Quantum theory of observation/Fundamental concepts and principles

The principles of quantum physics
Four principles (Dirac 1930, von Neumann 1932, Cohen-Tannoudji, Diu & Laloë 1973, Weinberg 2012) are enough:


 * The space of states of a quantum system is a complex Hilbert space, that is a complex vector space (cf. 1.1) equipped with a scalar product and complete for the norm defined by this product.


 * The evolution between two instants of an isolated system is determined by a unitary operator.


 * The state space of a composite system is the tensor product of the spaces of its components.


 * A last principle, the Born Rule, says how to calculate the probabilities of measurement results from the state vector of the observed system. It will be explained below (cf. 2.4). It gives a physical meaning to the scalar product in the Hilbert space (cf. 2.6).

The postulate of evolution has been formulated in its integral form. In its differential form, it is the Schrödinger equation, $$ i \hbar \frac{d}{dt} |\psi\rangle = H |\psi\rangle $$ It will not be used in this book because the integral form is better suited to the theory of observation.

The third principle can be considered as a consequence of the first. It is not a strictly logical consequence, but as soon as we accept the first principle, and as we conceive that a system may be composed of parts, which may be in different states, we are obliged to accept the third principle.

The first principle was stated in its current and slightly incorrect form. More precisely, it states that a physical state must be identified with a ray - a subspace of collinear vectors - of the Hilbert space, or, if probabilities are calculated, with the set of unit vectors of such a ray. The null vector, which is of null length, is therefore not a state vector. In practice, the difference between vector and ray does not pose difficulties in identifying quantum states.

In the first principle, the clause of completeness is necessary to reason on the state spaces of infinite dimension. In general, a clause of countability is added for the basis states. These clauses are not necessary when we reason, as in this book, on complex vector spaces of finite dimension, because they are always complete.

It is often added to these principles that physical quantities must be represented by observables, that is to say, Hermitian operators on the state space of the observed system (cf. 5.2). This addition is not necessary (Zeh, in Joos, Zeh &... 2003).

Another principle, the postulate of state vector (or wave function) collapse is often considered a quantum principle. It contradicts the principle of unitary evolution and Schrödinger's equation. It can not therefore be part of quantum physics, otherwise the theory would be contradictory. This postulate is, however, often considered necessary to give physical meaning to quantum mathematics, but Everett (1957) showed that it is not (cf. 4.4 and 4.5).

Ideal measurements
A measurement is determined by a basis of states of the measuring device: the $$ | i \rangle_A $$ are the pointer states. $$ i $$ indexes the possible results of the measurement. When a measurement is ideal (von Neumann 1932), there exists an orthonormal basis of states of the observed system $$ | i, j \rangle_S $$ such that the interaction between the detected system and the detector is described by:

$$U|i,j\rangle_S|ready\rangle_A = |i,j\rangle_S|i\rangle_A$$

for all $$ i $$ and $$ j $$, where $$ U $$ is the evolution operator between the initial time, before the measurement, and the final time, when the measurement is complete, $$ | ready \rangle_A $$ is the initial state of the detector, $$ | i, j \rangle_S $$ are the eigenstates associated with the result $$ i $$. $$ j $$ indexes the eigenstates associated with the same result.

The eigenstates of a measurement result are those for which observation certainly leads to this result. When a measurement is ideal, and only in this case, if the observed system is in an eigenstate then it remains in the same state, it is not disturbed by the measurement process.

When a single eigenstate is associated with a measurement result, it can be said that it is detected, or pointed to, by the measurement.

$$ | i, j \rangle_S | ready \rangle_A $$ is an abbreviated form of $$|i,j\rangle_S \otimes |ready\rangle_A$$ where $$\otimes$$ represents the tensor product of two vectors (cf.1.7).

