Quantum tomography
Encyclopedia
Quantum tomography
or quantum state tomography is the process of reconstructing the quantum state (density matrix
) for a source of quantum systems by measurements on the systems coming from the source. The source may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general mixed states. To be able to uniquely identify the state, the measurements must be tomographically complete. That is, the measured operators must form an operator basis
on the Hilbert space
of the system, providing all the information about the state. Such a set of observations is sometimes called a quorum.
In quantum process tomography on the other hand, known quantum states are used to probe a quantum process to find out how the process can be described. Similarly, quantum measurement tomography works to find out what measurement is being performed.
The general principle behind quantum state tomography is that by repeatedly performing many different measurements on quantum systems described by identical density matrices, frequency counts can be used to infer probabilities, and these probabilities are combined with Born's rule to determine a density matrix
which fits the best with the observations.
This can be easily understood by making a classical analogy. Let us consider a harmonic oscillator
(e.g. a pendulum). The position
and momentum
of the oscillator at any given point can be measured and therefore the motion can be completely described by the phase space
. This is shown in figure 1. By performing this measurement for a large number of identical oscillators we get a possibility distribution in the phase space
(figure 2). This distribution can be normalized (the oscillator at a given time has to be somewhere) and the distribution must be non-negative. So we have retrieved a function W(x,p) which gives a description of the chance of finding the particle at a given point with a given momentum.
For quantum mechanical particles the same can be done. The only difference is that the Heisenberg’s uncertainty principle
musn’t be violated, meaning that we cannot measure the particle’s momentum and position at the same time. However, we can measure only one of them. The particle’s momentum and its position are called quadratures (see Optical phase space
for more information) in quantum related states. By measuring one of the quadratures of a large number of identical quantum states will give us a probability density corresponding to that particular quadrature. This is called the marginal distribution
, pr(X) or pr(P) (see figure 3). In the following text we will see that this probability density is needed to characterize the particle’s quantum state, which is the whole point of quantum tomography.
Quantum tomography can be used for characterizing optical signals, including measuring the signal gain and loss of optical devices, as well as in quantum computing and quantum information theory to reliably determine the actual states of the qubits. One can imagine a situation in which a person Bob prepares some quantum states and then gives the states to Alice to look at. Not confident with Bob's description of the states, Alice may wish to do quantum tomography to classify the states herself.
which can then be used to express the pure state in the basis
of the measurement. Generally, being in a pure state is not known, and a state may be mixed. In this case, many different measurements will have to be performed, many times each. To fully reconstruct the density matrix
for a mixed state in a finite-dimensional Hilbert space
, the following technique may be used.
Born's rule states
, where 
is a particular measurement outcome projector
and
is the density matrix of the system.
Given a histogram
of observations for each measurement, one has an approximation
to 
for each 
.
Given linear operators
and 
, define the inner product
where
is representation of the 
operator as a column vector and 
a row vector such that 
is the inner product in 
of the two.
Define the matrix
as
.
Then applying this to
yields the probabilities:
.
Linear inversion corresponds to inverting this system using the observed relative frequencies
to derive 
(which is isomorphic to 
).
This system is not going to be square in general, as for each measurement being made there will generally be multiple measurement outcome projector
. For example, in a 2-D Hilbert space
with 3 measurements
, each measurement has 2 outcomes, leaving 
to be 6 x 4. To solve the system, multiply on the left by 
:
.
Now solving for
yields the pseudoinverse
:
.
This works in general only if the measurements were tomographically complete. Otherwise, the matrix
will not be invertible.
of light
, known as optical homodyne tomography
. Using balanced homodyne measurements, one can derive the Wigner function and a density matrix
for the state of the light
.
One approach involves measurements along different rotated directions in phase space
. For each direction
, one can find a probability distribution
for the probability density
of measurements in the
direction of phase space yielding the value 
. Using an inverse Radon transformation (the filtered back projection) on 
leads to the Wigner function, 
, which can be converted by an inverse fourier transform into the density matrix
for the state in any basis. A similar technique is often used in medical tomography
.
detector. This is explained by the following example.
