Direct quantum process tomography via measuring sequential weak values of incompatible observables

The weak value concept has enabled fundamental studies of quantum measurement and, recently, found potential applications in quantum and classical metrology. However, most weak value experiments reported to date do not require quantum mechanical descriptions, as they only exploit the classical wave nature of the physical systems. In this work, we demonstrate measurement of the sequential weak value of two incompatible observables by making use of two-photon quantum interference so that the results can only be explained quantum physically. We then demonstrate that the sequential weak value measurement can be used to perform direct quantum process tomography of a qubit channel. Our work not only demonstrates the quantum nature of weak values but also presents potential new applications of weak values in analyzing quantum channels and operations.


SUPPLEMENTARY NOTE 1 -DETAILS ON THE SEQUENTIAL WEAK VALUE MEASUREMENT SCHEME
In this note, we give more details on the schematic and the theory for the sequential weak value measurement for observableÂ andB. Note that the schematic is based on the proposal of realizing von-Neumann measurement with variable measurement strength via quantum erasure in Ref. [1]. As in the main text, we consider three qubits: a system qubit |ψ s , an ancillary qubit |ψ a , and a meter qubit |Φ m , where the system is a quantum state to be measured and the meter is used to register the measurement outcome. The ancilla is used to temporarily register the measurement outcome for the first observableÂ, the registered information of which is erased finally to leave all the measurement information in the meter qubit.
Initially, the total quantum state is prepared in |ψ s |0 a |0 m . In our scheme, the first interaction for observableÂ is a controlled-σ x (CNOT) type interaction between system and ancillary qubits. Hereσ x ,σ y andσ z stand for Pauli operators. The CNOT type interaction can be described as a unitary operationÛ A = (Î −Â) ⊗Î +Â ⊗σ x on the system and the ancilla qubits, whereÎ is the identity operation. After applyingÛ A , the total state evolves tô Next, we consider the measurement interaction for the observableB. That is the controlled-controlled-σ z (CCZ) operation with rotating operations on the meter qubit, where the unitary evolution for the second interaction is given asÛ whereΠ i a = |i i| is the projector on the ancillary qubit, the measurement strength is adjusted by g and the rotating matrix on meter qubit is givens asR The total states after applyingÛ B is given aŝ where we have used an approximation up to the first order of g with the assumption of the weak measurement condition g 1. From the last term it is clear that the action of the sequential observablesBÂ is registered on the meter qubit, but the superfluous second and third terms still remain. To remove these terms, we can consider to apply a projector on the ancilla qubitΠ + a = |+ +| where |+ ≡ (|0 + |1 )/ √ 2 (The projectorΠ + a can be viewed as a quantum erasure as it erases the measurement results registered in the ancilla qubit [1]). Then, the joint state of system and meter qubits after projecting ancilla qubit withΠ + a is given as Finally, the system qubit is projected on |φ s with a projectorΠ φ s = |φ φ|. Then, the remained meter state is given as where BÂ w ≡ φ|BÂ|ψ / φ|ψ is the sequential weak value for observablesÂ andB. The sequential weak value is registered in the amplitude of meter state, and it thus can be extracted by measuring the typical expectation values of Pauli observables, such asσ x andσ y : We further note that the quantum erasure is probabilistic as we only consider one POVM elementΠ + a . However, the eraser scheme can be deterministic if an additional interaction between the system and the meter qubits is allowed.
Let us consider the other POVM elementΠ − a =Î −Π + a = |− −|. If the ancillary state is projected byΠ − a , the system and meter state is given as An additional feed-forward interaction,Û C = (Î − 2Â) −1 ⊗Π 0 m −Î ⊗Π 1 m , is applied to the system and the meter qubits. After projecting the system state withΠ φ s , the meter state is given as Therefore, the sequential weak value can be deterministically extracted from both POVMs with Supplementary Equations (7-8).