The existence theorem of multiple destinies
According to the principle of unitary evolution, if the initial state of the observed system is $$ \sum_ {ij} \alpha_ {ij} | i, j \rangle_S $$, the final state after the measurement must be:

$$U(\sum_{ij} \alpha_{ij}|i,j\rangle_S)|ready\rangle_A = \sum_{ij} \alpha_{ij}U|i,j\rangle_S|ready\rangle_A = \sum_{ij} \alpha_{ij}|i,j\rangle_S|i\rangle_A$$

The $$ | i \rangle_A $$ correspond to different measurement results. $$ \sum_ {ij} \alpha_ {ij} | i, j \rangle_S | i \rangle_A $$ is therefore a superposition of measurement results.

The fundamental theorem of quantum measurement is thus obtained directly from the principles of quantum physics: if the observed system is initially in a superposition of measurement eigenstates, the complete system (observed system plus measuring device) is in a superposition of measurement results.

This theorem is very surprising. At the end of a measurement, a single result $$|i\rangle_A$$ is observed. A superposition of measurement results is not a measurement result. How can we understand that quantum physics predicts the existence of $$ \sum_ {ij} \alpha_ {ij} | i, j \rangle_S | i \rangle_A $$?

Hugh Everett III (1957) proposed the following answer:

$$ \sum_ {ij} \alpha_ {ij} | i, j \rangle_S | i \rangle_A $$ describes a superposition of the observer's destinies. An observer obtains a single measurement result because he knows of himself only one of his destinies. The other measurement results are also obtained, but in the observer's other destinies. Quantum physics describes a universe in which beings have very numerous destinies. The world as known by an observer is only a tiny part of quantum reality, a destiny among myriads of others. The fundamental theorem of quantum measurement can thus also be called the existence theorem of multiple destinies. It is a direct consequence of the principles of quantum physics. It has been stated in the particular case of ideal measurements but remains valid for all forms of quantum measurement. To deny it, one must deny that quantum physics correctly describes reality. It is possible that a new physics surpasses quantum physics and proves that these other destinies do not exist, but until now quantum physics has always proved its validity. No experimental results have ever disproved it.

The existence theorem of multiple destinies is empirically verifiable, but in a severely limited way. We will show later, in the chapter on quantum entanglement, that an observer can not observe his other destinies, but that a second observer can in principle observe them. It is in principle possible to observe that, after an observation, an observer system has several destinies, with experiments of the "Schrödinger's cat" type. But this conclusion is limited to reversible observation processes, because observation destroys information. Since the processes of life are irreversible, the simultaneous existence of the multiple destinies of a living being can not be observed.

The destruction of information by observation
The observation of a quantum superposition of states necessarily destroys the information carried by each of the superposed states.

Let $$ | P_i \rangle $$ be a basis of pointer states, $$ | ready \rangle $$ the initial state of the measuring device, $$ | S_j \rangle $$ a basis of states of the observed system and $$ U $$ the evolution operator for the measurement. For the superposition $$ \sum_j \alpha_j | S_j \rangle $$ to be observed, there should be a pointer state $$ | P_n \rangle $$ such that

$$U(\sum_{j} \alpha_{j}|S_j\rangle|ready\rangle) = |X \rangle |P_n \rangle$$

where $$ | X \rangle $$ is a state of the observed system.

If the observation does not destroy the information carried by each of the $$ | S_j \rangle $$ it must be possible to know the state of the observed system before the measurement from its state after the measurement, so we must have states $$ | S'_j \rangle $$, all orthogonal to each other such that

$$U(|S_j \rangle|ready\rangle )= |S'_j \rangle |Y_j \rangle $$ for each $$j$$, where the $$ | Y_j \rangle $$ are states of the measuring device.

These two conditions imply that all $$ | Y_j \rangle $$ must be equal to $$ | P_n \rangle $$.This means that the measuring apparatus measures nothing since it only jumps from $$ | ready \rangle $$ to $$ | P_n \rangle $$ for any state of the observed system. Hence a measuring apparatus which observes a superposition of states necessarily destroys the information carried by each of these states. This simple theorem can be extended to a more realistic model of a measuring apparatus where the initial and pointer states are in statistical ensembles.

