Fidelity deviation in quantum teleportation with a two-qubit state

TL;DR

This paper derives an exact formula for fidelity deviation in quantum teleportation with two-qubit states, providing conditions for universal teleportation and identifying optimal states with minimal fluctuations.

Contribution

It introduces a computable formula for fidelity deviation, establishing the universality condition and characterizing optimal two-qubit states for teleportation.

Findings

Exact formula for fidelity deviation derived

Universality condition for teleportation established

Existence of dispersion-free states for given fidelity

Abstract

Quantum teleportation with an arbitrary two-qubit state can be appropriately characterized in terms of maximal fidelity and fidelity deviation. The former quantifies optimality of the process and is defined as the maximal average fidelity achievable within the standard protocol and local unitary strategies, whereas the latter, defined as the standard deviation of fidelity over all input states, is a measure of fidelity fluctuations. The maximal fidelity for a two-qubit state is known and is given by a simple formula that can be exactly computed, but no such formula is known for the fidelity deviation. In this paper, we derive an exact computable formula for the fidelity deviation in optimal quantum teleportation with an arbitrary state of two qubits. From this formula, we obtain the dispersion-free condition, also known as the universality condition: the condition that all input states are teleported equally well and provide a necessary and sufficient condition for a state to be both useful (maximal fidelity larger than the classical bound) and universal (zero fidelity deviation). We also show that for any given maximal fidelity, larger than the classical bound, there always exist dispersion-free or universal states and argue that such states are the most desirable ones within the set of useful states. We illustrate these results with well-known families of two-qubit states: pure entangled states, Bell-diagonal states, and subsets of $X$ states.

Equations

$$\left|\psi ⟩⟨\psi \right|\otimes \rho \stackrel{\mathrm{LOCC}}{⟶}\tau \otimes \varsigma ,$$

$$⟨f_{\rho }⟩$$

$${\delta }_{\rho }$$

$${\delta }_{\rho }^2$$

$${\Delta }_{\rho }$$

$$\rho$$

$$\varrho$$

$$F_{\rho }$$

$${\mathcal{F}}_{\rho }$$

$${\mathcal{F}}_{\varrho }$$

$$F_{\varrho }$$

$${\Delta }_{\rho }$$

$$\begin{array}[]{ccc}\left|\Psi_{0}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|01\right\rangle-\left|10\right\rangle\right)&&\left|\Psi_{1}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|00\right\rangle-\left|11\right\rangle\right)\\ \left|\Psi_{2}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|00\right\rangle+\left|11\right\rangle\right)&&\left|\Psi_{3}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|01\right\rangle+\left|10\right\rangle\right)\end{array}$$

$${\varsigma }_k$$

$$p_k{\varsigma }_k$$

$${\sigma }_k\left(\mathbf{n}\cdot \mathbf{\sigma }\right){\sigma }_k$$

$$O_k^{†}$$

$$p_k{\varsigma }_k$$

$$f_{\psi ,\varrho }$$

$$f_{\psi ,\rho }$$

$$\sum _{k=0}^3{\mathbf{a}}^T{T}_k\mathbf{x}$$

$$f_{\psi ,\varrho }$$

$${\int }_G\mathrm{d}g{\mathcal{O}}_{g}X{\mathcal{O}}_g$$

$$⟨f_{\varrho }⟩$$

$$f_{\psi ,\varrho }^2$$

$$⟨f_{\varrho }^2⟩$$

$$\left({\mathbf{a}}^{T}T\mathbf{a}\right)\left({\mathbf{a}}^{T}T\mathbf{a}\right)$$

Full text

I Introduction

Quantum teleportation (Teleportation-1993, ) is a fundamental protocol to transmit quantum information using shared entanglement and Local Operations and Classical Communication (LOCC). The novelty of quantum teleportation lies in the fact that it allows us to transfer quantum states over quantum channels without physical transmission of quantum systems. Besides being successfully demonstrated experimentally (experiment-Boschi, ; Experiment-Ma, ; experiment-nolleke, ), quantum teleportation has also been studied in multiparty (Dur-multiparty-teleportation, ) and continuous variable systems (CV-Brausnstein, ).

