1 Introduction↩︎

The main goal of the original quantum teleportation protocol is to transfer the quantum state describing a single qubit in one location (Alice) onto another qubit in a different location (Bob) [1]–[6]. The qubit to be teleported (actually its “wave function”) is not physically sent from Alice to Bob and one does not even need to know its wave function for the protocol to work. The key ingredient needed to its implementation is a maximally entangled state shared between Alice and Bob (a two-qubit state), usually called the entangled resource or the quantum communication channel. They both must know which entangled state they share before starting the protocol to properly implement it. In addition to the entangled resource, Alice must be able to perform Bell measurements and Bob must be able to realize single qubit unitary operations at his share of the entangled resource to finish the protocol. These unitary “corrections” depend on which maximally entangled state Alice and Bob share as well as on the Bell measurement (BM) result obtained by Alice. As such, Alice and Bob should have access to a reliable classical communication channel, through which Alice informs Bob of her BM result (two bits of classical information). If all went well, at the end of one run of the protocol Bob’s qubit (his share of the two-qubit entangled state) will be exactly described by the state teleported by Alice (input). Alice’s input state, on the other hand, will be described by a maximally mixed state. No vestige of the original quantum state describing the input state remains at Alice’s.

Let us assume now that we do not know the quantum resource shared by Alice and Bob but do know the input state to be teleported. Can we apply the teleportation protocol to reconstruct the two-qubit state shared between Alice and Bob? Can this be done by measuring only the single qubit with Bob after the execution of the teleportation protocol? What is the minimum number of different known input states that Alice needs to teleport to Bob to fully reconstruct the shared state? Can this be extended to reconstruct a multipartite quantum state?

In this work we address all the previous questions, presenting what we call the teleportation-based quantum state tomography (QST) protocol. We explicitly show that arbitrary two- and three-qubit density matrices can be fully reconstructed using the teleportation protocol. We also show how to extend the present protocol to reconstruct an arbitrary \(n\)-qubit density matrix, investigating the minimal number of known quantum states needed to be teleported from Alice to Bob for the protocol to work.

The present protocol is a conceptually different tool aiming at the fully reconstruction of arbitrary density matrices, complementing the seminal strategies to deal with such a problem that do not make use of the teleportation protocol [7]–[23]. These tools are broadly known as QST protocols. See, for instance, Refs. [24], [25] for recent reviews addressing the most important QST protocols. We should also mention that one of the main goals of this article is to present the key ideas leading to the teleportation-based QST protocol. We are not addressing whether the present approach is more efficient than the standard ones. We want to show that the quantum teleportation protocol, whose operation is fundamentally based on Bell measurements and classical communication, can be tailored to reconstruct an arbitrary quantum state. The quantum state to be reconstructed being the state shared among all the participants in the teleportation protocol. In other words, by repurposing the quantum teleportation protocol, we can change a tool originally developed for transferring unknown quantum states around into one that reveals it. This exchange of the roles of what is known and unknown in the quantum teleportation protocol is the basic insight that led to the teleportation-based QST protocol.

2 Two-qubit states↩︎

2.1 The teleportation-based QST protocol↩︎

The quantum state shared between Alice and Bob, and which we want to fully reconstruct, is not in general a pure state. Therefore, we need to recast the teleportation protocol in the language of density matrices [26], [27], the appropriate objects to describe an arbitrary quantum state, be it a pure or a mixed state.

The quantum state shared between Alice and Bob are labeled \(1\) and \(2\), respectively, while Alice’s input state, i.e., the state to be teleported, is labeled \(A\) (see Fig. 1). The density matrices describing those qubits are \(\rho_{12}\) and \(\rho_{A}=|\psi\rangle\langle \psi|\). Alice’s input will always be a pure state, namely, \(|\psi\rangle=\alpha|0\rangle + \beta|1\rangle\), where \(|\alpha|^2+|\beta|^2=1\).

