Quantum theory of observation/General theory of quantum measurement

The theory of ideal measurement, initially proposed by von Neumann, is not entirely satisfactory for several reasons:

a) It assumes that there exists a basis of eigenstates of the observed system. These states produce results with certainty and are not disturbed by the measurement. But in general states which produce results with certainty do not exist. The detectors are not perfect. Their detection rates are almost never equal to 100%. Even when eigenstates exist, nothing prevents them from being disturbed by the measurement. The SWAP gate provides an example. It is also possible that the observed system is destroyed during the measurement. For example, most photodetectors absorb the photons they detect.

b) In order to have an ideal measurement, the detector must always be initially in the same state $$ | ready \rangle_A $$. But in general the initial state of the detector is not known with precision. It is also necessary that the final state $$ | i \rangle_A $$ be exactly determined. But in general, such a final state is not exactly known either.

Measurement operators respond to objections a. Measurement superoperators respond to objections b.

Measurement operators
As usual, the $$ | i \rangle_A $$ are the orthonormal basis of the pointer states of the measuring apparatus.

The $$ | j \rangle_S $$ are any orthonormal basis of the states of the observed system.

$$ U $$ is the evolution operator that describes the measurement process.

If $$ | \psi \rangle_S $$ is the initial state of the observed system, the final state is:

$$ U | \psi \rangle_S | ready \rangle_A = \sum_ {ij} \alpha (\psi) _ {ji} | j \rangle_S | i \rangle_A $$

The measurement operators $$ M (i) $$ are defined by:

$$ M (i) | \psi \rangle_S = \sum_ {j} \alpha (\psi) _ {ji} | j \rangle_S $$

These operators are determined from $$ U $$. They are linear because $$ U $$ is linear.

The probability that the result $$ i $$ is obtained is $$ \sum_ {j} | \alpha (\psi) _ {ji} | ^ 2 = | M (i) | \psi \rangle_S | ^ 2 $$. This is a generalization of the Born Rule. The state of the observed system after the measurement of $$ i $$ is $$ \frac {M (i) | \psi \rangle_S} {| M (i) | \psi \rangle_S |} $$. It is the state relative, in the sense of Everett, to the final state $$ | i \rangle_A $$ of the measuring apparatus.

$$ \frac {1} {| M (i) | \psi \rangle |} $$ is a factor of normalization.

To account for a measurement which destroys the observed system, it is enough to introduce a new state $$ | destroyed \rangle_S $$ to represent this destruction. Of course this is not a true quantum state, since it represents the state of a system that no longer exists. It is only a mathematical trick to adapt the formalism of measurement operators to the case of destructive measurements. A more rigorous approach is possible, with quantum field theory and annihilation operators.

 Examples: 

The CNOT gate

From :

$$CNOT(\alpha|0\rangle + \beta|1\rangle)|0\rangle = \alpha|00\rangle + \beta|11\rangle$$

we deduce :

$$ M (0) (\alpha | 0 \rangle + \beta | 1 \rangle) = \alpha | 0 \rangle $$

$$ M (1) (\alpha | 0 \rangle + \beta | 1 \rangle) = \beta | 1 \rangle $$

$$M(0)$$ and $$M(1)$$ are orthogonal projections on the states $$|0\rangle$$ and $$|1\rangle$$, respectively.

The SWAP gate From :

$$SWAP(\alpha|0\rangle + \beta|1\rangle)|0\rangle = \alpha|00\rangle + \beta|01\rangle$$

we deduce :

$$ M (0) (\alpha | 0 \rangle + \beta | 1 \rangle) = \alpha | 0 \rangle $$

$$ M (1) (\alpha | 0 \rangle + \beta | 1 \rangle) = \beta | 0 \rangle $$

Destructive and little effective measurement

It is assumed that if the observed system is in the state $$ | 1 \rangle $$ the detection rate is 10%. If it is in the state $$ | 0 \rangle $$ it is never detected. Such a measurement can be described by the evolution:

