A quantum state is the carrier of quantum information. One of the central topics in quantum science is the ability to characterize and manipulate quantum states, which would be applied to quantum communications, quantum cryptography, and quantum computation. Suppose that there is an unknown quantum state. It is impossible to create an identical copy of this state by the no-cloning theorem.^[1] However, the information of the unknown quantum state can be transmitted to a remote location by the process of quantum teleportation,^[2] with the help of classical communication and previously shared quantum entanglement. Now consider the following problem. There is a machine which can produce a large number of identical unknown quantum states. We are permitted to design any device to repeatedly measure this state and read out the probability distributions of the measurement outcomes. Can we infer all the information of the unknown state?

This is the problem of quantum state tomography. Two aspects are essential for this problem: the measurements and the method to reconstruct the unknown state via the measurement probabilities. Various measurements are considered for achieving quantum state tomography, such as Pauli measurements,^[3,4] projective measurement onto mutually unbiased bases,^[5–8] or onto general orthogonal bases,^[9–12] and symmetric informationally complete measurements.^[13–15] The data of measurement probabilities is collected after we take measurements on the ensemble of identical unknown quantum states. There are different methods applied on the data in order to calculate all the coefficients of the unknown state, such as linear inversion,^[16,17] maximum-likelihood estimation,^[9,18,19] Bayesian mean estimation,^[20] and the convex optimization program.^[12,21]

In this paper, we focus on the two-qubit pure state tomography. From the result of papers,^[9,22] the density matrix of an unknown two-qubit state can be determined with the expectation values of 16 Pauli measurements. It was proved that a two-qubit pure state |ϕ⟩ is uniquely determined among all states (UDA) with 11 Pauli measurements.^[4] That is to say, there does not exist any other state, pure or mixed, which has the same measurement results as |ϕ⟩. The projective measurements of five mutually unbiased bases have the same property.^[6] In Ref. [12], another set of five orthonormal bases are constructed by a sequence of orthogonal polynomials. The method of the convex optimization program can be used to search for a density matrix, whose measurement probabilities match the data obtained experimentally. It has been found that several projections onto the tensor products of Pauli eigenstates are enough for two-qubit pure states tomography. These tensor product states are arranged into five orthonormal bases. Furthermore, the projective measurements onto these bases is simulated by five quantum circuits. Each circuit is decomposed by the common quantum gates used in universal quantum computing,^[22] which is the footstone in building a universal quantum computer and has been researched on various materials. The measurement parts are the projective measurement {|0⟩⟨0|,|1⟩⟨1|} on each qubit. We can directly calculate the unknown two-qubit pure state with partial measurement probabilities of these bases.

Moreover, a necessary condition is given for the unsolved problem of whether three orthonormal bases can distinguish all two-qubit states. This problem dates back to the famous Pauli problem, whether or not the position and the momentum distributions can uniquely determine the wave function of a particle (up to a global phase).^[23] The answer to the Pauli problem is negative and many related topics have been studied. The finite version of Pauli problem is considered by Asher Peres.^[24] He noted that the probabilities of two orthonormal bases should be sufficient to determine a finite dimensional pure state up to a set of ambiguities. The findings in Ref. [25] suggest that the set is not a set of measure zero. A general question is the following. How many different orthonormal bases are needed at least in order to uniquely determine a pure quantum state in d-dimensional Hilbert space ? The minimal number of orthonormal bases is gradually clear except for d = 4.^[10,26–29] There exist four orthonormal bases which can distinguish any pair of different pure states in , but it is not clear whether three bases can do this. There is a partial answer for this problem. It is proved that three product orthonormal bases cannot distinguish all two-qubit states, not even four. The product basis here is restricted to the form of {ϕ_i⊗ψ_j:i, j = 1, 2}, where {ϕ₁, ϕ₂} and {ψ₁, ψ₂} are two orthonormal bases of . Given any three orthonormal bases of , the unitary transition matrices from to can be written as U₁ and U₂. We show that all 32 elements of U₁ and U₂ should not be zero. If not, these three bases are not enough. For example, if we choose the first basis as the standard basis and choose the second basis as the basis of Bell states, no matter how we choose a third basis, these three bases cannot distinguish all two-qubit pure states.