Initially, the state describing all qubits is \[\rho = \rho_{A} \otimes \rho_{12} = \left( \begin{array}{cc} |\alpha|^2 & \alpha\beta^* \\ \alpha^*\beta & |\beta|^2 \end{array} \right) \otimes \rho_{12}, \label{stepA}\tag{1}\] where \(*\) denotes complex conjugation. After one complete run of the protocol, Bob’s qubit becomes [26], [27] \[\varrho_{2}=(U_jTr_{A,1}[P_j \rho P_j]U_j^\dagger)/Q_j. \label{stepD}\tag{2}\] In Eq. (2 ), \(Tr_{A,1}\) is the partial trace on Alice’s qubits, \(j\) represents the Bell measurement (BM) result obtained by Alice (\(j=\Psi^-,\Psi^+,\Phi^-,\Phi^+\)), and \(P_j\) gives the projectors describing the four possible BM outcomes, \[\begin{align} P_{\Psi^{\pm}} = |\Psi^{\pm}\rangle \langle \Psi^{\pm}|, & P_{\Phi^{\pm}} = |\Phi^{\pm}\rangle \langle \Phi^{\pm}|, \label{projectorB} \end{align}\tag{3}\] with the Bell states being \[\begin{align} |\Psi^{\pm}\rangle=(|01\rangle \pm |10\rangle)/\sqrt{2}, &|\Phi^{\pm}\rangle=(|00\rangle \pm |11\rangle)/\sqrt{2}. \label{BellB} \end{align}\tag{4}\]

Alice’s probability to measure a given Bell state \(j\) depends on the input state as well on the shared state and it is given by [26], [27] \[Q_j=Q_j(\alpha,\beta) = Tr[{P_j \rho}], \label{prob}\tag{5}\] where we have explicitly highlighted its dependence on the input state for further convenience in the subsequent calculations.

The corrections that Bob must implement on his qubit after being informed of Alice’s BM outcome is \(U_j\). These unitary corrections are given by the identity operator and the standard Pauli matrices: \(\mathbb{1},\sigma^z,\sigma^x,\sigma^z\sigma^x\) [28]. Since in the teleportation-based QST we always know the state to be teleported, those corrections are not needed and we can set all of them to \(\mathbb{1}\). Therefore, at the end of each run of the teleportation protocol, Bob’s state will be given by \[\varrho_{2} = \frac{Tr_{A,1}[P_j \rho P_j]}{Q_j} =\frac{1}{Q_j}\left( \begin{array}{cc} \tilde{b}_{11} & \tilde{b}_{12} \\ \tilde{b}_{12}^* & \tilde{b}_{22} \end{array} \right) =\left( \begin{array}{cc} b_{11} & b_{12} \\ b_{12}^* & b_{22} \end{array} \right). \label{stepE}\tag{6}\] Equation (6 ) is the key result we need to build the teleportation-based QST protocol. Bob is supposed to be able to experimentally determine the matrix elements \(b_{ij}\) of his single qubit state. They are functions of the shared state \(\rho_{12}\) matrix elements as well as of the particular input state teleported by Alice. As we will see, by properly working with only four different inputs, we will obtain a complete system of linear equations on the matrix elements of \(\rho_{12}\) that allows us to express them in terms of the matrix elements \(b_{ij}\) of Bob’s state. It is also convenient to write \(\tilde{b}_{ij}=\tilde{b}_{ij}(\alpha,\beta)\), where we highlight its dependence on the input state.

Since any physical density matrix must be positive definite and Hermitian, the most general two-qubit state, entangled or not, that we want to reconstruct can be written as follows: \[\rho_{12} = \left( \begin{array}{cccc} m_{11} & m_{12} & m_{13} & m_{14}\\ m_{12}^* & m_{22} & m_{23} & m_{24} \\ m_{13}^* & m_{23}^* & m_{33} & m_{34}\\ m_{14}^* & m_{24}^* & m_{34}^* & m_{44} \end{array} \right), \label{rho12}\tag{7}\] Note that \(m_{jj}\) are all positive reals and \(\sum_{j}m_{jj}=1\), with the latter being the normalization condition. We have a total of \(16\) real parameters to determine (\(15\) if we use the normalization condition): the real and imaginary parts of \(m_{ij}\), if \(i \neq j\), and the four real quantities given by \(m_{jj}\).

For definiteness, and without loss of generality, we assume that Alice’s BM result is always the Bell state \(|\Psi^-\rangle\). The same analysis applies if she obtains any other of the three remaining Bell states. It will only change the way Bob’s coefficients \(b_{ij}\) are related to the coefficients \(m_{ij}\). See Appendix 7 for more details. In all four cases, though, we will always have a compatible system of linear equations allowing us to solve all \(m_{ij}\) as functions of \(b_{ij}\).