A laser
is directed onto a 50-50% beamsplitter, splitting the laserbeam into two beams. One is used as local oscillator
(LO) and the other is used to generate photons with a particular quantum state. The generation of quantum states can be realized, e.g. by directing the laser beam through a frequency doubling crystal
and then onto a parametric down-conversion crystal. This crystal generates two photons in a certain quantum state. One of the photons is used as a trigger signal used to trigger (start) the readout event of the homodyne tomography detector. The other photon is directed into the homodyne tomography detector, in order to reconstruct its quantum state. Since the trigger and signal photons are entangled (this is explained by the Spontaneous parametric down-conversion article), it is important to realize, that the optical mode of the signal state is created nonlocal only when the trigger photon impinges the photodector (of the trigger event readout module) and is actually measured. More simply said, it is only when the trigger photon is measured, that the signal photon can be measured by the homodyne detector.
Now let us consider the homodyne tomography
detector as depicted in figure 4. The signal photon (this is the quantum state we want to reconstruct) interferes with the local oscillator
, when they are directed onto a 50-50% beamsplitter. Since the two beams originate from the same so called master laser
, they have the same fixed phase
relation. The local oscillator must be intense, compared to the signal so it provides a precise phase reference. The local oscillator is so intense, that we can treat it classically (a = α) and neglect the quantum fluctuations.
The signal field is spatially and temporally controlled by the local oscillator, which has a controlled shape. Where the local oscillator is zero, the signal is rejected. Therefore, we have temporal-spatial mode selectivity of the signal.
The beamsplitter redirects the two beams to two photodetectors. The photodetectors generate a electric current
proportional to the photon number. The two detector currents are subtracted and the resulting current is proportional to the electric field operator in the signal mode, depended on relative optical phase of signal and local oscillator.
Since the electric field amplitude of the local oscillator is much higher than that of the signal the intensity or fluctuations in the signal field can be seen. The homodyne tomography system functions as an amplifier
. The system can be seen as an interferometer with such a high intensity reference beam (the local oscillator) that unbalancing the interference by a single photon in the signal is measurable. This amplification is well above the photodetectors noise floor
.
The measurement is reproduced a large number of times. Then the phase difference between the signal and local oscillator is changed in order to ‘scan’ a different angle
in the phase space
. This can be seen from figure 4. The measurement is repeated again a large number of times and a marginal distribution
is retrieved from the current difference. The marginal distribution
can be transformed into the density matrix
and/or the Wigner function. Since the density matrix
and the Wigner function give information about the quantum state of the photon, we have reconstructed the quantum state of the photon.
The advantage of this method is that this arrangement is insensitive to fluctuations in the frequency
of the laser
.
The quantum computations for retrieving the quadrature component from the current difference are performed as follows.
The photon number operator for the beams striking the photodetectors after the beamsplitter is given by:
,
where i is 1 and 2, for respectively beam one and two.
The mode operators of the field emerging the beamsplitters are given by:
The
denotes the annihilation operator of the signal and alpha the complex amplitude of the local oscillator.
The number of photon difference is eventually proportional to the quadrature and given by:
,
Rewriting this with the relation:
Results in the following relation:
,
where we see clear relation between the photon number difference and the quadrature component
. By keeping track of the sum current, one can recover information about the local oscillator’s intensity, since this is usually an unknown quantity, but an important quantity for calculating the quadrature component 
.
is that in general the computed solution will not be a valid density matrix. For example, it could give negative probabilities or probabilities greater than 1 to certain measurement outcomes. This is particularly an issue when fewer measurements are made.
Another issue is that in infinite dimensional Hilbert spaces, an infinite number of measurement outcomes would be required. Making assumptions about the structure and using a finite measurement basis leads to artifacts in the phase space density.
of giving the experimental results, it guarantees the state to be theoretically valid while giving a close fit to the data. The likelihood of a state is the probability that would be assigned to the observed results had the system been in that state.