$$U(\alpha|0\rangle + \beta|1\rangle)|0\rangle = \alpha|00\rangle + {\sqrt {0.9}}\beta|10\rangle + {\sqrt {0.1}}\beta|destroyed\rangle|1\rangle$$

where $$|0\rangle$$ is the initial state of the detector and $$ | 1 \rangle $$ its state when it detected the observed system. We can deduce :

$$M(0)(\alpha|0\rangle + \beta|1\rangle) = \alpha|0\rangle + {\sqrt {0.9}}\beta|1\rangle$$

$$M(1)(\alpha|0\rangle + \beta|1\rangle) = {\sqrt {0.1}}\beta|destroyed\rangle$$

For such a little effective measurement, there is no basis of eigenstates. The only eigenstate is $$ | 0 \rangle $$. All other states $$ \alpha | 0 \rangle + \beta | 1 \rangle $$ where $$\beta$$ is different from zero are not eigenstates because they do not lead to a result with certainty.

Observables and projectors
The principles of quantum physics are generally explained without reference to the quantum theory of measurement or to measurement operators. In order to account for the link between state vectors and observation, we say that physical quantities, that is to say measurable quantities, are represented by hermitian operators, which are called observables. The measurement results are the eigenvalues ​​of these operators. The eigenvectors associated with an eigenvalue are the states which lead with certainty to the measurement of this eigenvalue. Furthermore, the postulate of the collapse of the state vector is generally accepted, that is, when a measurement result is obtained, the state vector of the system is projected onto the subspace of the eigenstates of the measured result. Such an observation is therefore an ideal measurement, because if the system is in an eigenstate of the observable it is not perturbed by the measurement.

When a measurement is ideal, a measurement operator $$ M (i) $$ is precisely the projector $$ P (i) $$ on the subspace of the eigenstates of $$ i $$. This is why ideal measurements are also called projective measurements.

An observable, as an hermitian operator $$ O $$, can be defined from the $$ P (i) $$:

$$ O = \sum_ {i} iP (i) $$

The definition of observables by hermitian operators is therefore a special case, limited to ideal measurements, of the more general theory of measurement operators.

Uncertainty about the state of the detector and measurement superoperators
When the result $$ i $$ is obtained, it is assumed that the state of the detector is not known, but that it is known to be in a subspace $$ H (i) _A $$ of its state space. Since the measurement results must be distinguished, the $$ H (i) _A $$ must be mutually orthogonal. They are the subspaces of pointer states.

The $$ | i, j \rangle_A $$ are an orthonormal basis of pointer states of the measuring apparatus. $$ | i, j \rangle_A $$ is in $$ H (i) _A $$.

The initial state of the detector is assumed to be $$ | ready, k \rangle_A $$ with a probability $$ p_k $$. Before the measurement, the detector is therefore described by the density operator $$ \sum_k p_k | ready, k \rangle_A \langle ready, k | _A $$

The $$ | l \rangle_S $$ are any orthonormal basis of the states of the observed system.

$$ U $$ is the evolution operator that describes the measurement process.

If $$ | \psi_m \rangle_S | ready, k \rangle_A $$ is an initial state of the system, the final state is:

$$U|\psi_m\rangle_S|ready,k\rangle_A = \sum_{lij} \alpha(m,k)_{lij}|l\rangle_S|i,j\rangle_A$$

Let $$|\phi(i,m,k)\rangle = \sum_{lj} \alpha(m,k)_{lij}|l\rangle_S|i,j\rangle_A$$

The measurement superoperators $$ M (i) $$ operate on the set of density operators. They are defined by:

$$M(i)(|\psi_m\rangle_S \langle \psi_m|_S)= \sum_k p_k Tr_A(|\phi(i,m,k)\rangle \langle \phi(i,m,k)|)$$

for a pure state. What can be immediately generalized to:

$$M(i)(\sum_m p_m|\psi_m\rangle_S \langle \psi_m|_S)= \sum_m p_m M(i)(|\psi_m\rangle_S \langle \psi_m|_S) = \sum_{mk} p_m p_k Tr_A(|\phi(i,m,k)\rangle \langle \phi(i,m,k)|)$$

for a mixed state $$\rho_S = \sum_m p_m|\psi_m\rangle_S \langle \psi_m|_S$$.