SUPPLEMENTARY NOTE 2 -DIRECT QUANTUM STATE TOMOGRAPHY VIA SEQUENTIAL WEAK VALUES
A 2×2 input density matrixρ in in H/V basis is represented aŝ Each element of the density matrix is described as ρ mn = Tr[|a m a n |ρ in ] where |a 1 = |H and |a 2 = |V , i.e. ρ mn can be obtained from the expectation value ofρ in for the |a m a n | operator. In cases of off-diagonal elements, the operators are non-Hermitian operator which gives complex value as the expectation value.
The sequential weak values with incompatible observables allow to implement the non-Hermitian operator and measure the complex expectation value. The weak value for an input density matrixρ in is given as where p = Tr[Π φ sρin ] is the post-selection probability. Therefore, the elements of density matrix can be measured directly by setting observables and post-selection to be |a m a n |.
For the direct quantum state tomography (d-QST), the observablesÂ andB and post-selectionΠ φ s are set aŝ The |a i and |b i are orthonormal basis and mutually unbiased | a i |b j | = 1/ √ 2 [2]. In particular, for H/V basis, observables are set as |a 1 = |H , |a 2 = |V , |b 1 = |D , and |b 2 = |A . And, let us fixB = |b 1 b 1 |, and considerÂ = |a m a m | andΠ n s = |a n a n |. With this setting, the density matrix element can be directly measured from the sequential weak value BÂ mn w and post selection probability p n = Tr[Π n sρin ], In addition, the density matrix in different basis can be measured by setting the observables and post-selection in other ways. For example, the set of |a 1 = |D , |a 2 = |A , |b 1 = |H , and |b 2 = |V allow to do d-QST in D/A basis. We test the d-QST for a pure state |ψ in = (|H −i √ 3|V )/2 with our sequential weak value method, see Supplementary Figure 1. By setting the |a i , |b j for d-QST in D/A basis, we measure the sequential weak values corresponding to each density matrix elements. a ρ 11 , b ρ 12 , c ρ 21 and d ρ 22 in D/A basis. And, the p n is estimated from the ratio between coincidence counts in cases ofΠ 1 s = |D D| andΠ 2 s = |A A|. Because the |D D| and |A A| make complete set, the sum of post-selection probability is unity. So, the probability can be obtained from the coincidence counts ratio with the condition that the sum is unity. The e shows the result of d-QST with maximum likelihood method. The directly measured density matrix shows an excellent agreement with the ideal density matrix by the fidelity of

Quantum process tomography in Dirac basis
A quantum process E(ρ in ) in d-dimension Hilbert space can be represented as where {Ê i } is an operator basis set and the quantum process is fully characterized by the process matrix elements χ ij . A standard choice of {Ê i } is the Pauli basis set {Î,σ x ,σ y ,σ z } or a Kraus operator basis set {Ê i ≡ |a m a n |}, where {|a k } is an orthonormal state basis set and the index i is represented as new indexes {m, n} as i = (m − 1)d + n. We consider the Kraus basis set, and after changing the indexes i and j into {m, n} and {m , n } the quantum process is then rewritten as χ mn,m n |a m a n |ρ in |a n a m | We introduce another orthonormal state basis {|b l }, which is a complementary basis set to {|a k }, satisfying Thus, one can recognize {|b l } is mathematically the discrete Fourier transform of {|a k }: After substituting the relation of Supplementary Equation (17) into Supplementary Equation (15), then we have where we have defined a new operator basiŝ Supplementary Equation (18) is further simplified by substituting Finally, we have where kn and k n are the binary number representations of i and j, respectively. We call the basis {Ŝ i } as Dirac basis following the analogy with the Dirac distribution which is used to characterize a quantum state in the complementary basis [2,3]. In our direct quantum process tomography (d-QPT) scheme, the process matrix elements χ S ij is directly measured from the sequential weak values.