Suppose the measurements
have been observed with frequencies 
. Then the likelihood associated with a state 
is
where
is the probability of outcome 
for the state 
.
Finding the maximum of this function is non-trivial and generally involves iterative methods. The methods are an active topic of research.
of the n-dimensional Bloch sphere
. This can be seen as related to linear inversion giving states which lie outside the valid space (the Bloch sphere). MLE in these cases picks a nearby point that is valid, and the nearest points are generally on the boundary.
This is not physically a problem, the real state might have zero eigenvalues. However, since no value may be less than 0, an estimate of an eigenvalue being 0 implies that the estimator is certain the value is 0, otherwise they would have estimated some
greater than 0 with a small degree of uncertainty
as the best estimate. This is where the problem arises, in that it is not logical to conclude with absolute certainty after a finite number of measurements that any eigenvalue (that is, the probability of a particular outcome) is 0. For example, if a coin is flipped 5 times and each time heads was observed, it does not mean there is 0 probability of getting tails, despite that being the most likely description of the coin.
mean estimation (BME) is a relatively new approach which addresses the problems of maximum likelihood estimation. It focuses on finding optimal solutions which are also honest in that they include error bars in the estimate. The general idea is to start with a likelihood function
and a function describing the experimenter's prior knowledge (which might be a constant function), then integrate over all density matrices using the product of the likelihood function
and prior knowledge function as a weight.
Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional bloch sphere
. In the case of a coin flipped N times to get N heads described above, with a constant prior knowledge function, BME would assign
as the probability for tails.
BME provides a high degree of accuracy in that it minimizes the operational divergences of the estimate from the actual state.
Since a measurement can always be characterized by a set of POVM
's, the goal is to reconstruct the characterizing POVM
's
.
The simplest approach is linear inversion. Similar to in quantum state observation, use
.
Exploiting linearity as above, this can be inverted to solve for the
.
Not surprisingly, this suffers from the same pitfalls as in quantum state tomography. Namely, non-physical results, in particular negative probabilities. Here the
will not be valid POVM
's, as they will not be positive. Bayesian methods as well as Maximum likelihood estimation of the density matrix
can be used to restrict the operators to valid physical results.
As it was shown in a recent work, such a state reveals interesting quantum properties of the corresponding measurement such as its non-classicality
or its projectivity.
However, we cannot realize the tomography of this state with the usual methods based on measurements, since it needs non-destructive measurements which are some particularly measurements.
The experimental procedure, proposed in , is based on the retrodictive approach of quantum physics, in which we have an expression of retrodictive probabilities similar to Born's rule:
where
and 
are respectively the pre-measurement state, corresponding to the measurement characterized by some a POVM element 
, and a hermitian and positive operator corresponding to the preparation of the measured system in a state 
.
In the frame of the mathematical foundations of quantum physics, such a operator is a proposition about the state of the system, as a POVM element, and for having an exhaustive set of propositions, these operators must be a resolution of the Hilbert space:
From Born's, we can derive with Bayes' theorem, the expressions of the pre-measurement state
and proposition operators 
.
The pre-measurement state simply corresponds to the normalized POVM element:
and the proposition operators are linked to the possible preparations of the system by:
where
is the dimension of the Hilbert space and 
is the probability of preparing the state 
.
Thus, we can probe the measurement apparatus with a statistical mixture:
in order to measure the retrodictive probability
.
This mixture could be obtained by preparations based on random choices 'm' with the probabilities
.
Then, we replace the POVM elements describing the measurements in a usual method for the tomography of states by the operators
. The method will give the state giving the probabilities 
which are the most closest to those measured. This is the pre-measurement state with which we can have some interesting properties of the measurement giving the result 'n', as explained in.