This theorem shows to what condition the existence theorem of multiple destinies is empirically verifiable. If the measurement is irreversible, if the gathered information can not be erased, then the quantum superposition can not be observed. Since conscious destinies are successions of irreversible acquisitions of information (what is done cannot be undone, the past cannot be erased, the information acquired cannot be destroyed) their superpositions cannot be observed. But this does not forbid to verify the existence theorem of multiple destinies with reversible measurements. We can and do verify it on microscopic measuring systems: atoms, trapped photons or ions. A quantum computer is a way of verifying the existence of the multiple destinies of the qubits that constitute it.

The Born Rule
The complex numbers $$ \alpha_ {ij} $$ in the superposition $$\sum_{ij} \alpha_{ij}|i,j\rangle_S|i\rangle_A$$ are conceived as probability amplitudes. The probability of observing the result $$ i $$ is $$ \sum_ {j} | \alpha_ {ij} | ^ 2 $$. It can be admitted as a principle:

If the initial state of the observed system is $$ | \psi \rangle = \sum_ {ij} \alpha_ {ij} | i, j \rangle_S $$ where the $$ | i, j \rangle_S $$ are an orthonormal basis of measurement eigenstates, then the probability of the result $$ i $$ is $$ \sum_ {j} | \alpha_ {ij} | ^ 2 $$.

To apply this rule, $$ | \psi \rangle $$ must be normalized: $$ \sum_ {ij} | \alpha_ {ij} | ^ 2 = 1 $$.

One can try to prove this fourth principle from the first three (Everett 1957, Zurek 2003 ...). These proofs are controversial and will not be discussed here.

The Born rule was stated only for ideal measurements. It will be shown that it can be generalized to all forms of measurement (cf. 5.1 and 5.3).

Can we observe quantum states?
In order to observe, the state of the observing system (the detector, the measuring instrument) after the measurement must provide information on the state of the observed system before the measurement. An observation is perfect if one can deduce exactly the state of the detected system from the result of the measurement. Quantum measurements are never perfect. If the initial state of the detected system is not known in advance, the final state of the detector is never sufficient to know the state of the detected system, because many different states of the detected system can lead to the same result. It is sufficient that they have a non-zero probability of producing this result. The only information given by the observation is that the state of the observed system did not have a zero probability of producing this result. If the result $$ i $$ has been obtained, all we know about the initial state $$ \sum_ {ij} \alpha_ {ij} | i, j \rangle_S $$ of the observed system is that $$ \sum_ {j} | \alpha_ {ij} | ^ 2> 0 $$

How then can we know the state vector of a quantum system?

One can not detect the quantum state of an initially unknown system, produced spontaneously by Nature. On the other hand, material systems can be prepared in such a way that they are found in a single quantum state. If there is a measurement result whose eigenstate is unique, then it can be verified that this quantum state actually exists. Simply repeat the preparation a lot of times and verify that you always get the same measurement result.

Observation alone is not enough to know the quantum states. It is necessary to act on matter to prepare it in the states that one wishes to observe.

An ideal measurement is one way to prepare a state, when the measurement results each have a single eigenstate. If the result of the measurement is $$ i $$ then it is known with certainty that the observed system state is $$ | i \rangle_S $$ just after the measurement. This can be verified by repeating the measurement on the system just prepared.

In order to escape the existence theorem of multiple destinies, many physicists assert that state vectors are only theoretical tools for calculating probabilities, and that they do not really represent reality (Peres 1995). But when the observed system has been properly prepared, one can know with certainty its state vector, one can check it without any doubt being allowed. Physicists now know how to prepare, manipulate and observe the state vectors they imagine (for example, Haroche & Raimond 2006), is not it sufficient to assert that the state vector really represents the physical state of the observed system?

Orthogonality and incomplete discernability of quantum states
It is sometimes said, improperly, that when a material being is in a superposition of states like $$ \frac {1} {\sqrt 2} (| 0 \rangle + | 1 \rangle) $$, it is at the same time in the states $$ | 0 \rangle $$ and $$ | 1 \rangle $$. For example, when commenting on Young's experiment, it is said that the photon passes through both slits at the same time. That sounds absurd. If the photon is in a slit, it can not be in the other. To say that it is in both simultaneously is therefore a contradiction. It is said that it passes through the two slits only to say that if one were to detect its passage, one would find it in one slit or the other. But it will never be found in both at the same time.

