1.4 Example: Observables and States of a Qubit
Let us consider the case where the linear span
of the convex set of states is four dimensional, i.e. . For later convenience we choose the parameters in the set of states ,
Proposition 1.4.1 is isomorphic to the set of states of a qubit"
$
are the self-adjoint trace class operators on
.
Proof: Such an isomorphism is given by
"'
where the components of are the three Pauli spin operators
* * * * *
>*
(1.36) Here * and represent an orthonormal basis of the Hilbert space
. “bra” is the linear form corresponding to via the inner product
on , i.e.
. The identity and the three Pauli spin operators form a basis of s"
$
. Since the linear mapping maps the canonical basis " $ of V onto a basis of s"
$
, it is an isomorphism. Restricted on the isomorphism reads:
and the eigenvalues of ":
$
can thus be identified with the set of states of a qubit. In the general formulation
is essentially given by a translated three dimensional unit-sphere which is obviously isomor-phic to the the unit sphere in
. The representation of qubit states by means of vectors ,
is called Bloch representation and is the “Bloch vector”. The pure states are the extremal ele-ments of , i.e. the vectors which lie on the surface of the three dimensional sphere (
).
They correspond to density operators which are rank one projectors:
Proposition 1.4.2 The density operator " $ '+,B"
12 Generalised Measurements and the Axioms of Q.M.
Proof: is a rank one projector in
, if and only if (iff) it has Eigenvalues
and
*
.
According to Eq. (1.38) this again is the case, iff ) .
All mixed states can be represented by convex combinations of projectors and correspond to Bloch vectors which lie in the interior of the unit sphere (
). Figure 1.1 shows the state space of a qubit as represented by the unit-sphere in
(Bloch sphere).
PSfragreplacements * " * $
" * $
Figure 1.1: State space of a qubit as represented by the Bloch sphere. The pure states can be expressed by unit vectors , which correspond to Bloch vectors of unit norm. Mixed states ( ) cannot be expressed by vectors in but only by density operators. In the centre of the Bloch sphere lies the “total” mixture .
Proposition 1.4.3 The set of event vectors for is isomorphic to the set of positive operators on
with being a positive operator (in short notation ( (* ). The operators are the so-called “Effects” of a qubit (see below).
Proof: Consider the isomorphism given in proposition 1.4.1
"
is satisfied, if and only if the eigenvalues
"
TheEffects play in Quantum Mechanics the role of the event vectors in our construction.
Each measurement outcome is associated with an Effect and the probability to obtain
1.4 Example: Observables and States of a Qubit 13 this outcome is given by the so-called expectation value of
(cp. box “Effects and Observables”) with respect to the state of the observed system before the measurement:
%&"
$/
tr
(1.40)
This reflects the dual structure of event vectors and states, since the trace generates an inner product on s"
The correspondence between states and event vectors in the general statistical description of a measurement and the states and Effects of a qubit is extraordinary. We will see below that there is also a correspondence for more complex systems than qubits. From the point of view of classical physics Quantum Mechanics possesses very surprising features which seem to be irreconcilable with classical physics. It is remarkable that simple statistical considera-tions like the ones above can lead to a model which provides these non-classical features. What separates classical models from quantum models? Where did we step over the “quantum me-chanical threshold”? For a start we admitted an ambiguity in the decomposition of mixed states into pure states. If mixed states are uniquely decomposable into pure states, the correspond-ing statistical models only apply to classical physics and not quantum physics, cp. subsection 1.5. Another possibility to draw the line between classical and quantum seems to be to say, that in contrast to classical physics Quantum Mechanics provides pure states which cannot be represented as orthogonal vectors. Since the general description of measurements from above foresees non-orthogonal pure states it comprises quantum mechanical features, as can be seen at the superposition principle of Quantum Mechanics.
Pure states can be represented, as we have seen in the case of qubits, by rank-one projectors on a Hilbert space. They can also be represented by the rays on which they project in the Hilbert space (or by normed vectors
, being representatives of the rays). In fact each rank-one projector (or normed vector
, ) corresponds to a pure state. This is expressed in the superposition principle of Quantum Mechanics:
Superposition principle7: Any superposition
is a again a pure state.
7The superposition principle is not always valid: there are e.g. no superpositions of states corresponding to different masses or electrical charges.