Each of the techniques listed above are known as indirect methods for characterization of quantum dynamics, since they require the use of quantum state tomography to reconstruct the process. In contrast, there are direct methods such as direct characterization of quantum dynamics (DCQD) which provide a full characterization of quantum systems without any state tomography.
, can be described by a completely positive map
,
where
, the bounded operators on Hilbert space
; with operation elements
satisfying 
so that 
.
Let
be an orthogonal basis for 
. Write the 
operators in this basis
.
This leads to
,
where
.
The goal is then to solve for
, which is a positive superoperator
and completely characterizes
with respect to the 
basis.
linearly independent inputs 
, where 
is the dimension of the Hilbert space 
. For each of these input states 
, sending it through the process gives an output state 
which can be written as a linear combination of the 
, i.e. 
. By sending each 
through many times, quantum state tomography can be used to determine the coefficients 
experimentally.
Write
,
where
is a matrix of coefficients.
Then
.
Since
form a linearly independent basis,
.
Inverting
gives 
:
.
Tomography
Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in tomography is called a tomograph, while the image produced is a tomogram. The method is used in radiology, archaeology, biology, geophysics, oceanography, materials science,...
or quantum state tomography is the process of reconstructing the quantum state (density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
) for a source of quantum systems by measurements on the systems coming from the source. The source may be any device or system which prepares quantum states either consistently into quantum pure states or otherwise into general mixed states. To be able to uniquely identify the state, the measurements must be tomographically complete. That is, the measured operators must form an operator basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
on the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
of the system, providing all the information about the state. Such a set of observations is sometimes called a quorum.
In quantum process tomography on the other hand, known quantum states are used to probe a quantum process to find out how the process can be described. Similarly, quantum measurement tomography works to find out what measurement is being performed.
The general principle behind quantum state tomography is that by repeatedly performing many different measurements on quantum systems described by identical density matrices, frequency counts can be used to infer probabilities, and these probabilities are combined with Born's rule to determine a density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
which fits the best with the observations.
This can be easily understood by making a classical analogy. Let us consider a harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
(e.g. a pendulum). The position
Position
Position may refer to:* Position , a player role within a team* Position , the orientation of a baby prior to birth* Position , a mathematical identification of relative location...
and momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
of the oscillator at any given point can be measured and therefore the motion can be completely described by the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
. This is shown in figure 1. By performing this measurement for a large number of identical oscillators we get a possibility distribution in the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
(figure 2). This distribution can be normalized (the oscillator at a given time has to be somewhere) and the distribution must be non-negative. So we have retrieved a function W(x,p) which gives a description of the chance of finding the particle at a given point with a given momentum.
For quantum mechanical particles the same can be done. The only difference is that the Heisenberg’s uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
musn’t be violated, meaning that we cannot measure the particle’s momentum and position at the same time. However, we can measure only one of them. The particle’s momentum and its position are called quadratures (see Optical phase space
Optical phase space
In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an optical system. For any such system, a plot of the quadratures against each other, possibly as...
for more information) in quantum related states. By measuring one of the quadratures of a large number of identical quantum states will give us a probability density corresponding to that particular quadrature. This is called the marginal distribution
Marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. The term marginal variable is used to refer to those variables in the subset of variables being retained...
, pr(X) or pr(P) (see figure 3). In the following text we will see that this probability density is needed to characterize the particle’s quantum state, which is the whole point of quantum tomography.
What quantum state tomography is used for
Quantum tomography is applied on a source of systems, to determine what the quantum state is of the output of that source. Unlike a measurement on a single system, which determines the system's current state after the measurement (in general, the act of making a measurement alters the quantum state), quantum tomography works to determine the state(s) prior to the measurements.Quantum tomography can be used for characterizing optical signals, including measuring the signal gain and loss of optical devices, as well as in quantum computing and quantum information theory to reliably determine the actual states of the qubits. One can imagine a situation in which a person Bob prepares some quantum states and then gives the states to Alice to look at. Not confident with Bob's description of the states, Alice may wish to do quantum tomography to classify the states herself.