If a being is in the state $$ | 0 \rangle $$ it is not in the state $$ | 1 \rangle $$ and vice versa. When it is in the state $$ \frac {1} {\sqrt 2} (| 0 \rangle + | 1 \rangle) $$, it is not in the state $$ | 0 \rangle $$ or in the state $$ | 1 \rangle $$, but in a third state, different from the previous two (Griffiths 2004).

If a being is prepared in the state $$ | 0 \rangle $$ it can not be detected immediately after preparation in the state $$ | 1 \rangle $$ because $$ | 0 \rangle $$ and $$ | 1 \rangle $$ are orthogonal, i.e., their scalar product $$\langle 1|0\rangle$$ is zero. On the other hand, if a being is prepared in the state $$ \frac {1} {\sqrt 2} (| 0 \rangle + | 1 \rangle) $$, there is a probability one half of detecting it in the state $$ | 0 \rangle $$ and the same probability to detect it in the state $$ | 1 \rangle $$, because $$ \left| \langle 0 | \frac {1} {\sqrt 2} (| 0 \rangle + | 1 \rangle) \right| ^ 2 = \left| \frac {1} {\sqrt 2} \langle 0| 0\rangle \right| ^ 2 = \frac {1} {2} $$. When two states are orthogonal, they are completely discernible, because they can both be eigenstates of the same measuring instrument. When they are not orthogonal, their difference is partially blurred, especially if their scalar product is close to 1. They are not completely discernible, in the sense that it is not possible to make a measurement which would enable to distinguish them, because they can not be eigenstates of the same measuring instrument associated with different values.

It is sometimes said that $$ | \langle \phi | \psi \rangle | ^ 2 $$ is the probability that a being in the state $$ | \psi \rangle $$ is in the state $$ | \phi \rangle $$. This sounds like an absurdity, since if $$ | \psi \rangle $$ and $$ | \phi \rangle $$ are different, a being can not be in both states at the same time. But if we hear it charitably, we understand that it only means that a being prepared in the state $$ | \psi \rangle $$ has a probability $$ | \langle \phi | \psi \rangle | ^ 2 $$ to be detected in the state $$ | \phi \rangle $$. This is why we are tempted to say that if $$ | \phi \rangle $$ is not orthogonal to $$ | \psi \rangle $$, a being in the $$ | \psi \rangle $$ state is partially in the $$ | \phi \rangle $$ state.

The incompatibility of quantum measurements
If $$ | 0 \rangle $$ and $$ | 1 \rangle $$ are eigenstates of a measurement, they can be said to be observed or pointed states. On the other hand, superpositions of $$ | 0 \rangle $$ and $$ | 1 \rangle $$ are not states which can be observed by such a measurement.

Let $$|x^+\rangle = \frac{1}{\sqrt 2}(|0\rangle + |1\rangle)$$ and $$|x^-\rangle = \frac{1}{\sqrt 2}(|0\rangle - |1\rangle)$$ be two new basis vectors (the names $$x^+$$ and $$x^-$$ come from the spin 1/2 theory).

$$|x^+\rangle$$ et $$|x^-\rangle$$ can also be eigenstates of a measurement, and hence the states observed by this measurement. The measurements of {$$ 0,1 $$} on the one hand, and {$$|x^+\rangle$$, $$|x^-\rangle$$} on the other hand are incompatible. There is no state of the observed system such as $$ | 0x^+ \rangle $$ because no quantum state is an eigenstate of both measurements. If one measurement is made immediately after another, random results will always be found. It is not possible to prepare the observed system in a state which provides a certain result for the two successive measurements. If the result of the first measurement is certain, the latter can not be.