The $$ M (i) \rho_S $$ are not usually density operators because their trace is not equal to one.

Like measurement operators, measurement superoperators determine the probabilities of observations and the final states of the observed system (generalized Born Rule):

If the initial state is $$ \rho_S $$, pure or mixed, the probability of observing the result $$ i $$ is $$Tr(M(i)\rho_S)$$. The final state, usually mixed, after the measurement of $$ i $$ is represented by the operator $$\frac{M(i)\rho_S}{Tr(M(i)\rho_S)}$$.

It can be concluded that the probabilities of measurement results depend only on $$ \rho_S $$. Different mixtures of states which determine the same $$ \rho_S $$ can not be distinguished by observations.

The final state of the system observed after the measurement of $$i$$ is the state relative, in Everett's sense, to the final state $$\frac{\sum_{mk} p_m p_k Tr_S(|\phi(i,m,k)\rangle \langle \phi(i,m,k)|)}{Tr[\sum_{mk} p_m p_k Tr_S(|\phi(i,m,k)\rangle \langle \phi(i,m,k)|)]}$$ of the measuring apparatus.

The generalized Born rule shows how the main theorems on the ideal measurements (the existence theorem of multiple destinies, calculation of probabilities of measurement results, apparent reduction of the state vector) can be generalized to all measurement processes.

Proof of the generalized Born Rule :

$$Tr_A(|\phi(i,m,k)\rangle \langle \phi(i,m,k)|) = Tr_A(\sum_{ljl'j'} \alpha(m,k)_{lij}\alpha^*(m,k)_{l'ij'}
 * l\rangle_S|i,j\rangle_A \langle l'|_S\langle i',j'|_A$$

$$= \sum_{ljl'} \alpha(m,k)_{lij}\alpha^*(m,k)_{l'ij} |l\rangle_S \langle l'|_S$$

Hence :

$$Tr(M(i)\rho_S) = \sum_{ljkm} p_m p_k |\alpha(m,k)_{lij}|^2$$

The Born rule allows us to conclude that it is the probability $$ Pr (i) $$ to obtain the result $$ i $$.

After the measurement, the SA system is in a mixture of the states $$\frac{|\phi(i,m,k)\rangle}{||\phi(i,m,k)\rangle|}$$ with probabilities $$Pr(i,m,k)=||\phi(i,m,k)\rangle|^2 p_m p_k$$ for all values of $$i$$, $$m$$ and $$k$$.

Knowing that the result $$i$$ is obtained, SA is in the mixture of the states $$\frac{|\phi(i,m,k)\rangle}{||\phi(i,m,k)\rangle|}$$ with probabilities $$\frac{Pr(i,m,k)}{Pr(i)} = \frac{||\phi(i,m,k)\rangle|^2 p_m p_k}{Tr(M(i)\rho_S)}$$ for all values of $$m$$ and $$k$$. It is therefore represented by the density operator:

$$\rho^f_{SA} = \sum_{m,k}\frac{p_m p_k}{Tr(M(i)\rho_S)}|\phi(i,m,k)\rangle \langle \phi(i,m,k)|$$

The density operator $$\rho^f_S$$ of the observed system is thus :

$$\rho^f_S = Tr_A(\rho^f_{SA}) = \frac{\sum_{mk} p_m p_k Tr_A(|\phi(i,m,k)\rangle \langle \phi(i,m,k)|)} {Tr(M(i)\rho_S)} = \frac{M(i)\rho_S}{Tr(M(i)\rho_S)}$$

