View source for Physics:Permutationally invariant quantum state tomography

{{Short description|Efficient reconstruction of quantum states based on measurements}}
'''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 [[Physics:Quantum state|state]] of a system consisting of <math>N</math> subsystems is increasing exponentially with <math>N.</math> For instance, for an <math>N</math>-[[Qubit|qubit]] system, <math>2^N-2</math> real parameters are needed to describe the [[Physics:Quantum state|state vector]] of a pure state, or
<math>2^{2N} -1 </math> real parameters are needed to describe the  [[Density matrix|density matrix]] of a [[Physics:Quantum state#Mixed states|mixed state]]. ''[[Physics:Quantum tomography|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 [[Physics:Quantum tomography|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 <math>\varrho_{\rm PI},</math> 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 <math>\varrho_{\rm PI}</math> is close to the  density matrix of the system.
Even if the state is not permutationally invariant, <math>\varrho_{\rm PI}</math> 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 <math>\varrho_{\rm PI}</math> for <math>N</math> qubits  scales as <math>\sim N^3.</math> The number of local measurement settings scales as  <math>\sim N^2. </math> Thus, permutationally invariant quantum tomography is considered manageable even for large <math>N</math>. 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 ion quantum computer|trapped cold ions]] or systems in [[Physics:Ultracold atom|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 <ref name="PRL10">{{cite journal |last1=Tóth |first1=G. |last2=Wieczorek |first2=W. |last3=Gross |first3=D. |last4=Krischek |first4=R. |last5=Schwemmer |first5=C. |last6=Weinfurter |first6=H. |title=Permutationally Invariant Quantum Tomography |journal=Physical Review Letters |date=16 December 2010 |volume=105 |issue=25 |pages=250403 |doi=10.1103/PhysRevLett.105.250403|pmid=21231565 |arxiv=1005.3313 |bibcode=2010PhRvL.105y0403T |s2cid=21786571 }}</ref>

:<math>\varrho_{\rm PI}=\frac{1}{N!} \sum_k \Pi_k \varrho \Pi_k^\dagger,</math>

where <math>\Pi_k</math> denotes the ''k''th permutation. For instance, for <math>N=2</math> we have two permutations. <math>\Pi_1</math> leaves the order of the two particles unchanged. <math>\Pi_2</math> exchanges the two particles. In general, for <math>N</math> particles, we have <math>N!</math> permutations.

It is easy to see that <math> \varrho_{\rm PI} </math> 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 <math> N </math> 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|degrees of freedom]] of <math> \varrho_{\rm PI} </math> scales polynomially with the number of particles.  For a system of <math>N</math> qubits (spin-<math> 1/2</math> particles) the number of real degrees of freedom is <ref name="PRL14"/>

:<math>\binom{N+3}{N}-1.</math>

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

To determine these degrees of freedom,<ref name="PRL10"/>

:<math>\binom{N+2}{N}</math>

''local measurement settings'' are needed. Here, a local measurement settings means that the operator <math>A_j</math> 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 [[Qubit|qubit]]s.

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 <math>N</math>-qubit state in a <math>2^N\times2^N</math> density matrix is not scalable. However, <math>\varrho_{\rm PI}</math> is a blockdiagonal matrix due to its permutational invariance and thus it can be stored much more efficiently.<ref name="NJP12"/>

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 [[Definite matrix|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|convex optimization]], which also guarantees that the solution is a global optimum.<ref name="NJP12">{{cite journal |last1=Moroder |first1=Tobias |last2=Hyllus |first2=Philipp |last3=Tóth |first3=Géza |last4=Schwemmer |first4=Christian |last5=Niggebaum |first5=Alexander |last6=Gaile |first6=Stefanie |last7=Gühne |first7=Otfried |last8=Weinfurter |first8=Harald |title=Permutationally invariant state reconstruction |journal=New Journal of Physics |date=1 October 2012 |volume=14 |issue=10 |pages=105001 |doi=10.1088/1367-2630/14/10/105001|arxiv=1205.4941 |bibcode=2012NJPh...14j5001M |s2cid=73720137 }}</ref>

== Characteristics of the method ==

PI tomography is commonly used in experiments involving permutationally invariant states. If the density matrix <math>\varrho_{\rm PI}</math> obtained by PI tomography is [[Quantum entanglement|entangled]], then density matrix of the system, <math>\varrho</math> is also entangled. For this reason, the usual methods for entanglement verification, such as [[Entanglement witness|entanglement witness]]es or the Peres-Horodecki criterion,  can be applied to <math>\varrho_{\rm PI}</math>. 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 <math>\varrho</math> and for <math>\varrho_{\rm PI}.</math> 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 <math>\varrho_{\rm PI}.</math>

== 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. [[Physics:Quantum tomography|Compressed sensing]] is especially suited for low rank states.<ref>{{cite journal |last1=Gross |first1=David |last2=Liu |first2=Yi-Kai |last3=Flammia |first3=Steven T. |last4=Becker |first4=Stephen |last5=Eisert |first5=Jens |title=Quantum State Tomography via Compressed Sensing |journal=Physical Review Letters |date=4 October 2010 |volume=105 |issue=15 |pages=150401 |doi=10.1103/PhysRevLett.105.150401|pmid=21230876 |arxiv=0909.3304 |bibcode=2010PhRvL.105o0401G |s2cid=19029700 }}</ref> Matrix product state tomography is most suitable for, e.g., [[Cluster state|cluster state]]s and ground states of spin models.<ref>{{cite journal |last1=Cramer |first1=Marcus |last2=Plenio |first2=Martin B. |last3=Flammia |first3=Steven T. |last4=Somma |first4=Rolando |last5=Gross |first5=David |last6=Bartlett |first6=Stephen D. |last7=Landon-Cardinal |first7=Olivier |last8=Poulin |first8=David |last9=Liu |first9=Yi-Kai |title=Efficient quantum state tomography |journal=Nature Communications |date=December 2010 |volume=1 |issue=1 |pages=149 |doi=10.1038/ncomms1147|pmid=21266999 |arxiv=1101.4366 |bibcode=2010NatCo...1..149C |s2cid=17325851 }}</ref> 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.<ref name="PRL14">{{cite journal |last1=Schwemmer |first1=Christian |last2=Tóth |first2=Géza |last3=Niggebaum |first3=Alexander |last4=Moroder |first4=Tobias |last5=Gross |first5=David |last6=Gühne |first6=Otfried |last7=Weinfurter |first7=Harald |title=Experimental Comparison of Efficient Tomography Schemes for a Six-Qubit State |journal=Physical Review Letters |date=24 July 2014 |volume=113 |issue=4 |pages=040503 |doi=10.1103/PhysRevLett.113.040503|pmid=25105604 |arxiv=1401.7526 |bibcode=2014PhRvL.113d0503S |s2cid=26493608 }}</ref>

== Experiments ==

Permutationally invariant tomography has been tested experimentally for a four-qubit symmetric Dicke state,<ref name="PRL10"/> and also for a six-qubit symmetric Dicke in photons, and has been compared to full state tomography and compressed sensing.<ref name="PRL14"/> 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.<ref name="NJP12"/> The hybrid method combining permutationally invariant tomography and compressed sensing has also been tested.<ref name="PRL14"/>

== References ==
<references/>

[[Category:Quantum mechanics]]
[[Category:Tomography]]

{{Sourceattribution|Permutationally invariant quantum state tomography}}