When two measurements have a common basis of eigenstates, they are said to be compatible. If they are ideal measurements, they do not disturb each other. One can be done just before the other without affecting its result.

The existence of incompatible measurements is an immediate consequence of the principle of quantum superposition. It is a typically quantum effect which has no equivalent in classical physics.

What can be said about the reality of $$ x^+ $$ and $$ x^- $$ when the observed system is in the state $$|0\rangle$$ ?

Since $$|0\rangle = \frac{1}{\sqrt 2}(|x^+\rangle + |x^-\rangle)$$ the system is in a superposition of $$ x^+ $$ and $$ x^- $$. Neither is real. It is only their superposition which is real.

Heisenberg showed that the position and momentum measurements of a quantum particle are incompatible (Heisenberg 1930). They can only disrupt each other. It is therefore impossible to attribute simultaneously exactly defined position and momentum to the same particle. This incompatibility is mathematically translated by the relation $$\Delta x \Delta p \ge \hbar$$ where $$\Delta x$$ is the indeterminacy of the position measurement, $$\Delta p$$ the indeterminacy of the momentum measurement and $$ \hbar $$ the Planck constant $$ h $$ divided by $$ 2 \pi $$. If this relation is not respected, no quantum state can be an eigenstate of both measurements.

It must be called Heisenberg's relation of indeterminacy rather than a relation of uncertainty, because the latter expression suggests that we do not know simultaneously the position and the momentum of a particle but that they could be known. For there to be uncertainty, there must be something to know that we do not know. But the incompatibility of quantum measurements does not say that there is more to know than what we observe. On the contrary, it says that there is no real state for which position and momentum are exactly defined. Such states can not be observed because they do not exist. Heisenberg's relation does not come from our ignorance, or from our uncertainty, but from the indeterminacy of reality. The quantum states can not be simultaneously eigenstates of all possible measurements, because of their incompatibility.

Uncertainty and density operators
A density operator $$ \rho $$ is used to describe situations where the quantum state of a system is not known accurately. It is defined from a set of states $$ | j \rangle $$, supposed to be normalized, but not necessarily orthogonal, each assigned a probability $$ p_j $$. By definition $$ \rho = \sum_j p_j | j \rangle \langle j | $$ where $$ | j \rangle \langle j | $$ is the orthogonal projection on the state $$ | j \rangle $$.

If the $$ | j \rangle $$ are orthogonal, then $$ p_j $$ is the weight of $$ | j \rangle $$ in $$ \rho $$. With a slight abuse of language, we can say that $$ p_j $$ is the density of presence in the state $$ | j \rangle $$.

It is said of $$ \rho $$ that it describes a mixture of states, or a mixed state. The same density operator can be defined from different mixtures, but we shall show later (4.3) that such mixtures can not be distinguished by observation. A density operator contains all available information on the state of the observed system.

If all $$ p_j $$ are equal to zero except $$ p_0 = 1 $$, the state vector $$ | 0 \rangle $$ is known with exactness. The system is then said to be in a pure state. In this case $$ \rho = | 0 \rangle \langle 0 | $$. The formalism of density operators can therefore be applied to both mixed states and pure states.

The trace of a density operator is always equal to one:

$$Tr(\rho)=Tr(\sum_j p_j|j\rangle \langle j|)=\sum_j p_j Tr(|j\rangle \langle j|)=\sum_j p_j=1$$

If a system is prepared in the mixed state $$ \rho $$ the probability that it is detected in a pure state $$ | \psi \rangle $$ is $$ Tr (\rho | \psi \rangle \langle \psi |) $$.

Proof: $$Tr(\rho | \psi \rangle \langle \psi | ) = Tr(\sum_j p_j |j\rangle \langle j| \psi \rangle \langle \psi | ) = \sum_j p_j \langle j|  \psi \rangle Tr(|j\rangle \langle \psi | ) = \sum_j p_j | \langle j |\psi \rangle |^2 $$

As a density operator determines the probabilities of detection of all quantum states, it determines the probabilities of all the results of all possible measurements. In this sense, it completely determines the physical state of the system.

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