The rest of this paper is organized as follows. In Section 2, we describe the five product bases and construct five quantum circuits to simulate the projective measurements onto these bases. Then, we show the method to directly calculate the unknown pure state with partial measurement probabilities. In Section 3, we give a necessary condition that three orthonormal bases must satisfy in order to distinguish all two-qubit pure states. We will end up with a conclusion in Section 4.

As a warm up, let us consider the one-qubit state tomography. It is known in many textbooks that three orthonormal bases uniquely determine an unknown qubit state,^[22] which can be the eignenbases of Pauli operators σ_x, σ_y, and σ_z. Denote the eignenbases as {|+⟩, |−⟩}, {|L⟩, |R⟩}, and {|0⟩,|1⟩} correspondingly. Here

Let U be a d × d unitary operation. If we want to measure a d-dimensional pure state |ϕ⟩ with the projective measurements onto the orthonormal bases {U|0⟩,..., U|d − 1⟩}, we can take the unitary operation U^† on the state |ϕ⟩, then measure U^†|ϕ⟩ with the projective measurements onto the orthonormal bases {|0⟩,..., |d − 1⟩}. As the probabilities of the j-th outcome for both sides are equal, |⟨ψ|U|j⟩|² = |⟨j|U^†|ψ⟩|², where j = 0,..., d − 1.

Let V₁ and V₂ be the unitary transition from basis {|0⟩, |1⟩} to basis {|+⟩, |−⟩} and {|L⟩, |R⟩}. Then V₁ = H, ; V₂ = |L⟩⟨0| + |R⟩⟨1|, . We can simulate the projective measurements onto these Pauli eigenbases with the circuits in Fig. 1.

Fig. 1.

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	Fig. 1. Three circuits for one qubit tomography. The measurement parts are the projective measurement {|0⟩⟨0|, |1⟩⟨1|}. Before measurement, we take the unitary operation H, , and I on the unknown qubit state respectively.

Now, we construct five product orthonormal bases to directly determine an unknown two-qubit pure state, which are as follows:

Fig. 2.

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	Fig. 2. Circuit for the projective measurement onto . Let the unknown two-qubit pure state be |ϕ⟩. We directly measure each qubit with the projective measurement {|0⟩⟨0|, |1⟩⟨1|}. The state would collapse to |00⟩, |01⟩, |10⟩, or |11⟩.

Next we use the circuits to simulate the projective measurements onto the bases , which are depicted in Fig. 3. Let U_j be the unitary transition from the basis to the basis , j = 1, 2, 3, 4. The task is to decompose the gates by the common gates in universal quantum computation. For example, . If we take the operation on the states of the basis , we will obtain the states of the basis .

Fig. 3.

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Fig. 3. (color online) Circuits for the projective measurements onto bases . The circuits at top left, top right, bottom left, and bottom right simulate the projective measurements onto , and respectively. There are two two-qubit operations in these circuits. The first one is the swap operation, which is depicted in Fig. 4. The other one is a controlled unitary operation, . The one-qubit operations are the Hadamard gate H and the gate that appeared in the circuit for one qubit tomography. At the end of each circuit, we measure each qubit with the projective measurement {|0⟩⟨0|, |1⟩⟨1|}.

An arbitrary unknown two-qubit pure state, separable or entangled, can be expressed with the vector form a|00⟩ + b|01⟩ + c|10⟩ + d|11⟩, where |a|² + |b²| + |c|² + |d|² = 1. Any complex number g can be expressed as |g|e^iθ_g. So we can write the unknown state with a new form

Fig. 4.

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	Fig. 4. Simulation of swap operation. The swap operation is: |00⟩⟨00| + |10⟩⟨ 01| + |01⟩⟨10| + |11⟩⟨11|. If we take the swap operation on input state |jk⟩, the output state will be |kj⟩, j, k = 0, 1. It can be decomposed into three control-not operations. The bases states of and would be transformed into the bases states of and by this operation.