Linear inversion
Using Born's rule, one can derive the simplest form of quantum tomography. If it is known in advance that the state is represented by a pure state, a single measurement can be performed repeatedly to build up a histogramHistogram
In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson...
which can then be used to express the pure state in the basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
of the measurement. Generally, being in a pure state is not known, and a state may be mixed. In this case, many different measurements will have to be performed, many times each. To fully reconstruct the density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
for a mixed state in a finite-dimensional Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, the following technique may be used.
Born's rule states
, where is a particular measurement outcome projectorProjection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
and
is the density matrix of the system.Given a histogram
Histogram
In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson...
of observations for each measurement, one has an approximation
to for each .Given linear operators
and , define the inner productwhere
is representation of the operator as a column vector and a row vector such that is the inner product in of the two.Define the matrix
as.Then applying this to
yields the probabilities:.Linear inversion corresponds to inverting this system using the observed relative frequencies
to derive (which is isomorphic to ).This system is not going to be square in general, as for each measurement being made there will generally be multiple measurement outcome projector
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
. For example, in a 2-D Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
with 3 measurements
, each measurement has 2 outcomes, leaving to be 6 x 4. To solve the system, multiply on the left by :.Now solving for
yields the pseudoinversePseudoinverse
In mathematics, and in particular linear algebra, a pseudoinverse of a matrix is a generalization of the inverse matrix. The most widely known type of matrix pseudoinverse is the Moore–Penrose pseudoinverse, which was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951 and...
:
.This works in general only if the measurements were tomographically complete. Otherwise, the matrix
will not be invertible.Continuous variables and quantum homodyne tomography
In infinite dimensional Hilbert spaces, e.g. in measurements of continuous variables such as position, the methodology is somewhat more complex. One notable example is in the tomographyTomography
Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in tomography is called a tomograph, while the image produced is a tomogram. The method is used in radiology, archaeology, biology, geophysics, oceanography, materials science,...
of light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
, known as optical homodyne tomography
Tomography
Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in tomography is called a tomograph, while the image produced is a tomogram. The method is used in radiology, archaeology, biology, geophysics, oceanography, materials science,...
. Using balanced homodyne measurements, one can derive the Wigner function and a density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
for the state of the light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
.
One approach involves measurements along different rotated directions in phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
. For each direction
, one can find a probability distributionProbability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
for the probability densityProbability density
Probability density may refer to:* Probability density function in probability theory* The product of the probability amplitude with its complex conjugate in quantum mechanics...
of measurements in the
direction of phase space yielding the value . Using an inverse Radon transformation (the filtered back projection) on leads to the Wigner function, , which can be converted by an inverse fourier transform into the density matrixDensity matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
for the state in any basis. A similar technique is often used in medical tomography
Tomography
Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in tomography is called a tomograph, while the image produced is a tomogram. The method is used in radiology, archaeology, biology, geophysics, oceanography, materials science,...
.
Example homodyne tomography.
What you would want, is to measure field amplitudes or quadratures with high efficiencies, which we can by using photodetectors, together with temporal mode selectivity. Balanced homodyne tomography is a reliable technique of reconstructing quantum states in the optical domain. This technique combines the advantages of the high efficiencies of photodiodes in measuring the intensity or photon number of light, together with measuring the quantum features of light by a clever set-up called the homodyne tomographyTomography
Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in tomography is called a tomograph, while the image produced is a tomogram. The method is used in radiology, archaeology, biology, geophysics, oceanography, materials science,...
detector. This is explained by the following example.
A laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
is directed onto a 50-50% beamsplitter, splitting the laserbeam into two beams. One is used as local oscillator
Local oscillator
A local oscillator is an electronic device used to generate a signal normally for the purpose of converting a signal of interest to a different frequency using a mixer. This process of frequency conversion, also referred to as heterodyning, produces the sum and difference frequencies of the...