The selection of pointer states and environmental pressure
Quantum physics does not a priori impose any particular state basis on the Hilbert space of a material system. Any basis can do the trick. There are no fundamental states from which the others would be obtained by superposition. In particular, $$ | 0 \rangle $$ and $$ | 1 \rangle $$ are as much superposed states as $$\frac{1}{\sqrt 2}(|0\rangle + |1\rangle) = |x^+\rangle$$ and $$\frac{1}{\sqrt 2}(|0\rangle - |1\rangle) = |x^-\rangle$$, since $$|0\rangle = \frac{1}{\sqrt 2}(|x^+\rangle + |x^-\rangle)$$ and $$|1\rangle = \frac{1}{\sqrt 2}(|x^+\rangle - |x^-\rangle)$$.

The formalism of operators and superoperators of measurement does not impose a priori any basis of pointer states. For any basis $$ | i \rangle_A $$ of pointer states, and any initial state $$ | ready \rangle_A $$, we can define the measurement operators $$ M (i) $$. For any decomposition of $$ H_A $$ into mutually orthogonal subspaces $$ H (i) _A $$ and any density operator which represents an initial state, the measurement superoperators $$ M (i) $$ can be defined.

But a superposition of measurement results is not a measurement result. What compels us to choose a basis of pointer states rather than another, obtained with superpositions of the preceding ones? (Zurek 1981)

If the measuring devices are macroscopic, the choice of pointer states, or pointer subspaces, is naturally necessary, because the non-localized macroscopic states are very fragile (cf. 4.18) and generally unobservable. For a measurement to provide a defined result, this result must have a minimum duration, at least the time required to save it, on a hard disk, a sheet of paper, or simply in our memory. If the measurement result is destroyed as soon as it is obtained, without being recorded, it is not a result. That is why the pointer states of the macroscopic apparatuses are always, or almost always, localized states, and that the pointer subspaces contain only localized states. There is no other choice unless the decoherence by the environment is sufficiently low for the observation of non-localized macroscopic states to be possible.

The pointer states of microscopic probes
A photon, and more generally any particle or molecule, can be considered as a measuring device, a microscopic probe. In this case the decoherence by the environment does not act, or little. What then is the basis of pointer states? It may be a priori arbitrary. It is sufficient that these pointer states are observable, that they themselves are states pointed by another measuring apparatus, for a microscopic probe to function as an observation instrument. There is, however, a criterion which is sometimes sufficient to select the right basis of pointer states: we want the state of the observer system to give us as much information as possible about the observed system.

For example, in a CNOT gate, if we want the target qubit to give us the best information about the control qubit, the {$$ | 0 \rangle, | 1 \rangle $$} basis is required. We must also choose $$ | 0 \rangle $$ or $$ | 1 \rangle $$ as the initial state of the target qubit. Any other choice of a basis of pointer states or of its initial state would prevent the target qubit from achieving an ideal measurement. In particular, if we select {$$ | x ^ + \rangle, | x ^ - \rangle $$} as a basis of pointer states, then the target qubit can not provide any information on the control qubit.

In general, when an interaction makes possible an ideal measurement, there is only one pointer state basis for which the measurement is ideal. Any other choice of basis of pointer states would hamper observation by preventing it from being ideal.

The interaction between the observed system and the measuring device is sometimes sufficient to select a preferred base of pointer states. But it is not always the case. For a SWAP interaction, any choice of basis of pointer states is a priori as good as any other.

A double constraint for the design of observation instruments
As all material beings interact with other material beings, they can all be considered as instruments of observation, and used for this purpose. But when we design measuring instruments, we want to optimize their operation. Two constraints then guide us: of course, the instrument of observation must collect in the best way the desired information on the observed system, but it is also necessary that the collected information be accessible to us and that we can record it before it is erased. The interaction between the observed system and the observing instrument enables the second to obtain information about the first. The accessibility of the information thus obtained depends on the interaction between the observing instrument and its environment. These two constraints together determine the choice of the basis of the pointer states. If they can not be satisfied simultaneously then there is no good instrument of observation.

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