put aside, then that rather popular approach could be seen to accord in a number of respects with at least one understanding of the Copenhagen interpretation. That option is assisted by both the vagueness of Bohr’s presentation and the fact that, in Heisenberg’s clearer articulation, quantum mechanics serves only to predict the results of measurements, which are required to be only classically describable, rather than to describe the microscopic world between measurements, say as a means for explaining what is observed." (emphasis added)

Jaeger, G. (2009). Entanglement, Information, and the Interpretation of Quantum Mechanics. Springer.

We can turn to journal articles:

"In the standard and Copenhagen interpretations, property ascription is determined by an observable that represents the measurement of a physical quantity and that in turn defines the preferred basis. However, any Hermitian operator can play the role of an observable, and thus any given state has the potential for an infinite number of different properties whose attribution is usually mutually exclusive unless the corresponding observables commute sin which case they share a common eigenbasis, which preserves the uniqueness of the preferred basisd. What then determines the observable that is being measured?"

Schlosshauer, M. (2005). Decoherence, the measurement problem, and interpretations of quantum mechanics. Reviews of Modern Physics, 76(4), 1267.

"In short, the mathematical properties of the wave functions are completely in accord with the idea that they describe the evolution of the probabilities of the actual things, not the actual things themselves. The idea that they describe also the evolution of the actual things themselves leads to metaphysical monstrosities."

Stapp, H. P. (1972). The Copenhagen interpretation. American Journal of Physics, 40(8), 1098-1116.

"The point of view of most physicists is rather pragmatic: it is a physical theory with a definite mathematical background which finds excellent agreement with experiment. So, from a technical point of view, quantum mechanics (QM) is a set of mathematically formulated presecriptions that deserves for calculations of probabilities of different measurement outcomes."

Caponigro, M. (2010). Interpretations of Quantum Mechanics: A Critical Survey. Prespacetime Journal, 1(5).

"if the present basic tenets of what we lossely term the Copenhagen interpretation are correct, statistical explanation is intrinsic to the theory: since, to repeat, only in the act of observation does the event come into being, and it is a consequence in the theory that there can be no predictive statement of microscopic events."

Schlegel, R. (1970). Statistical explanation in physics: The Copenhagen interpretation. Synthese, 21(1), 65-82.

And finally from the useful Compendium of Quantum Physics:

"one cannot say that the Born probabilities are either subjective (i.e. Bayesian, or due to ignorance) or objective (i.e. fundamentally ingrained in nature and independent of the observer). Instead, the situation is more subtle and has no counterpart in classical physics or probability theory: the choice of a particular classical description is subjective, but once it has been made the ensuing probabilities are objective and the particular outcome of an experiment compatible with the chosen classical context is unpredictable."

Landsman, N. P. (2009). The Born rule and its interpretation.

As I noted in my first reply, there is a remarkable degree of divergence over what, exactly, the orthodox position is (and how it relates to the Copenhagen interpretation), but the literature make clear that insofar as any "orthodox" interpretation is actually the one held at least in practice by most physicists, it must be one in which the question "how does the state-vector correspond to the state of a physical system?" the answer is "don't ask" (or, alternatively, and in particular if one examines the formulation of the CI by Heisenberg and the later formulations of von Neumann and Wigner, the state of a quantum system is a relation between experimental preparation and outcome).

That the state-vector contains "all the information needed" or "all the information there is" (and various other forms this postulate takes depending upon the author) says nothing about the correspondence between the mathematical representation and the physical system and more importantly says nothing about our knowledge of a physical system (as, apart from anything else, were it correct it would blatantly violate the uncertainty principle, which holds that there must exist a degree of uncertainty for any state/property of any quantum system). What we have is all the information required for yielding probabilities.

Further, the instrumentalist/pragmatic "interpretation" of QM is about as orthodox as it gets. The "shut up and calculate" approach no longer dominates as it did once, but the plurality of views (particularly by philosophers and theoretical physicists) have tended to ensure the practicing physicists retain in spirit this approach, particularly given the fact that state-vector and wave-functions become replaced by fields and a wholly new level of mathematical abstraction in modern physics (from the standard model to supersymmetry).

Come on, Andrew, it’s not that complicated! Let me help you to sort out this pile of citations you poured here.

Let’s talk about a collapse of a wave function ψ. Imagine that before the measurement happening at the moment t, ψ represents a superposition ψ1+ψ2 so that at the moment t the wave function ψ of the quantum system (or other entity) collapses to ψ1.

Now, if we stand on the position that ψ is in one-to-one correspondence to elements of the reality Y such as the linear mapping ψ-->Y, then for us this collapse process is supposed to be a real one: At the moment t something real (like a wave) suddenly changes from (ψ1+ψ2) -->Ybefore to ψ1-->Yafter, where Ybefore ≠ Yafter.

On the other hand, if we support the point of view that ψ only describes the state of our knowledge about the reality, then for us the quantum collapse is not a real process but the change in our knowledge. This implies that (ψ1+ψ2) and ψ1 are subjective states of information about the system and hence correspond to the same element of reality Y, i.e., (ψ1+ψ2) -->Y, ψ1-->Y (which obviously would mean that there’s no one-to-one correspondence between ψ and Y).

That’s it, that is all. This – in the nutshell – is the problem of the status of the wave function in quantum mechanics.

This problem of course isn’t only about the point of view – it’s much deeper since different answer leads to different physics. For example, dynamical collapse theories (the Dynamical Reduction Program), which insist on the reality of collapsing process, posit that the Schrödinger equation should be modified by the addition of stochastic and nonlinear terms.

A quantum state can be represented as a vector in Hilbert space. For the state of the simplest quantum system, a single qubit, this Hilbert space is 2 dimensional. That's because once you normalize a 3 dimensional vector so it has the real world property of all probabilities summing to one, it can be specified by 2 dimensions, a single complex number with norm 1. It can then be directly mapped onto the surface of a sphere, specifically the Block sphere. This is an elementary concept for those working in quantum computing. (I don't!)

Superposition of qubit state, coherence of qubits, decoherence by measurement, are all real-world properties necessary for quantum computers to function. So I think its safe to say that for systems consisting of only qubits, the state vector represents a state of reality. It should then also apply to all 2-particle entangled systems, such as every delayed choice experiment ever done.

Yikes, I didn't realize this post is 4 years old!