(LO) and the other is used to generate photons with a particular quantum state. The generation of quantum states can be realized, e.g. by directing the laser beam through a frequency doubling crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
and then onto a parametric down-conversion crystal. This crystal generates two photons in a certain quantum state. One of the photons is used as a trigger signal used to trigger (start) the readout event of the homodyne tomography detector. The other photon is directed into the homodyne tomography detector, in order to reconstruct its quantum state. Since the trigger and signal photons are entangled (this is explained by the Spontaneous parametric down-conversion article), it is important to realize, that the optical mode of the signal state is created nonlocal only when the trigger photon impinges the photodector (of the trigger event readout module) and is actually measured. More simply said, it is only when the trigger photon is measured, that the signal photon can be measured by the homodyne detector.
Now let us consider the homodyne tomography
Tomography
Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in tomography is called a tomograph, while the image produced is a tomogram. The method is used in radiology, archaeology, biology, geophysics, oceanography, materials science,...
detector as depicted in figure 4. The signal photon (this is the quantum state we want to reconstruct) interferes with the local oscillator
Local oscillator
A local oscillator is an electronic device used to generate a signal normally for the purpose of converting a signal of interest to a different frequency using a mixer. This process of frequency conversion, also referred to as heterodyning, produces the sum and difference frequencies of the...
, when they are directed onto a 50-50% beamsplitter. Since the two beams originate from the same so called master laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
, they have the same fixed phase
Phase
-In physics:*Phase , a physically distinctive form of a substance, such as the solid, liquid, and gaseous states of ordinary matter**Phase transition is the transformation of a thermodynamic system from one phase to another*Phase...
relation. The local oscillator must be intense, compared to the signal so it provides a precise phase reference. The local oscillator is so intense, that we can treat it classically (a = α) and neglect the quantum fluctuations.
The signal field is spatially and temporally controlled by the local oscillator, which has a controlled shape. Where the local oscillator is zero, the signal is rejected. Therefore, we have temporal-spatial mode selectivity of the signal.
The beamsplitter redirects the two beams to two photodetectors. The photodetectors generate a electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
proportional to the photon number. The two detector currents are subtracted and the resulting current is proportional to the electric field operator in the signal mode, depended on relative optical phase of signal and local oscillator.
Since the electric field amplitude of the local oscillator is much higher than that of the signal the intensity or fluctuations in the signal field can be seen. The homodyne tomography system functions as an amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
. The system can be seen as an interferometer with such a high intensity reference beam (the local oscillator) that unbalancing the interference by a single photon in the signal is measurable. This amplification is well above the photodetectors noise floor
Noise floor
In signal theory, the noise floor is the measure of the signal created from the sum of all the noise sources and unwanted signals within a measurement system, where the noise is defined as any signal other than the one being monitored....
.
The measurement is reproduced a large number of times. Then the phase difference between the signal and local oscillator is changed in order to ‘scan’ a different angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
in the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
. This can be seen from figure 4. The measurement is repeated again a large number of times and a marginal distribution
Marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. The term marginal variable is used to refer to those variables in the subset of variables being retained...
is retrieved from the current difference. The marginal distribution
Marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. The term marginal variable is used to refer to those variables in the subset of variables being retained...
can be transformed into the density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
and/or the Wigner function. Since the density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
and the Wigner function give information about the quantum state of the photon, we have reconstructed the quantum state of the photon.
The advantage of this method is that this arrangement is insensitive to fluctuations in the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of the laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
.
The quantum computations for retrieving the quadrature component from the current difference are performed as follows.
The photon number operator for the beams striking the photodetectors after the beamsplitter is given by:
,where i is 1 and 2, for respectively beam one and two.