Direct quantum process tomography via sequential weak values
In this subsection, we elaborate how our sequential weak value measurement scheme can be used for characterizing a unknown quantum process in Dirac basis. We consider a quantum process in the qubit Hilbert space (d = 2) and two complementary basis sets : {|a 1 = |0 , |a 2 = |1 } and {|b 1 = |+ , |b 2 = |− }. The initial input state is prepared in a pure stateρ n in = |a n a n |, the first observableÂ is set asÂ k = |b k b k |, an arbitrary quantum process E(·) is placed between observablesÂ andB, the second observableB is set asB n = |a n a n |, and finally we consider a projectorΠ k s = |b k b k |. With these settings, the sequential weak value is given as where is the post-selection probability.
where we have used a i |b j = (−1) δi2δj2 / √ 2 and δ i2 is the Kronecker delta. Likewise, we calculate further aŝ Then, we finally get Therefore, the process matrix elements in Dirac basis can be directly obtained from the sequential weak value B nÂk n k w as χ S ij ≡ χ S nk,n k = B nÂk n k w × 4p n k × (−1) δ k2 δ n 2 (−1) δ k 2 δn2 .
Note that nk and n k are the binary number representations of i and j, respectively.

Constraints on the process matrix in Dirac basis
For the qubit Hilbert space (d = 2), the process matrix is a 4 × 4 matrix and thus it has 32 real elements. However, since the process matrix is Hermitian, it has at most 16 independent real parameters. We can further reduce the number of independent parameters by invoking that the post-selection probability is given by Supplementary  Equation (25). The post-selection probability p n k depends on the input stateρ n in , the process E, and the final projectorΠ k s . Thus, there are four possible cases for p n k , and they are explicitly calculated as p 11 = (χ 11 + χ * 12 + χ * 13 + χ * 14 )/4, Since p n k is probability, it should be a real number. Therefore, we obtain four constraints for imaginary parts of process matrix elements as Im(χ 12 ) + Im(χ 13 ) + Im(χ 14 ) = 0, Im(χ 12 ) + Im(χ 23 ) + Im(χ 24 ) = 0, Im(χ 13 ) − Im(χ 23 ) + Im(χ 34 ) = 0, Im(χ 14 ) − Im(χ 24 ) − Im(χ 34 ) = 0.
Note that these constraints must hold regardless of whether the quantum process is a trace preserving map or a trace non-preserving map [4].

Fidelity estimation via d-QPT
The most common method to compare two quantum process is to evaluate the fidelity between two quantum process matrices. The fidelity is defined by where χ S exp is the experimentally measured quantum process matrix and χ S ideal is a target ideal operation. The direct quantum process tomography via sequential weak values allows one to estimate the fidelity between a target ideal operation and an actually implemented operation without a standard full quantum process tomography.
In the main text, we have considered two cases, the Hadamard operation and R x -gate operation which corresponds to a quarter wave plate (QWP) at 45 • . The quantum process matrix for the Hadamard operation is given in Dirac basis as As defined by Supplementary Equation (32), it is straightforward to show that the fidelity for Hadamard operation is evaluated as Likewise, the ideal R x -gate operation is given as And, the fidelity is evaluated as where we have used the constraints of Supplementary Equation (31) and the trace preserving condition.
Compressive sensing quantum process tomography with sequential weak values The standard full quantum process tomography (s-QPT) is based on the mathematical linear inversion. Therefore, the number of experimental configurations must be larger than the number of independent parameters in the process matrix. In other words, the experimental configurations should be tomographically complete. However, when the dimension of system gets larger, the s-QPT becomes unfavorable as the number of configurations scales exponentially. To resolve this problem, a mathematical technique, known as compressive sensing, has been employed to the quantum process tomography [5,6].
In the compressive sensing quantum process tomography (cs-QPT), experimentally measured outcomes P = {P 1 , P 2 , · · · , P n } with n different configurations are used to reconstruct a process matrix, where d is the dimension of the system and thus the process matrix has m ≡ d 4 independent parameters. Let us represent the process matrix elements in a vectorized form χ, which has at least m elements. P and χ have a relation as where the subscripts denote the dimension and Λ is a n×m matrix given by the experimental configurations. Therefore, if n < m, the experimental configurations are tomographically incomplete, thus χ becomes underdetermined. However, the compressive sensing technique allows one to find χ with incomplete set of measurements, where χ is assumed to be sparse. In quantum information processing, it is typically aimed to implement a quantum gate, which is a unitary process and thus the sparsity assumption is valid. The idea of compressive sensing is to solve the following convex optimization problem [5,6],