Using Eqs. (1 ), (3 ), and (7 ), it is not difficult to see that Alice’s probability to measure the Bell state \(|\Psi^-\rangle\) is \[\begin{align} Q_{\Psi^{\small{-}}}(\alpha,\beta) &=&[(m_{33}+m_{44})|\alpha|^2 + (m_{11}+m_{22})|\beta|^2]/2 \nonumber \\ &&-Re[(m_{13}+m_{24})\alpha^*\beta]. \label{AliceProb} \end{align}\tag{8}\] And if we also employ Eq. (6 ), we obtain \[\begin{align} \tilde{b}_{11}(\alpha,\beta) & =& (m_{33}|\alpha|^2 +m_{11}|\beta|^2)/2 -Re[m_{13}\alpha^*\beta], \tag{9} \\ \tilde{b}_{22}(\alpha,\beta) & =& (m_{44}|\alpha|^2 +m_{22}|\beta|^2)/2 -Re[m_{24}\alpha^*\beta], \tag{10} \\ 2\tilde{b}_{12}(\alpha,\beta) & =& m_{34}|\alpha|^2 \!+\!m_{12}|\beta|^2 \!-\!m_{14}\alpha^*\beta \!-\! m_{23}^*\alpha\beta^*\!.\tag{11} \end{align}\]

Looking at Eqs. (9 )-(11 ), we realize that if Alice teleports the state \(|\psi\rangle=|0\rangle\) (\(\alpha,\beta=1,0\)), we can determine the coefficients \(m_{11},m_{22}\), and \(m_{12}\). Analogously, if she teleports the state \(|\psi\rangle=|1\rangle\) (\(\alpha,\beta=0,1\)) we obtain \(m_{33},m_{44}\), and \(m_{34}\). Finally, teleporting the states \(|\psi\rangle=(|0\rangle+|1\rangle)/\sqrt{2}\) (\(\alpha,\beta=1/\sqrt{2},1/\sqrt{2}\)) and \(|\psi\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}\) (\(\alpha,\beta=1/\sqrt{2},i/\sqrt{2}\)), we arrive at a system of linear equations on the real and imaginary parts of the remaining coefficients (\(m_{13},m_{14},m_{23},m_{24}\)) that allows us to express them in terms of the previous coefficients and of the matrix elements of Bob’s states associated with the latter two teleported states by Alice. Note that Bob’s states are obtained experimentally (measured) by him.¹ The matrix elements of his states are the piece of data needed to fully reconstruct the two-qubit state shared between Alice and Bob. The solution to all those equations can be written as follows: \[\begin{align} m_{11} &=& 2 \tilde{b}_{11}(0, 1),\;\; m_{22} = 2 \tilde{b}_{22}(0, 1), \label{1of16}\\ m_{33} &=& 2 \tilde{b}_{11}(1, 0),\;\; m_{44} = 2 \tilde{b}_{22}(1, 0), \\ m_{12} &=& 2 \tilde{b}_{12}(0, 1),\;\; m_{34} = 2 \tilde{b}_{12}(1, 0),\\ m_{13} &=& (1 - i) [\tilde{b}_{11}(0, 1) + \tilde{b}_{11}(1, 0)] \nonumber \\ &+& 2 i \tilde{b}_{11}(1/\sqrt{2}, i/\sqrt{2}) - 2 \tilde{b}_{11}(1/\sqrt{2}, 1/\sqrt{2}), \\ m_{24} &=&(1 - i) [\tilde{b}_{22}(0, 1) + \tilde{b}_{22}(1, 0)] \nonumber \\ &+& 2 i \tilde{b}_{22}(1/\sqrt{2}, i/\sqrt{2}) - 2 \tilde{b}_{22}(1/\sqrt{2}, 1/\sqrt{2}), \end{align}\tag{12}\] \[\begin{align} m_{14} &=&(1 - i) [\tilde{b}_{12}(0, 1) + \tilde{b}_{12}(1, 0)] \nonumber \\ &+& 2 i \tilde{b}_{12}(1/\sqrt{2}, i/\sqrt{2}) - 2 \tilde{b}_{12}(1/\sqrt{2}, 1/\sqrt{2}), \\ m_{23} &=& (1 - i) [\tilde{b}_{12}^*(0, 1) + \tilde{b}_{12}^*(1, 0)] \nonumber \\ &+& 2 i \tilde{b}_{12}^*(1/\sqrt{2}, i/\sqrt{2}) - 2 \tilde{b}_{12}^*(1/\sqrt{2}, 1/\sqrt{2}).\label{16of16} \end{align}\tag{13}\]

