Permutationally invariant quantum state tomography

Last updated September 29, 2024

Permutationally invariant quantum state tomography (PI quantum state tomography) is a method for the partial determination of the state of a quantum system consisting of many subsystems.

In general, the number of parameters needed to describe the quantum mechanical state of a system consisting of $N$ subsystems is increasing exponentially with $N.$ For instance, for an $N$ -qubit system, $2^{N}-2$ real parameters are needed to describe the state vector of a pure state, or $2^{2N}-1$ real parameters are needed to describe the density matrix of a mixed state. Quantum state tomography is a method to determine all these parameters from a series of measurements on many independent and identically prepared systems. Thus, in the case of full quantum state tomography, the number of measurements needed scales exponentially with the number of particles or qubits.

For large systems, the determination of the entire quantum state is no longer possible in practice and one is interested in methods that determine only a subset of the parameters necessary to characterize the quantum state that still contains important information about the state. Permutationally invariant quantum tomography is such a method. PI quantum tomography only measures $\varrho _{\rm {PI}},$ the permutationally invariant part of the density matrix. For the procedure, it is sufficient to carry out local measurements on the subsystems. If the state is close to being permutationally invariant, which is the case in many practical situations, then $\varrho _{\rm {PI}}$ is close to the density matrix of the system. Even if the state is not permutationally invariant, $\varrho _{\rm {PI}}$ can still be used for entanglement detection and computing relevant operator expectations values. Thus, the procedure does not assume the permutationally invariance of the quantum state. The number of independent real parameters of $\varrho _{\rm {PI}}$ for $N$ qubits scales as $\sim N^{3}.$ The number of local measurement settings scales as $\sim N^{2}.$ Thus, permutationally invariant quantum tomography is considered manageable even for large $N$ . In other words, permutationally invariant quantum tomography is considered scalable.

The method can be used, for example, for the reconstruction of the density matrices of systems with more than 10 particles, for photonic systems, for trapped cold ions or systems in cold atoms.

The permutationally invariant part of the density matrix

PI state tomography reconstructs the permutationally invariant part of the density matrix, which is defined as the equal mixture of the quantum states obtained after permuting the particles in all the possible ways ^[1]

\varrho _{\rm {PI}}={\frac {1}{N!}}\sum _{k}\Pi _{k}\varrho \Pi _{k}^{\dagger },

where $\Pi _{k}$ denotes the kth permutation. For instance, for $N=2$ we have two permutations. $\Pi _{1}$ leaves the order of the two particles unchanged. $\Pi _{2}$ exchanges the two particles. In general, for $N$ particles, we have $N!$ permutations.

It is easy to see that $\varrho _{\rm {PI}}$ is the density matrix that is obtained if the order of the particles is not taken into account. This corresponds to an experiment in which a subset of $N$ particles is randomly selected from a larger ensemble. The state of this smaller group is of course permutationally invariant.

The number of degrees of freedom of $\varrho _{\rm {PI}}$ scales polynomially with the number of particles. For a system of $N$ qubits (spin- $1/2$ particles) the number of real degrees of freedom is ^[2]

{\binom {N+3}{N}}-1={\frac {1}{6}}(N^{3}+6N^{2}+11N).

The measurements needed to determine the permutationally invariant part of the density matrix

To determine these degrees of freedom,^[1]

{\binom {N+2}{N}}={\frac {(N+2)(N+1)}{2}}={\frac {1}{2}}(N^{2}+3N+2)

local measurement settings are needed. Here, a local measurement settings means that the operator $A_{j}$ is to be measured on each particle. By repeating the measurement and collecting enough data, all two-point, three-point and higher order correlations can be determined.

Efficient determination of a physical state

So far we have discussed that the number of measurements scales polynomially with the number of qubits.

However, for using the method in practice, the entire tomographic procedure must be scalable. Thus, we need to store the state in the computer in a scalable way. Clearly, the straightforward way of storing the $N$ -qubit state in a $2^{N}\times 2^{N}$ density matrix is not scalable. However, $\varrho _{\rm {PI}}$ is a blockdiagonal matrix due to its permutational invariance and thus it can be stored much more efficiently.^[3]

Moreover, it is well known that due to statistical fluctuations and systematic errors the density matrix obtained from the measured state by linear inversion is not positive semidefinite and it has some negative eigenvalues. An important step in a typical tomography is fitting a physical, i. e., positive semidefinite density matrix on the tomographic data. This step often represents a bottleneck in the overall process in full state tomography. However, PI tomography, as we have just discussed, allows the density matrix to be stored much more efficiently, which also allows an efficient fitting using convex optimization, which also guarantees that the solution is a global optimum.^[3]

Characteristics of the method

PI tomography is commonly used in experiments involving permutationally invariant states. If the density matrix $\varrho _{\rm {PI}}$ obtained by PI tomography is entangled, then density matrix of the system, $\varrho$ is also entangled. For this reason, the usual methods for entanglement verification, such as entanglement witnesses or the Peres-Horodecki criterion, can be applied to $\varrho _{\rm {PI}}$ . Remarkably, the entanglement detection carried out in this way does not assume that the quantum system itself is permutationally invariant.