If we measure |ϕ⟩ with the projective measurement onto the basis , the state will collapse to |00⟩, |01⟩, |10⟩, or |11⟩. Denote the measurement probabilities as , , , and respectively. The coefficients of {|a|,|b|,|c|,|d|} can be calculated by the probabilities of the basis . The following equations hold:

At least one probability in is not equal to zero. Next we divide it into four cases and calculate the remaining phases {θ_a, θ_b, θ_c, θ_d} at each case. Record the probability as if the state collapses to the k-th element of basis .

For example, the probabilities of the basis are the following: , , , and . Then |a| = 1, b = c = d = 0. We do not need to calculate the phases {θ_b, θ_c, θ_d} and the unknown state is |00⟩ for the freedom choice of the global phase. The nonzero probability may be another one. We list the nonzero probability and the corresponding state in the following table (Table 1).

Table 1.

Table 1.

Table 1. The nonzero probability and the corresponding state. . Nonzero probability Unknown state |00⟩ |01⟩ |10⟩ |11⟩

Table 1. The nonzero probability and the corresponding state. .

There are subcases totally. For example, . The unknown state can be written as . We only need to calculate the phase θ_b. This task can be fulfilled by considering the probabilities that the state collapses to |0 + ⟩ and |0L⟩. The following equations hold:

When , the unknown state is |a||00⟩ + |c|e^iθ_c|10⟩. In order to calculate the phase θ_c, we can consider the probabilities that the state collapses to | + 0⟩ and |L0⟩.

When , the unknown state is |a||00⟩ + |d|e^iθ_d|11⟩. In order to calculate the phase θ_d, we can consider the probabilities that the state collapses to |L − ⟩ and |LR⟩.

When , the unknown state is |b||01⟩ + |c|e^iθ_c|10⟩. In order to calculate the phase θ_d, we can consider the probabilities that the state collapses to | − L⟩ and |RL⟩.

When , the unknown state is |b||01⟩ + |d|e^iθ_d|11⟩. In order to calculate the phase θ_d, we can consider the probabilities that the state collapses to | + 1⟩ and |L1⟩.

When , the unknown state is |c||10⟩ + |d|e^iθ_d|11⟩. In order to calculate the phase θ_d, we can consider the probabilities that the state collapses to |1 + ⟩ and |1L⟩.

There are four subcases in total. For example, . The unknown state can be written as |a||00⟩ + |b|e^iθ_b|01⟩ + |c|e^iθ_c|10⟩. The phases θ_b and θ_c can be determined by Eqs. (5) and (6).

When , the unknown state is |a||00⟩ + |b|e^iθ_b|01⟩ + |d|e^iθ_d|11〉. The phases θ_b can be determined by Eq. (5). The phase difference θ_d − θ_b can be determined by considering the probabilities that the state collapses to | + 1⟩ and |L1⟩.

When , the unknown state is |a||00⟩ + |c|e^iθ_c|10⟩ + |d|e^iθ_d|11⟩. The phase θ_c can be determined by Eq. (6). The phase difference θ_d − θ_c can be determined by considering the probabilities that the state collapses to |1 + ⟩ and |1L⟩.

When , the unknown state is |b||01⟩ + |c|e^iθ_c|10⟩ + |d|e^iθ_d|11⟩. The phase θ_d can be determined by Eq. (9). The phase difference θ_d − θ_c can be determined by considering the probabilities that the state collapses to |1 + ⟩ and |1L⟩.

We write the unknown two-qubit pure state as |a||00⟩ + |b|e^iθ_b|01⟩ + |c|e^iθ_c|10⟩ + |d|e^iθ_d|11⟩. We can use Eqs. (5) and (6) to determine the phases θ_b and θ_c. The phase difference θ_d − θ_b can be determined by Eq. (11). Then all the phases are clear.