Some important remarks are in order now. First, the four states teleported by Alice to Bob leading to the full reconstruction of the shared state between them are not unique. For instance, instead of the states \(|\psi\rangle=(|0\rangle+|1\rangle)/\sqrt{2}\) and \(|\psi\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}\), she can use the states \(|\psi\rangle=(|0\rangle-|1\rangle)/\sqrt{2}\) and \(|\psi\rangle=(|0\rangle-i|1\rangle)/\sqrt{2}\). The key point here is that we must use at least four states. Each state leads to a different density matrix describing Bob’s qubit at the end of the teleportation protocol. And since each density matrix possesses four independent real parameters (the two real diagonal entries and the complex non-diagonal one), four different teleported states will lead to \(4\times 4 = 16\) real independent parameters. These \(16\) independent parameters with Bob match exactly the number of real parameters needed to completely describe a two-qubit state. That is why we obtained \(16\) linear equations in the real parameters of the two-qubit state that we wanted to reconstruct and whose solutions are compactly expressed in Eqs. (12 )-(13 ). Also, we will show next that the extension of the previous analysis to an \(n\)-qubit state is what leads to the generalization of the present protocol to multipartite qubit systems.

Second, since we are assuming that Bob, and eventually Alice, can fully determine single qubit states, they can use this ability to determine some of the parameters of the two-qubit state without using the teleportation-based QST protocol. This is accomplished by measuring \(\rho_1\) and \(\rho_2\), the single qubit states obtained by tracing out one or the other qubit from \(\rho_{12}\).

Third, the previous solution, Eqs. (12 )-(13 ), is also valid for low rank matrices. For example, if the rank of \(\rho_{12}\) is only three, such that the fourth row and column are all zero, Eqs. (12 )-(13 ) are still valid.

2.2 The standard QST protocol↩︎

Let us now briefly compare the present approach with the usual way of reconstructing a two-qubit state [10], [24]. Our goal here is to illustrate the main conceptual differences between both strategies. In Ref. [10], a two-qubit density matrix is reconstructed by measuring the expectation values associated with a set of projectors that span the two-qubit Hilbert space. Being more explicit, we need to be able to make \(16\) projective measurements. These \(16\) projective measurements are given by \(\Pi_j\otimes \Pi_k\), where \(j,k=0,1,2,3\), and \(\Pi_j\) are the following single qubit projectors: \(\Pi_0=|0 \rangle\langle 0|,\Pi_1=|1 \rangle\langle 1|,\Pi_2=|D \rangle\langle D|, \Pi_3=|R \rangle \langle R|\). Here \(|D \rangle=(|0\rangle+|1\rangle )/\sqrt{2}\) and \(|R \rangle=(|0\rangle-i|1\rangle)/\sqrt{2}\). This set of projectors is not unique and the ones just reported are usually employed in optical experiments [10]. Conceptually, the standard QST protocol probes the unknown two-qubit state from different local bases. The collected probabilities provide enough information to determine all independent parameters of the two-qubit density matrix. Once these probabilities are known, the density matrix is reconstructed through a linear inversion procedure.

In the teleportation-based QST protocol, the goal is the same. We need to obtain a complete set of probabilities from which the density matrix can be fully reconstructed. The difference is that the information is extracted through local Bell state measurements rather than through local projective measurements in different bases. Typically, BMs with ancillary states are combined with all the steps of the teleportation protocol. These ingredients convert information about the unknown state into BM outcomes. The resulting probabilities play the same role as the coincidence probabilities in the standard protocol, giving a tomographically complete data set from which the density matrix can be reconstructed.

The fundamental distinction between both approaches lies in the measurement primitive employed to access the information contained in the unknown state. In the standard QST protocol, tomography is achieved through a collection of local projective measurements performed in several complementary bases directly onto the two qubits. In the teleportation-based QST protocol, the same information is extracted through Bell state projections using the qubits of the unknown state to be reconstructed and the four ancillary states previously described. The conceptual novelty of the teleportation-based QST protocol is therefore not in the mathematical reconstruction itself, but in demonstrating that the entire tomographic information can be acquired using BMs as the sole measurement resource, with the information associated with the unknown state teleported to a single qubit, the one with Bob. See also Sec. 3 for further details showing why only BMs are enough to fully reconstruct a two-qubit state.