The mode operators of the field emerging the beamsplitters are given by:
The
denotes the annihilation operator of the signal and alpha the complex amplitude of the local oscillator.The number of photon difference is eventually proportional to the quadrature and given by:
,Rewriting this with the relation:
Results in the following relation:
,where we see clear relation between the photon number difference and the quadrature component
. By keeping track of the sum current, one can recover information about the local oscillator’s intensity, since this is usually an unknown quantity, but an important quantity for calculating the quadrature component .Problems with linear inversion
One of the primary problems with using linear inversion to solve for the density matrixDensity matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
is that in general the computed solution will not be a valid density matrix. For example, it could give negative probabilities or probabilities greater than 1 to certain measurement outcomes. This is particularly an issue when fewer measurements are made.
Another issue is that in infinite dimensional Hilbert spaces, an infinite number of measurement outcomes would be required. Making assumptions about the structure and using a finite measurement basis leads to artifacts in the phase space density.
Maximum likelihood estimation
Maximum likelihood estimation (also known as MLE or MaxLik) is a popular technique for dealing with the problems of linear inversion. By restricting the domain of density matrices to the proper space, and searching for the density matrix which maximizes the likelihoodLikelihood
Likelihood is a measure of how likely an event is, and can be expressed in terms of, for example, probability or odds in favor.-Likelihood function:...
of giving the experimental results, it guarantees the state to be theoretically valid while giving a close fit to the data. The likelihood of a state is the probability that would be assigned to the observed results had the system been in that state.
Suppose the measurements
have been observed with frequencies . Then the likelihood associated with a state iswhere
is the probability of outcome for the state .Finding the maximum of this function is non-trivial and generally involves iterative methods. The methods are an active topic of research.
Problems with maximum likelihood estimation
Maximum likelihood estimation suffers from some less obvious problems than linear inversion. One problem is that it makes predictions about probabilities that cannot be justified by the data. This is seen most easily by looking at the problem of zero eigenvalues. The computed solution using MLE often contains eigenvalues which are 0, i.e. it is rank deficient. In these cases, the solution then lies on the boundaryBoundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of the n-dimensional Bloch sphere
Bloch sphere
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....
. This can be seen as related to linear inversion giving states which lie outside the valid space (the Bloch sphere). MLE in these cases picks a nearby point that is valid, and the nearest points are generally on the boundary.
This is not physically a problem, the real state might have zero eigenvalues. However, since no value may be less than 0, an estimate of an eigenvalue being 0 implies that the estimator is certain the value is 0, otherwise they would have estimated some
greater than 0 with a small degree of uncertaintyUncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
as the best estimate. This is where the problem arises, in that it is not logical to conclude with absolute certainty after a finite number of measurements that any eigenvalue (that is, the probability of a particular outcome) is 0. For example, if a coin is flipped 5 times and each time heads was observed, it does not mean there is 0 probability of getting tails, despite that being the most likely description of the coin.
Bayesian methods
BayesianBayesian average
A Bayesian average is a method of estimating the mean of a population consistent with Bayesian interpretation, where instead of estimating the mean strictly from the available data set, other existing information related to that data set may also be incorporated into the calculation in order to...
mean estimation (BME) is a relatively new approach which addresses the problems of maximum likelihood estimation. It focuses on finding optimal solutions which are also honest in that they include error bars in the estimate. The general idea is to start with a likelihood function
Likelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...
and a function describing the experimenter's prior knowledge (which might be a constant function), then integrate over all density matrices using the product of the likelihood function
Likelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...
and prior knowledge function as a weight.
Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional bloch sphere
Bloch sphere
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....
. In the case of a coin flipped N times to get N heads described above, with a constant prior knowledge function, BME would assign
as the probability for tails.BME provides a high degree of accuracy in that it minimizes the operational divergences of the estimate from the actual state.
Quantum measurement tomography
One can imagine a situation in which an apparatus performs some measurement on quantum systems, and determining what particular measurement is desired. The strategy is to send in systems of various known states, and use these states to estimate the outcomes of the unknown measurement. Also known as "quantum estimation", tomography techniques are increasingly important including those for quantum measurement tomography and the very similar quantum state tomography.Since a measurement can always be characterized by a set of POVM
POVM
In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...
