Direct measurement methods of density matrix of an entangled quantum state

The proposed methods are feasible based on previous experimental works.

“The similarity and feasibility of those two methods are discussed by considering the previous experimental works.”

• Also stated in: § IV. CONCLUSION AND REMARKS · plain

  “In this study we investigated how to reconstruct the unknown density operator of two component entangled quantum state by using postselected weak measurement method and three sequential measurements where each complementary to the last, respectively, and discussed its feasibility by taking into account the recent related experimental works.”

• cites [26] Zhang, 2016, Phys. Rev. A sourceSupported
• cites [30] Fedorov, 2011, New J. Phys. sourcePartly supported

  The Conclusions (p. unavailable) propose experimental reconstruction schemes for both qutrits and ququarts, and the appendix describes the ququart method as similar to the qutrit method. Our copy does not establish the methods’ feasibility by considering previous experimental works, so that basis for the claim is unsupported. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

The matrix elements of a quantum system can be obtained by considering the weak measurement of an observable composed of three incompatible projection operators Π_ij = π_j π_D π_i.

“As Lundeen and his co-workers [11] studied, the matrix elements of a quantum system can be obtained by considering the weak measurement of an observable composed of three incompatible projection operator:”

• cites [11] Lundeen, 2012, Phys. Rev. Lett. sourceSupported

The real and imaginary parts of the weak value of π1_i π2_j can be written in terms of detection probabilities P1...P6 as Eq 25 and Eq 26.

“Finally, the real and imaginary parts of weak value of π 1 i π 2 j ( Eq.( 13)) can be written as”

Every matrix element of ρ in B' basis can be expressed in terms of the weak value of joint projection operator π1_i π2_j in A' basis (Eq 9).

“Furthermore, we can define the density matrix ρ of an entangled state in B ′ as well, and the expressions of its matrix elements can be written as”

The weak average of π_DD π_ij can be obtained as Eq 41 using the method introduced in Refs.[11,32].

“If we assume that the polarizers represented by the projection operator π ij and π DD causing displacement along x and y directions, respectively, we can get the weak average of π DD π ij by using the method introduced in Refs.[11,32] as”

• cites [11] Lundeen, 2012, Phys. Rev. Lett. sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

• cites [32] Ochoa, 2018, Sci. Rep. sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

• AssumptionThe authors state this as an assumption: The polarizers represented by the projection operators π_ij and π_DD cause displacement along x and y directions, respectively.

The weak value of joint operators π1_i π2_j can be found from the modular values of π1_i, π2_j, and π1_i+π2_j via Eq 13.

“From this theoretical result we can deduce that if we can measure the modular values of projection operators π 1 i , π 2 j and π 1 i + π 2 j , respectively, the weak value of joint operators π 1 i π 2 j can be found easily”

Every matrix element of ρ in A' basis can be expressed in terms of the weak value of joint projection operator π1_i π2_j in A' basis (Eq 8).

“By using the definition of weak value of joint operators, every matrix element of ρ which is written in Eq.( 5) can be expressed in terms of the weak value of joint project operator π 1 i π 2 j in A ′ basis as”

In the first method, the real and imaginary parts of a matrix element can be read out from detection probability after taking appropriate projection operations before detection on the final state of the pointer.

“Based on the theoretical analysis of Ref. [30], the real and imaginary parts of a matrix element can be readout from detection probability after taking appropriate pro-jection operations before detection on the final state of the pointer.”

• cites [30] Fedorov, 2011, New J. Phys. sourceNot checked yet

  A check is scheduled.

By measuring the modular values of π1_i, π2_j, and π1_i+π2_j, one can determine the matrix elements ρ_ij,kl and ρ_αβ,α'β' of density matrix ρ using Eq 8 and Eq 9.

“after that we can determine the matrix elements ρ ij,kl and ρ αβ,α ′ β ′ of density matrix ρ by using Eq.( 8) and Eq.( 9).”

In the second method, three sequential measurements on three projection operators of two subsystems, each complementary to the last, are taken to find the matrix elements of a two-photon entangled state.

“Three sequential measurements on three projection operators of two subsystems where each complementary to the last are taken to find the matrix elements of a two photon entangled state.”

In the second method, the real and imaginary parts of corresponding matrix elements can be found by calculating the joint positions and momentum shifts of the final pointer state.

“Thus, the real and imaginary parts of corresponding matrix elements can be found by calculating the joint positions and momentum shifts of the final pointer state.”

To reconstruct the density matrix of a two-dimensional unknown state by weak measurement, one must scan through all elements in both A and B bases.