In other words, the reconstruction stage follows the same philosophy as the standard QST protocol: a tomographically complete set of probabilities is first obtained and then linearly inverted to recover the density matrix. The difference lies in how these probabilities are acquired and ultimately where the parameters characterizing the unknown state are located. Whereas the standard QST protocol relies on local projective measurements in multiple bases, with the information associated with the unknown state spread into the unknown two-qubit state, the teleportation-based QST protocol extracts the same tomographic information exclusively from Bell state measurement outcomes, with the information related to the unknown two-qubit state being transferred to Bob’s qubit [see Eq. (6 ) and Eqs. (9 )-(11 )].

3 One-qubit states↩︎

We can adapt the present protocol to reconstruct a single qubit state. In other words, the ability to implement Bell measurements (BMs) and to prepare those four states, namely, \(|0\rangle,|1\rangle,(|0\rangle+|1\rangle)/\sqrt{2}\), and \((|0\rangle+i|1\rangle)/\sqrt{2}\), suffices to reconstruct a single qubit state. Moreover, by using the two-qubit QST protocol discussed above or the general \(n\)-qubit protocol presented later, we see that an arbitrary \(n\)-qubit state can be fully reconstructed using BMs alone, provided that the four states introduced above can be prepared and that the outcomes of all BMs performed by Alice are communicated classically to Bob. Indeed, once the teleportation-based QST protocol is executed, the only remaining task on Bob’s side is the determination of single-qubit states. Crucially, this step does not require any measurement beyond BMs, since each single-qubit state can itself be reconstructed using the Bell-state-measurement-only single qubit reconstruction protocol described next. Therefore, every stage of the reconstruction procedure can be carried out exclusively through BMs, with no need for any other type of measurement.

Let an arbitrary single qubit be given by \[\varrho_{1} = \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{12}^* & a_{22} \end{array} \right). \label{stepF}\tag{14}\] The analogous to Eq. (1 ) is now \(\rho=\rho_A\otimes \varrho_1\). If we make a Bell measurement on the two-qubit state \(\rho\), the probability to obtain the Bell state \(|\Psi^-\rangle\) is \[Q_{\Psi^{\small{-}}}(\alpha,\beta) = [a_{22}|\alpha|^2 + a_{11}|\beta|^2]/2 -Re[a_{12}\alpha^*\beta]. \label{qsingle}\tag{15}\]

Looking at Eq. (15 ), we realize that if \(\rho_A\) is the density matrix representing the state \(|0\rangle\), we obtain \(a_{22}\) as a function of the probability to measure the Bell state \(|\Psi^-\rangle\). Similarly, using the state \(|1\rangle\) gives \(a_{11}\) and using both \((|0\rangle + |1\rangle)/\sqrt{2}\) and \((|0\rangle + i|1\rangle)/\sqrt{2}\) gives \(a_{12}\). Writing the solution explicitly we have \[\begin{align} a_{11} &=& 2 Q_{\Psi^{\small{-}}}(0,1), \, a_{22} = 2 Q_{\Psi^{\small{-}}}(1,0), \\ a_{12} &=& (1-i)[Q_{\Psi^{\small{-}}}(1,0)+Q_{\Psi^{\small{-}}}(0,1)] \nonumber \\ &+& 2iQ_{\Psi^{\small{-}}}(1/\sqrt{2},i/\sqrt{2}) -2Q_{\Psi^{\small{-}}}(1/\sqrt{2},1/\sqrt{2}). \label{single} \end{align}\tag{16}\] Before we move on to the three-qubit state protocol, we should mention that the choice for \(|\Psi^-\rangle\) as the BM result above is just for convenience and definiteness. Similar results follow for the three other possible outcomes of a BM. There is nothing particularly special about the Bell state \(|\Psi^-\rangle\) when it comes to the quantum state reconstruction protocols being studied here.

4 Three-qubit states↩︎

The three-qubit teleportation-based QST is very similar to the two-qubit case. Most of the equations are formally the same. The main changes are given by Eq. (1 ), which now becomes \[\begin{align} \rho &=& \rho_{A_{1}} \otimes \rho_{123} \otimes \rho_{A_{3}} \nonumber \\ &=& \left( \begin{array}{cc} |\alpha|^2 & \alpha\beta^* \\ \alpha^*\beta & |\beta|^2 \end{array} \right) \otimes \rho_{123} \otimes \left( \begin{array}{cc} |\gamma|^2 & \gamma\delta^* \\ \gamma^*\delta & |\delta|^2 \end{array} \right). \label{stepA3} \end{align}\tag{17}\] In other words, Alice now teleports a pair of input states, namely, \(|\psi\rangle_{A_1}=\alpha|0\rangle + \beta|1\rangle\) and \(|\psi\rangle_{A_3}=\gamma|0\rangle + \delta|1\rangle\), where \(|\alpha|^2+|\beta|^2=1\) and \(|\gamma|^2+|\delta|^2=1\) (see Fig. 2).