14 Generalised Measurements and the Axioms of Q.M.
Effects and Observables
The probability of the outcome of a quantum mechanical measurement is represented by means of a positive operator (Effect) and the state of the m-system immediately before the measurement:
%&"
$/
tr
Here tr can be interpreted as the expectation value of the characteristic random vari-able which assigns the value to the event and the value * to all other events. A gen-eralisation of the concept of random variables is obtained by replacing classical random variables which assign to each “elementary” event , a real number (classical
num-ber) " $ by random variables which assign to each elementary event , an operator
(quantum number) "
where without “hat” is a real valued random variable, and is the Effect assigned to the event . Since we are concerned with measurements, the elementary events ,
are the possible outcomes of a measurement and " $ represents the reading of the pointer of the measurement apparatus. This generalisation of the concept of random variables incor-porates the statistics of quantum mechanical measurements quite naturally. It stands close to a formulation of probability theory without population by Benz et al. in [BP80]. This formulation refers to characteristic random variables with support on certain events rather than to the events themselves. Benz et al. mention the possibility to comprise quantum mechanical statistics by transiting to operator-valued random variables.
The mean value (expectation value) of the pointer readings can be expressed as
5 2 " $
, where the Effects belonging to the mea-surement have to be named explicitly because the decomposition of
into a sum of posi-tive operators is in general not unique. The self-adjoint operator with Effects can be interpreted as representative of an observable, i.e. a measurable quantity, which assumes, when measured, the values "
$
with probabilities related to the Effects
. The physical meaning of such an observable is in general not obvious. In the special case when the Effects are orthogonal projectors we are dealing with a spectral decomposition of , which is unique.
then represents what is called anordinary observable. Some of the or-dinary observables correspond to observables in classical physics as for example position, linear- or angular momentum and energy.
1.4 Example: Observables and States of a Qubit 15
The superposition principle has peculiar consequences.
A superposition of states
,
being a pure state does not allow for an either-or interpretation. This is counterintuitive if the states
,
correspond to classically exclusive properties like an electron passing a charged wire on the left and on the right hand side, respectively (cp. M¨ollenstedt’s Biprism [MD56]). In a measurement of the electron’s position when it passes the wire, it could be detected either on the left-hand side or on the right-hand side with a certain probability. Nevertheless in the absence of such a position measurement both components
and
are present at the same time because only together they lead to an interference pattern on a phosphorising screen be-hind the wire, which has been detected, cp. [MD56]. The probability for the electron to generate a light spark at position on the phosphorising screen (Effect
) is a convex combination of the probabilities if the state was either
or
is the state of the electron immediately before it hits the screen.
1 " $ ;=
1 is the so-called wavefunction in position representation of the state
1 ,
Re[z] marks the real part of the complex argument , and! means the complex conjugate of .
Two pure states represented by the normed vectors
,
, cannot be distinguished by the outcome of a single measurement if they are non-orthogonal, i.e.
? *
. This means there is no measurement which shows with probability equal to a certain measurement result – say “+”– if the system is in state
and a different result – say
“-”– if the system is in state
. This can be understood as follows. Like the Effects of a qubit, the Effect corresponding to the result “ ” has to satisfy ( (+* . In order
have to be eigenvectors of to the eigenvalues and* , respectively. Since
is positive and thus self-adjoint, its eigenvectors to different eigenvalues have to be orthogonal, contradicting the assumption
?6*
.
The non-distinguishability of pure states is extra-ordinary from the point of view of clas-sical physics. There, two pure states correspond to different properties of a single system which can be distinguished at least in principle8 in a measurement. In order to decide whether a quantum mechanical system is in one of two non-orthogonal states, one needs to make measurements on many systems which are all prepared in the state in question.
8Of course each real measurement has a finite resolution. But the resolution is in principle not bounded, thus any finite resolution can be reached.
16 Generalised Measurements and the Axioms of Q.M.
There is no measurement which is sensitive to the state of the m-system (i.e. the proba-bilities of the outcomes depend on the initial state) and which at the same time does not change any initial state cf. [Bus97]. In other words, contrary to measurements in classical physics there is no information gain without disturbance in quantum measurements.
The outcome of a quantum measurement is in general not predictable.
All these counter-intuitive features vanish if all pure states are pairwise orthogonal. The state space is then a simplex, i.e. each mixed state can beuniquelydecomposed into a convex com-bination of pure states, and one obtains the statistics of measurements in classical physics.