Quantum tomography
Quantum
In quantum process tomography on the other hand, known
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
This can be easily understood by making a classical analogy. Consider a
For quantum mechanical particles the same can be done. The only difference is that the Heisenberg's uncertainty principle mustn't be violated, meaning that we cannot measure the particle's momentum and position at the same time. 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, or (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 the quantum state 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,
Methods of quantum state tomography
Linear inversion
Using
Given
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
- .
Here Ei is some fixed list of individual measurements (with binary outcomes), and A does all the measurements at once.
Then applying this to yields the
- .
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 projectors . For example, in a 2-D Hilbert space with 3 measurements , each measurement has 2 outcomes, each of which has a projector Ei, for 6 projectors, whereas the real dimension of the space of density matrices is (2⋅22)/2=4, leaving to be 6 x 4. To solve the system, multiply on the left by :
- .
Now solving for yields the
- .
This works in general only if the measurement list Ei is tomographically complete. Otherwise, the matrix will not be
Continuous variables and quantum homodyne tomography
In infinite dimensional
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
Example: single-qubit state tomography
The density matrix of a single qubit can be expressed in terms of its
- .
The single-qubit state tomography can be performed by means of single-qubit Pauli measurements:[8]
- First, create a list of three quantum circuits, with the first one measuring the qubit in the computational basis (Z-basis), the second one performing a Hadamard gate before measurement (which makes the measurement in X-basis), and the third one performing the appropriate phase shift gate (that is ) followed by a Hadamard gate before measurement (which makes the measurement in Y-basis);
- Then, run these circuits (typically thousands of times), and the counts in the measurement results of the first circuit produces , the second circuit , and the third circuit ;
- Finally, if , then a measured Bloch vector is produced as , and the measured density matrix is ; If , it'll be necessary to renormalize the measured Bloch vector as before using it to calculate the measured density matrix.
This algorithm is the foundation for qubit tomography and is used in some quantum programming routines, like that of Qiskit.[9][10]
Example: homodyne tomography.
Electromagnetic field amplitudes (quadratures) can be measured with high efficiency using
Quantum homodyne tomography is understood by the following example. A
Now consider the
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
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 balanced detection method is that this arrangement is insensitive to fluctuations in the intensity of the laser.
The quantum computations for retrieving the quadrature component from the current difference are performed as follows.
The
- ,
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
Problems with linear inversion
One of the primary problems with using linear inversion to solve for the
Another issue is that in infinite dimensional
Maximum likelihood estimation
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.[12][13] 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
This is not physically a problem, the real state might have zero
Bayesian methods
Bayesian 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.[4]
BME provides a high degree of accuracy in that it minimizes the operational divergences of the estimate from the actual state.[4]
Methods for incomplete data
The number of measurements needed for a full quantum state tomography for a multi-particle system scales exponentially with the number of particles, which makes such a procedure impossible even for modest system sizes. Hence, several methods have been developed to realize quantum tomography with fewer measurements.
The concept of matrix completion and compressed sensing have been applied to reconstruct density matrices from an incomplete set of measurements (that is, a set of measurements which is not a quorum). In general, this is impossible, but under assumptions (for example, if the density matrix is a pure state, or a combination of just a few pure states) then the density matrix has fewer degrees of freedom, and it may be possible to reconstruct the state from the incomplete measurements.[14]
Permutationally Invariant Quantum Tomography[15] is a procedure that has been developed mostly for states that are close to being permutationally symmetric, which is typical in nowadays experiments. For two-state particles, the number of measurements needed scales only quadratically with the number of particles. [16] Besides the modest measurement effort, the processing of the measured data can also be done efficiently: It is possible to carry out the fitting of a physical density matrix on the measured data even for large systems. [17] Permutationally Invariant Quantum Tomography has been combined with compressed sensing in a six-qubit photonic experiment.[18]
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's, the goal is to reconstruct the characterizing POVM's . The simplest approach is linear inversion. As 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.[19]
Quantum process tomography
Quantum process tomography (QPT) deals with identifying an unknown quantum dynamical process. The first approach, introduced in 1996 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.[20] Other techniques include ancilla-assisted process tomography (AAPT) and entanglement-assisted process tomography (EAPT) which require an extra copy of the system.[21]
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.[22]
The number of experimental configurations (state preparations and measurements) required for full quantum process tomography grows exponentially with the number of constituent particles of a system. Consequently, in general, QPT is an impossible task for large-scale quantum systems. However, under weak decoherence assumption, a quantum dynamical map can find a sparse representation. The method of compressed quantum process tomography (CQPT) uses the compressed sensing technique and applies the sparsity assumption to reconstruct a quantum dynamical map from an incomplete set of measurements or test state preparations.[23]
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 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.[21][22]
Standard quantum process tomography
SQPT approaches this using
Write
- ,
where is a matrix of coefficients. Then
- .
Since form a linearly independent basis,
- .
Inverting gives :
- .
References
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- ^ Bradben. "Pauli measurements - Azure Quantum". docs.microsoft.com. Retrieved 2022-04-16.
- ^ "Quantum State Tomography — Qiskit Experiments 0.2.0 documentation". qiskit.org. Retrieved 2022-04-11.
- ^ "Quantum Tomography — Qiskit 0.36.0 documentation". qiskit.org. Retrieved 2022-04-11.
- ^ Online Encyclopedia of Laser Physics and Technology. "Frequency Doubling". Archived from the original on 2016-06-03. Retrieved 2015-08-16.
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