After Alice implements a BM on qubits \(1\) and \(A_1\), another one on qubits \(3\) and \(A_3\), and tells Bob her results, Bob qubit is given by an expression similar to Eq. (6 ), where we now need two indexes to properly label the \(16\) possible outcomes of the two BMs implemented by Alice: \[\varrho_{2} = \frac{Tr_{A_1,1,A_3,3}[P_{i,j} \rho P_{i,j}]}{Q_{i,j}} =\left( \begin{array}{cc} b_{11} & b_{12} \\ b_{12}^* & b_{22} \end{array} \right). \label{stepE3}\tag{18}\] In Eq. (18 ), \(Q_{i,j}\) is Alice’s probability to measure Bell states \(i\) and \(j\) and \(P_{i,j}\) is the projector describing the two independent BMs. Being more explicit, \[\begin{align} Q_{i,j} &=& Tr[{P_{i,j} \rho}], \label{prob3} \\ P_{i,j} &=& P_{i} \otimes \mathbb{1} \otimes P_{j}, \end{align}\tag{19}\] where \(P_{i}\) and \(P_{j}\) are given by Eq. (3 ).

Similarly to the two-qubit state protocol, we assume that Bob can experimentally determine the matrix elements \(b_{ij}\) of his single qubit state, which are now functions of the matrix elements of \(\rho_{123}\) and of the pair of input states teleported by Alice. As we show in Appendix 8, using all \(16\) pairs of inputs that can be formed out of the states \(|0\rangle, |1\rangle, (|0\rangle+|1\rangle)/\sqrt{2},\) and \((|0\rangle+i|1\rangle)/\sqrt{2}\), we can express the \(64\) real parameters that define \(\rho_{123}\) in terms of the matrix elements \(b_{ij}(\alpha,\beta,\gamma,\delta)\) of Bob’s states.

5 \(n\)-qubit states↩︎

The remarkable feature of the three-qubit protocol is that it works perfectly by Alice teleporting only separable pairs of qubits. There is no need to teleport entangled states (a Bell state, for instance). Also, there is no need to implement more sophisticated joint measurements involving all four qubits that are with Alice (\(1, A_1, 3, A_3\)). For the three-qubit protocol, the \(16\) possible choices of pairs of qubits to be teleported from Alice to Bob lead to \(16\) different density matrices at Bob’s, one different density matrix at the end of each one of the \(16\) different teleportation protocols that can be implemented using a particular pair as input. We are considering, for definiteness, only the cases where Alice obtains \(|\Psi^-\rangle_{A_1,1}|\Psi^-\rangle_{A_3,3}\) as the result of her two independent BMs. Each density matrix has four independent real parameters, leading to a total of \(4\times 16\) real parameters with Bob. These \(64\) real parameters are connected to \(64\) linear equations that can be solved to give the \(64\) real parameters defining \(\rho_{123}\).

If we now have an \(n\)-qubit state to be reconstructed, Alice needs to teleport \(n-1\) single qubits, using as input all \(n-1\) arrangements of states made out of the following four qubits, namely, \(|0\rangle, |1\rangle, (|0\rangle+|1\rangle)/\sqrt{2},\) and \((|0\rangle+i|1\rangle)/\sqrt{2}\) (see Fig. 3).

Indeed, an \(n\)-qubit density matrix has \(2^n\times 2^n\) elements. The \(n\) diagonal elements are real and the remaining ones are complex, with only half of the latter independent since the density matrix is Hermitian. There are, thus, a total of \(4^n\) real parameters that we must know to fully reconstruct it. On the other hand, with four different single qubit states, we have a total of \(4^{n-1}\) different arrangements comprising \(n-1\) qubits. Each one of these arrangements will lead to one different density matrix with Bob at the end of the teleportation protocol (after the \(n-1\) BMs that Alice implements as described in Fig. 3). As always, we are assuming, for definiteness, a particular sequence of BM results obtained by Alice (for instance, \(|\Psi^-\rangle_{A_1,1}\otimes\cdots \otimes|\Psi^-\rangle_{A_j,j}\otimes\cdots\otimes|\Psi^-\rangle_{A_{n-1},n-1}\)), but the present discussion applies to any other sequence of BMs as well. Since a single qubit density matrix provides four real parameters, Bob will end up with \(4\times 4^{n-1} = 4^n\) real parameters. This number matches exactly the number of independent real parameters characterizing an \(n\)-qubit density matrices, allowing Bob to solve a system of \(4^n\) linear equations that eventually gives all the matrix elements of the \(n\)-qubit state as functions of the matrix elements of all Bob’s single qubit density matrices.