“However, if we want to reconstruct the density matrix of a two dimensional unknown state by using weak measurement technique, we have to take scan through all elements in both A and B bases since its unknown parameters more than the corresponding pure state.”

Any matrix element of an entangled state can be obtained efficiently via proper weak and strong measurements in the proposed schemes.

“In our schemes any matrix elements of an entangled state can be obtained efficiently via proper weak and strong measurements.”

In the first method, since the weak measurement of joint projection operators of two subsystems cannot be measured directly, it is found in terms of the modular values of corresponding operators.

“Since the weak measurement of joint projection operators of two subsystems can not be measured directly, it is founded in terms of the modular values of corresponding operators.”

The Eq 41 is still valid for non-commuting observables if they are measured sequentially as measuring π_DD followed by π_ij.

“Here, we have to note that π ij and π DD are non-commute, but as showed in Ref. [34] the Eq.( 41) still valid for non-commuting observables if they are measured sequentially as measuring the π DD followed by π ij .”

• cites [34] Gross, 2015, Phys. Rev. A sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

In the second method, after taking two sequential weak measurements with projection operators complementary to each other, followed by a strong measurement on another projection operator complementary to the last, the weak average equals the expectation values of products of annihilation operators of four Gaussian pointer states.

“After taking the two sequential weak measurements with projection operators where complementary each other, and followed by a strong measurement on another projection operator where complementary to the last, the weak average equal to the expectation values of products of annihilation operators (can be defined in terms of position and momentum operator) of four Gaussian pointer states.”

• AssumptionThe authors state this as an assumption: The initial state of the total system is |Φ Φ| ⊗ ρ, where the pointer state is composed of two Gaussian beams of two photons with x and y transverse spatial distributions.
• AssumptionThe authors state this as an assumption: The spatial degree of freedom of every pointer of two subsystems has x and y directional zero mean Gaussian distribution, and initially there is no correlation between them.

To reconstruct the density matrix of a two-component entangled state in A' basis, more strong measurement steps in B' basis are needed than to determine matrix elements in B' basis.

“From the above examples we can deduce that the determination of matrix elements ρ ij,kl in A ′ basis need more experimental steps than the matrix elements ρ αβ,α ′ β ′ in B ′ basis.”

In recent study of Guang-Can-Guo [30], they showed that the joint weak values π1_i π_j^w_αβ can be found in terms of modular values.

“However, in recent study of Guang-Can-Guo [30], they showed that the joint weak values π 1 i π j w αβ can be found in terms of modular values.”

• cites [30] Fedorov, 2011, New J. Phys. sourceNot checked yet

In the same measurement process, method one may need more resources than method two.

“Thus, in the same measurement process the method one may need more resources than method two.”

The elements of density operator of two entangled photon state can be reconstructed by determining the expectation values f in Eq 46 and Eq 47 via ABCDPdτ = ABCD_f.

“Then, the elements of density operator of two entangled photon state can be reconstructed by determining the expectation values f in Eq.( 46) and Eq.( 47) via ABCDPdτ = ABCD f , respectively.”

During readout of matrix elements in the Lab of a two entangled photon process, more and some complicated experimental setups and more resources are needed in method one rather than method two.

“Since the Hilbert space of two entangled photon state is larger than single photon case, during the readout of the matrix elements in the Lab of a two entangled photon processes we would need more and some complicated experimental setups and need more resources in method one rather than method two.”

In the second method, the result of these sequential measurements is proportional to the value of matrix elements.

“It is found that the result of these sequential measurements proportional to the value of matrix elements.”

The direct measurement of density matrix of an entangled system by using weak measurement technique has not been explicitly studied until now.

“However, in general, the state of a quantum system is represented by density operator, and the direct measurement of density matrix of an entangled system by using weak measurement technique have not been explicitly studied until now.”

In their work to get the complex amplitude C_ij in Eq 2 we only need to scan a definite element of B' basis, i.e. C_ij = χ π1_i π2_j^w_DD.

“In their work to get the complex amplitude C ij in Eq.( 2) we only need to scan a definite element of B ′ basis, i.e.”

• cites [30] Fedorov, 2011, New J. Phys. sourceNot checked yet

The direct measurement of a photon polarization state in two-dimensional system and direct measurement of density matrix of a single photon polarization state in pure and mixed state cases showed the power of weak measurement technique in state determination processes.

“In particular, the direct measurement of a photon polarization state in two dimensional system [14] and direct measurement of density matrix of a single photon polarization state in pure and mixed state cases [26] showed the power of weak measurement technique in state determination processes.”

• cites [14] Salvail, 2013, Nat. Photonics sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

• cites [26] Zhang, 2016, Phys. Rev. A sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.