's, the goal is to reconstruct the characterizing POVM
POVM
In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...
's
.The simplest approach is linear inversion. Similar to in quantum state observation, use
.Exploiting linearity as above, this can be inverted to solve for the
.Not surprisingly, this suffers from the same pitfalls as in quantum state tomography. Namely, non-physical results, in particular negative probabilities. Here the
will not be valid POVMPOVM
In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...
's, as they will not be positive. Bayesian methods as well as Maximum likelihood estimation of the density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
can be used to restrict the operators to valid physical results.
Quantum tomography of pre-measurement states
The main tool of the retrodictive approach of quantum physics is the pre-measurement state which allows predictions about state preparations of the measured system leading to a given measurement result.As it was shown in a recent work, such a state reveals interesting quantum properties of the corresponding measurement such as its non-classicality
Nonclassical light
Nonclassical light is any state of light that cannot be described using classical electromagnetism; its characteristics are described by the quantised electromagnetic field and quantum mechanics...
or its projectivity.
However, we cannot realize the tomography of this state with the usual methods based on measurements, since it needs non-destructive measurements which are some particularly measurements.
The experimental procedure, proposed in , is based on the retrodictive approach of quantum physics, in which we have an expression of retrodictive probabilities similar to Born's rule:
where
and are respectively the pre-measurement state, corresponding to the measurement characterized by some a POVM element , and a hermitian and positive operator corresponding to the preparation of the measured system in a state .In the frame of the mathematical foundations of quantum physics, such a operator is a proposition about the state of the system, as a POVM element, and for having an exhaustive set of propositions, these operators must be a resolution of the Hilbert space:
From Born's, we can derive with Bayes' theorem, the expressions of the pre-measurement state
and proposition operators .The pre-measurement state simply corresponds to the normalized POVM element:
and the proposition operators are linked to the possible preparations of the system by:
where
is the dimension of the Hilbert space and is the probability of preparing the state .Thus, we can probe the measurement apparatus with a statistical mixture:
in order to measure the retrodictive probability
.This mixture could be obtained by preparations based on random choices 'm' with the probabilities
.Then, we replace the POVM elements describing the measurements in a usual method for the tomography of states by the operators
. The method will give the state giving the probabilities which are the most closest to those measured. This is the pre-measurement state with which we can have some interesting properties of the measurement giving the result 'n', as explained in.Quantum process tomography
Quantum process tomography (QPT) deals with identifying an unknown quantum dynamical process. The first approach, introduced in 1997 and sometimes known as standard quantum process tomography (SQPT) involves preparing an ensemble of quantum states and sending them through the process, then using quantum state tomography to identify the resultant states. Other techniques include ancilla-assisted process tomography (AAPT) and entanglement-assisted process tomography (EAPT) which require an extra copy of the system.Each of the techniques listed above are known as indirect methods for characterization of quantum dynamics, since they require the use of quantum state tomography to reconstruct the process. In contrast, there are direct methods such as direct characterization of quantum dynamics (DCQD) which provide a full characterization of quantum systems without any state tomography.
Quantum dynamical maps
A quantum process, also known as a quantum dynamical map,, can be described by a completely positive map,where
, the bounded operators on Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
; with operation elements
satisfying so that .Let
be an orthogonal basis for . Write the operators in this basis.This leads to
,where
.The goal is then to solve for
, which is a positive superoperatorSuperoperator
In physics, a superoperator is a linear operator acting on a vector space of linear operators.Sometimes the term refers more specially to a completely positive map which does not increase or preserves the trace of its argument....
and completely characterizes
with respect to the basis.Standard quantum process tomography
SQPT approaches this using linearly independent inputs , where is the dimension of the Hilbert space . For each of these input states , sending it through the process gives an output state which can be written as a linear combination of the , i.e. . By sending each through many times, quantum state tomography can be used to determine the coefficients experimentally.Write
,where
is a matrix of coefficients.Then
.Since
form a linearly independent basis,.Inverting
gives :.