Now we draw a conclusion. The coefficients {|a|, |b|, |c|, |d|} can be calculated by the probabilities , which are obtained from the projective measurement onto the basis . The remaining task is to calculate the phases for the nonzero amplitudes. The number of nonzero amplitudes may be k as listed in the case k, where k ∈ {1, 2, 3, 4}. We can let one phase be zero for the freedom choice of the global phase. For the cases 2, 3, and 4, we use two, four, and six measurement probabilities respectively to determine the phases. These probabilities are some parts obtained from the projective measurements onto the bases , where each basis state is the tensor product of Pauli eigenstates. Until we determine the amplitudes and the phases, the unknown two-qubit state will be clear.

Let be m orthonormal bases of the d-dimensional Hilbert space , where . Given any pair of different pure states ρ₁ and ρ₂, these bases can distinguish them if for some basis states . The minimal number of orthonormal bases needed to distinguish all d-dimensional pure states is clear except for d = 4.^[10] Four orthonormal bases can distinguish all two-qubit pure states; however, the problem remains unsolved whether three orthonormal bases are enough. Here we give a necessary condition that three orthonormal bases should satisfy. If not, three bases cannot distinguish all two-qubit pure states.

Denote three orthonormal bases as . We record the bases , , and as {|1⟩, |2⟩, |3⟩, |4⟩}, {|ϕ₁⟩,..., |ϕ₄⟩}, {|ψ₁⟩,..., |ϕ₄⟩} respectively. The unitary transition matrices from to are written as U₁ and U₂. U₁ = (x_{i, j}), U₂ = (y_{i, j}), i, j = 1,..., 4. In other words, the bases of and are the following:

Here we give the proof. Without loss of generality, the zero matrix element is x₁₁. We may as well let the first rows of matrix U₁ and U₂ be real numbers. Namely, x_k1 and y_k1 are all real, k = 1, 2, 3, 4. This is because the probabilities that state |ϕ⟩ collapses to the basis states |ϕ_k⟩ or c|φ_k⟩ are equal, |c| = 1. So the bases {|ϕ₁⟩, |φ₂⟩, |ϕ₃⟩, |ϕ₄⟩}, and {c₁|ϕ₁⟩, c₂|ϕ₂⟩, c₃|ϕ₃⟩, c₄|ϕ₄⟩} have the same power of discrimination, where |c_k| = 1 for k = 1, 2, 3, 4.

We can construct a pair of unnormalized vector states:

It is easy to prove that their probability distributions onto the first basis are the same.

For the second basis ,

Define . With a similar analysis, if for k = 1, 2, 3, 4, their probability distributions onto the basis are the same one.

Now we consider the following linear equations:

Then,

The zero element may appear at the j-th row of the matrices U₁ or U₂. Then the two vector states can be constructed as: |Φ₁⟩ = |j⟩ + ∑_{k ≠ j} [Re(c_k) + i· Im(c_k)]|k⟩, |Φ₂⟩ = |j⟩ − ∑_{k ≠ j}[Re(c_k) + i · Im(c_k)]|k⟩. With a similar analysis, there exists a nontrivial solution so that the probability distributions onto the three orthonormal bases are equal. This completes the proof.

This necessary condition can give us some insights to the minimal number of orthonormal bases which can distinguish all two-qubit pure states. If the minimal number is actually three, we should find them with the constraint of the condition. For example, if we choose basis as the standard basis and choose basis as the basis of Bell states, no matter how we choose a third basis, these three bases cannot distinguish all two-qubit pure states. As there are zero elements in matrix U₁, if any three orthonormal bases which satisfy this condition cannot distinguish all two-qubit pure states, the minimal number is four.

In this paper, we show that any unknown two-qubit pure state can be directly reconstructed by the measurement probabilities of projections onto 16 tensor products of Pauli eigenstates. These projections are arranged into five orthonormal bases. Moreover, we construct five quantum circuits to simulate the projective measurements onto the states of these bases. In the end, we focus on the unsolved problem of whether three orthonormal bases can distinguish all two-qubit pure states. A necessary condition is given. If the condition is not satisfied, three orthonormal bases are not sufficient to distinguish all two-qubit pure states.