6 Conclusion↩︎

We showed that the quantum teleportation protocol can be used to fully reconstruct an arbitrary \(n\)-qubit state. The teleportation-based quantum state tomography (QST) protocol here developed works by using the basic “primitives” of the teleportation protocol, namely, Bell state measurements and classical communication. If the quantum states to be teleported from Alice to Bob are known, this knowledge can be used to reconstruct the quantum state shared between them and through which the teleportation occurs. We also showed that the minimum number of known states that Alice must teleport is four and that they all lie on the equator and the poles of the Bloch sphere [28].

For the protocol to work, Alice should be able to prepare several copies of those four states and she should be able to realize independent Bell measurements (BMs). In other words, she needs to project those qubits and the ones coming from the state shared with Bob onto Bell states. Bob should be able to determine his single qubit at the end of the teleportation protocol. This can be accomplished by any well-established QST protocol [7]–[23] or by simply using the Bell-state-measurement-only single qubit reconstruction protocol also discussed in this work. Each different teleported state by Alice will lead to a different single qubit density matrix for Bob. These several density matrices with Bob contain all the data needed to reconstruct the shared state between them. They provide us with a complete set of linear equations where the density matrix elements of the shared state are the unknown variables. By solving this set of equations, we obtain the shared state density matrix elements in terms of the density matrix elements of Bob’s single qubits.

The teleportation-based QST protocol here developed works even if the shared state is distributed among different parties located in different places (see Figs. 2 and 3). If all parties cooperate, i.e., realize their BMs and communicate classically their results to Bob, he will be able to fully reconstruct the shared state by operating locally on his qubit as prescribed by the teleportation-based QST protocol. Moreover, if we have some prior knowledge of the state shared among all parties, the number of states needed to be teleported by Alice and the resources needed to execute the protocol are reduced. An interesting extension of this work would be the investigation of the minimal resources needed to reconstruct matrix product states [29], [30] or nearly pure states, which require much less resources to be reconstructed using traditional QST techniques [16], [17], [20]–[22]. Also, the investigation of the possible connection between quantum data hiding [31], [32] and local QST protocols may lead to a deeper understanding of the inherent limitations of tomography schemes based on local operations and classical communication.

Finally, we would like to mention that the present protocol naturally leads to the following two open problems. First, is it possible to extend the present analysis for higher spins? For instance, can we adapt the spin-1 teleportation protocol [33] to fully reconstruct density matrices describing single and multipartite spin-1 systems? Second, can the continuous-variable (CV) teleportation protocol [2], [3], [6] be modified to reconstruct CV-states, such as a two-mode or \(n\)-mode Gaussian state, where only local operations and classical communication are allowed in the reconstruction protocol [34], [35]?

Note: The extension of the present protocol to spin-1 systems is reported in Ref. [36].

7 The two-qubit case↩︎

If Alice obtains the Bell state \(j\) after her Bell measurement, the unnormalized state \(\tilde{\varrho}_{2,j}\) with Bob is [see Eq. (6 )] \[\tilde{\varrho}_{2,j} = Tr_{A,1}[P_j \rho P_j] =\left( \begin{array}{cc} \tilde{b}_{11,j} & \tilde{b}_{12,j} \\ \tilde{b}_{12,j}^* & \tilde{b}_{22,j} \end{array} \right). \label{ustepE}\tag{20}\]