Moreover, the expectation value of any permutaionally invariant operator is the same for $\varrho$ and for $\varrho _{\rm {PI}}.$ Very relevant examples of such operators are projectors to symmetric states, such as the Greenberger–Horne–Zeilinger state, the W state and symmetric Dicke states. Thus, we can obtain the fidelity with respect to the above-mentioned quantum states as the expectation value of the corresponding projectors in the state $\varrho _{\rm {PI}}.$

The quantum fidelity of $\varrho _{\rm {PI}}$ and $\varrho$ can be bounded from below as^[1]

F(\varrho ,\varrho _{\rm {PI}})\geq \langle P_{s}\rangle _{\varrho }^{2},

where $P_{s}$ is the projector to the symmetric subspace. For symmetric states, ${\langle }P_{s}\rangle =1$ holds. This way, we can lower bound the difference knowing only $\varrho _{\rm {PI}}.$

Links to other approaches

There are other approaches for tomography that need fewer measurements than full quantum state tomography. As we have discussed, PI tomography is typically most useful for quantum states that are close to being permutionally invariant. Compressed sensing is especially suited for low rank states.^[4] Matrix product state tomography is most suitable for, e.g., cluster states and ground states of spin models.^[5] Permutationally invariant tomography can be combined with compressed sensing. In this case, the number of local measurement settings needed can even be smaller than for permutationally invariant tomography.^[2]

Experiments

Permutationally invariant tomography has been tested experimentally for a four-qubit symmetric Dicke state,^[1] and also for a six-qubit symmetric Dicke in photons, and has been compared to full state tomography and compressed sensing.^[2] A simulation of permutationally invariant tomography shows that reconstruction of a positive semidefinite density matrix of 20 qubits from measured data is possible in a few minutes on a standard computer.^[3] The hybrid method combining permutationally invariant tomography and compressed sensing has also been tested.^[2]

Related Research Articles

Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. The sender does not have to know the particular quantum state being transferred. Moreover, the location of the recipient can be unknown, but to complete the quantum teleportation, classical information needs to be sent from sender to receiver. Because classical information needs to be sent, quantum teleportation cannot occur faster than the speed of light.

Quantum entanglement is the phenomenon of a group of particles being generated, interacting, or sharing spatial proximity in such a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.

The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.

In quantum mechanics, a density matrix is a matrix that describes an ensemble of physical systems as quantum states. It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed ensembles. Mixed ensembles arise in quantum mechanics in two different situations:

when the preparation of the systems lead to numerous pure states in the ensemble, and thus one must deal with the statistics of possible preparations, and
when one wants to describe a physical system that is entangled with another, without describing their combined state; this case is typical for a system interacting with some environment. In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.

Quantum decoherence is the loss of quantum coherence. Quantum decoherence has been studied to understand how quantum systems convert to systems which can be explained by classical mechanics. Beginning out of attempts to extend the understanding of quantum mechanics, the theory has developed in several directions and experimental studies have confirmed some of the key issues. Quantum computing relies on quantum coherence and is one of the primary practical applications of the concept.

In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.

In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

<span class="mw-page-title-main">Quantum tomography</span> Reconstruction of quantum states based on measurements

Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. The source of these states 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. The term tomography was first used in the quantum physics literature in a 1993 paper introducing experimental optical homodyne tomography.

The Peres–Horodecki criterion is a necessary condition, for the joint density matrix $of two quantum mechanical systems and, to be separable. It is also called the PPT criterion, for positive partial transpose . In the 2\times2 and 2\times3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply. The theorem was discovered in 1996 by Asher Peres and the Horodecki family$

In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix $ρ$ , the von Neumann entropy is

Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement and quantum squeezing. This field promises to develop measurement techniques that give better precision than the same measurement performed in a classical framework. Together with quantum hypothesis testing, it represents an important theoretical model at the basis of quantum sensing.

Squashed entanglement, also called CMI entanglement, is an information theoretic measure of quantum entanglement for a bipartite quantum system. If $is the density matrix of a system composed of two subsystems and, then the CMI entanglement of system is defined by$

In the case of systems composed of $subsystems, the classification of quantum-entangled states is richer than in the bipartite case. Indeed, in multipartite entanglement apart from fully separable states and fully entangled states, there also exists the notion of partially separable states.$

In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.

Entanglement distillation is the transformation of N copies of an arbitrary entangled state $into some number of approximately pure Bell pairs, using only local operations and classical communication.$

In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distanceT is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions.

The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a quantum system:

The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information. It is one of the central quantities used to qualify the utility of an input state, especially in Mach–Zehnder interferometer-based phase or parameter estimation. It is shown that the quantum Fisher information can also be a sensitive probe of a quantum phase transition. The quantum Fisher information $of a state with respect to the observable is defined as$

In quantum information, the gnu code refers to a particular family of quantum error correcting codes, with the special property of being invariant under permutations of the qubits. Given integers g, n, and m, the two codewords are

In quantum metrology in a multiparticle system, the quantum metrological gain for a quantum state is defined as the sensitivity of phase estimation achieved by that state divided by the maximal sensitivity achieved by separable states, i.e., states without quantum entanglement. In practice, the best separable state is the trivial fully polarized state, in which all spins point into the same direction. If the metrological gain is larger than one then the quantum state is more useful for making precise measurements than separable states. Clearly, in this case the quantum state is also entangled.

References

1 2 3 4 Tóth, G.; Wieczorek, W.; Gross, D.; Krischek, R.; Schwemmer, C.; Weinfurter, H. (16 December 2010). "Permutationally Invariant Quantum Tomography". Physical Review Letters. 105 (25): 250403. arXiv: 1005.3313 . Bibcode:2010PhRvL.105y0403T. doi:10.1103/PhysRevLett.105.250403. PMID 21231565. S2CID 21786571.
1 2 3 4 Schwemmer, Christian; Tóth, Géza; Niggebaum, Alexander; Moroder, Tobias; Gross, David; Gühne, Otfried; Weinfurter, Harald (24 July 2014). "Experimental Comparison of Efficient Tomography Schemes for a Six-Qubit State". Physical Review Letters. 113 (4): 040503. arXiv: 1401.7526 . Bibcode:2014PhRvL.113d0503S. doi:10.1103/PhysRevLett.113.040503. PMID 25105604. S2CID 26493608.
1 2 3 Moroder, Tobias; Hyllus, Philipp; Tóth, Géza; Schwemmer, Christian; Niggebaum, Alexander; Gaile, Stefanie; Gühne, Otfried; Weinfurter, Harald (1 October 2012). "Permutationally invariant state reconstruction". New Journal of Physics. 14 (10): 105001. arXiv: 1205.4941 . Bibcode:2012NJPh...14j5001M. doi:10.1088/1367-2630/14/10/105001. S2CID 73720137.
↑ Gross, David; Liu, Yi-Kai; Flammia, Steven T.; Becker, Stephen; Eisert, Jens (4 October 2010). "Quantum State Tomography via Compressed Sensing". Physical Review Letters. 105 (15): 150401. arXiv: 0909.3304 . Bibcode:2010PhRvL.105o0401G. doi:10.1103/PhysRevLett.105.150401. PMID 21230876. S2CID 19029700.
↑ Cramer, Marcus; Plenio, Martin B.; Flammia, Steven T.; Somma, Rolando; Gross, David; Bartlett, Stephen D.; Landon-Cardinal, Olivier; Poulin, David; Liu, Yi-Kai (December 2010). "Efficient quantum state tomography". Nature Communications. 1 (1): 149. arXiv: 1101.4366 . Bibcode:2010NatCo...1..149C. doi:10.1038/ncomms1147. PMID 21266999. S2CID 17325851.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[PRL10-1] 1 2 3 4 Tóth, G.; Wieczorek, W.; Gross, D.; Krischek, R.; Schwemmer, C.; Weinfurter, H. (16 December 2010). "Permutationally Invariant Quantum Tomography". Physical Review Letters. 105 (25): 250403. arXiv: 1005.3313 . Bibcode:2010PhRvL.105y0403T. doi:10.1103/PhysRevLett.105.250403. PMID 21231565. S2CID 21786571.

[PRL14-2] 1 2 3 4 Schwemmer, Christian; Tóth, Géza; Niggebaum, Alexander; Moroder, Tobias; Gross, David; Gühne, Otfried; Weinfurter, Harald (24 July 2014). "Experimental Comparison of Efficient Tomography Schemes for a Six-Qubit State". Physical Review Letters. 113 (4): 040503. arXiv: 1401.7526 . Bibcode:2014PhRvL.113d0503S. doi:10.1103/PhysRevLett.113.040503. PMID 25105604. S2CID 26493608.

[NJP12-3] 1 2 3 Moroder, Tobias; Hyllus, Philipp; Tóth, Géza; Schwemmer, Christian; Niggebaum, Alexander; Gaile, Stefanie; Gühne, Otfried; Weinfurter, Harald (1 October 2012). "Permutationally invariant state reconstruction". New Journal of Physics. 14 (10): 105001. arXiv: 1205.4941 . Bibcode:2012NJPh...14j5001M. doi:10.1088/1367-2630/14/10/105001. S2CID 73720137.

[4] Gross, David; Liu, Yi-Kai; Flammia, Steven T.; Becker, Stephen; Eisert, Jens (4 October 2010). "Quantum State Tomography via Compressed Sensing". Physical Review Letters. 105 (15): 150401. arXiv: 0909.3304 . Bibcode:2010PhRvL.105o0401G. doi:10.1103/PhysRevLett.105.150401. PMID 21230876. S2CID 19029700.

[5] Cramer, Marcus; Plenio, Martin B.; Flammia, Steven T.; Somma, Rolando; Gross, David; Bartlett, Stephen D.; Landon-Cardinal, Olivier; Poulin, David; Liu, Yi-Kai (December 2010). "Efficient quantum state tomography". Nature Communications. 1 (1): 149. arXiv: 1101.4366 . Bibcode:2010NatCo...1..149C. doi:10.1038/ncomms1147. PMID 21266999. S2CID 17325851.

[1]

[2]

[3]

[4]

[5]