Using Eqs. (1 ), (3 ), (7 ), and (20 ), we get that Alice’s probability to measure the Bell state \(j\) is \[\begin{align} Q_{\Psi^{\small{-}}}(\alpha,\beta) &=&[(m_{33}+m_{44})|\alpha|^2 +(m_{11}+m_{22})|\beta|^2]/2 \nonumber \\ &&-Re[(m_{13}+m_{24})\alpha^*\beta], \\ Q_{\Psi^{\tiny{+}}}(\alpha,\beta) &=& Q_{\Psi^{\small{-}}}(-\alpha,\beta), \\ Q_{\Phi^{\small{-}}}(\alpha,\beta) &=& Q_{\Psi^{\small{-}}}(\beta,\alpha) \\ Q_{\Phi^{\tiny{+}}}(\alpha,\beta) &=& Q_{\Psi^{\small{-}}}(-\beta,\alpha), \label{AliceProb1} \end{align}\tag{21}\] and the corresponding Bob’s unnormalized state is \[\begin{align} \tilde{\varrho}_{2,\Psi^{\small{-}}}(\alpha,\beta) &=& \left( \begin{array}{cc} \tilde{b}_{11,\Psi^{\small{-}}}(\alpha,\beta) & \tilde{b}_{12,\Psi^{\small{-}}}(\alpha,\beta) \\ \tilde{b}_{12,\Psi^{\small{-}}}^*(\alpha,\beta) & \tilde{b}_{22,\Psi^{\small{-}}}(\alpha,\beta) \end{array} \right),\\ \tilde{\varrho}_{2,\Psi^{\tiny{+}}}(\alpha,\beta) &=& \tilde{\varrho}_{2,\Psi^{\small{-}}}(-\alpha,\beta), \\ \tilde{\varrho}_{2,\Phi^{\small{-}}}(\alpha,\beta) &=& \tilde{\varrho}_{2,\Psi^{\small{-}}}(\beta,\alpha), \\ \tilde{\varrho}_{2,\Phi^{\tiny{+}}}(\alpha,\beta) &=& \tilde{\varrho}_{2,\Psi^{\small{-}}}(-\beta,\alpha), \end{align}\] where \[\begin{align} 2\tilde{b}_{11,\Psi^{\small{-}}}(\alpha,\beta) & =& m_{33}|\alpha|^2 +m_{11}|\beta|^2 -2Re[m_{13}\alpha^*\beta], \tag{22} \\ 2\tilde{b}_{22,\Psi^{\small{-}}}(\alpha,\beta) & =& m_{44}|\alpha|^2 +m_{22}|\beta|^2 -2Re[m_{24}\alpha^*\beta], \tag{23} \\ 2\tilde{b}_{12,\Psi^{\small{-}}}(\alpha,\beta) & =& m_{34}|\alpha|^2 \!+\!m_{12}|\beta|^2 \!-\!m_{14}\alpha^*\beta \!-\! m_{23}^*\alpha\beta^*\!.\tag{24} \end{align}\]

8 The three-qubit case↩︎

The three-qubit state that we want to reconstruct can be written as follows: \[\rho_{123} = \left( \begin{array}{cccccccc} m_{11} & m_{12} & m_{13} & m_{14} & m_{15} & m_{16} & m_{17} & m_{18}\\ m_{12}^* & m_{22} & m_{23} & m_{24} & m_{25} & m_{26} & m_{27} & m_{28} \\ m_{13}^* & m_{23}^* & m_{33} & m_{34} & m_{35} & m_{36} & m_{37} & m_{38}\\ m_{14}^* & m_{24}^* & m_{34}^* & m_{44} & m_{45} & m_{46} & m_{47} & m_{48} \\ m_{15}^* & m_{25}^* & m_{35}^* & m_{45}^* & m_{55} & m_{56} & m_{57} & m_{58} \\ m_{16}^* & m_{26}^* & m_{36}^* & m_{46}^* & m_{56}^* & m_{66} & m_{67} & m_{68} \\ m_{17}^* & m_{27}^* & m_{37}^* & m_{47}^* & m_{57}^* & m_{67}^* & m_{77} & m_{78} \\ m_{18}^* & m_{28}^* & m_{38}^* & m_{48}^* & m_{58}^* & m_{68}^* & m_{78}^* & m_{88} \end{array} \right), \label{rho123}\tag{25}\] where \(m_{jj}\) are positive reals and \(\sum_{j}m_{jj}=1\). We have \(64\) real parameters to determine (\(63\) if we use the normalization condition): the real and imaginary parts of \(m_{ij}\), if \(i \neq j\), and the eight real quantities given by \(m_{jj}\).

For definiteness, we assume that Alice’s two BM results are always given by \(|\Psi^-\rangle_{A_1,1}|\Psi^-\rangle_{A_3,3}\). The same analysis applies for any other of the remaining \(15\) pairs of Bell states that she can measure. Similar to what we showed for the two-qubit case (Appendix 7), we can obtain anyone of those other cases by properly changing the signs and/or exchanging the values of \(\alpha,\beta,\gamma,\delta\) in the expressions for the case we will be dealing with from now on.

References↩︎