The basic protocol of quantum teleportation is rather simple and can be understood as an instance of quantum state transfer

[TABLE]

where $\rho$ is a two-qubit entangled state initially shared between the sender (Alice) and the receiver (Bob), $\left|\psi\right\rangle$ is the input state unknown to Alice, $\tau$ is the final state of Alice’s qubits, and $\varsigma$ is the final state of Bob’s qubit–the output state of quantum teleportation. Note that, (1) describes not only the standard protocol (Teleportation-1993, ) but also any other protocol within the paradigm of LOCC (MPR-Horodecki-1999, ; VV-2003, ).

The standard figure of merit for quantum teleportation for a two-qubit state $\rho$ is given by the average fidelity (RMP-Horodecki-96, )

[TABLE]

where $f_{\psi,\rho}=\left\langle\psi\left|\varsigma\right|\psi\right\rangle$ is the fidelity between an input-output pair $\left(\left|\psi\rangle\langle\psi\right|,\varsigma\right)$, and the integral is over a uniform distribution ${\rm d}\psi$ (normalized Haar measure, $\int{\rm d}\psi=1$) of all input states $\psi=\left|\psi\right\rangle\left\langle\psi\right|$. The average fidelity, unless mentioned otherwise, is computed with respect to the standard protocol (Teleportation-1993, ). Perfect teleportation requires $\rho$ to be maximally entangled but this is possible only when Alice and Bob have access to a noiseless quantum channel. In practice, however, the available channels are noisy leading to mixed or noisy entangled states. Consequently, teleportation will not be perfect and the average fidelity will be less than one.

Recently (Bang-et-al-2018, ) it has been pointed out that the average fidelity alone is not sufficient to fully characterize quantum teleportation because it does not give us any information on the fluctuations in fidelity, if any, over the input states, although one expects fluctuations to be present in general either due to imperfections in experiments, or even as a property inherent to the resource states. This suggests that one must also consider, in addition to the average fidelity, a physically meaningful measure of fluctuations. Fidelity deviation is one such measure, which is defined as the standard deviation of fidelity over all input states (Bang-et-al-2012, ; Bang-et-al-2018, ):

[TABLE]

where $\left\langle f_{\rho}^{2}\right\rangle=\int f_{\psi,\rho}^{2}{\rm d}\psi$. Physically, fidelity deviation is a measure of spread of fidelity values around the average. Note that,

[TABLE]

Thus $0\leq\delta_{\rho}\leq\frac{1}{2}$, where $\delta_{\rho}=0$ iff $f_{\psi,\rho}=\left\langle f_{\rho}\right\rangle$ for all $\left|\psi\right\rangle$. Evidently, the ordered pair $\left(\left\langle f_{\rho}\right\rangle,\delta_{\rho}\right)$ is a more informative performance-measure of quantum teleportation.

Motivation and Results

Soon after the original work on quantum teleportation it was realized that the average fidelity obtained within the standard protocol is not always optimal (RMP-Horodecki-96, ). One can, in fact, maximize the average fidelity over local unitary (LU) strategies. The maximal fidelity $F_{\rho}$ is defined as the maximal average fidelity achievable within the standard protocol and LUs (RMP-Horodecki-96, ; Badziag-2000, ), and any protocol that achieves the maximal value is said to be optimal. In particular, an optimal protocol is where Alice and Bob first apply a local unitary operation to transform $\rho$ to the canonical form $\varrho$ and then use $\varrho$ for quantum teleportation following the standard protocol (Badziag-2000, ). The optimality of the protocol follows from the equality $F_{\rho}=F_{\varrho}=\left\langle f_{\varrho}\right\rangle$ which shows that the maximal fidelity for a given $\rho$ is the same as the average fidelity obtained for its canonical representative $\varrho$ within the standard protocol. The details of this equivalence and the canonical form will be discussed in the next section. We say that $\rho$ is useful for quantum teleportation if and only if $F_{\rho}>\frac{2}{3}$ (MPR-Horodecki-1999, ; Massar_popescu-1995, ), where $\frac{2}{3}$ is the maximum average fidelity obtained in classical protocols.

In this paper, our goal is to obtain the fidelity deviation in optimal quantum teleportation with an arbitrary two-qubit state $\rho$. Noting that the maximal fidelity $F_{\rho}$ is realized with the canonical $\varrho$, we accordingly define the fidelity deviation as

[TABLE]

Remark*.*

In general, $\Delta_{\rho}$ is not the same as the minimum of $\delta_{\rho}$, where the minimum is taken over all LU strategies. This is because any strategy that maximizes $\left\langle f_{\rho}\right\rangle$ may not minimize $\delta_{\rho}$ and vice versa. But we will see that there exist states where $\Delta_{\rho}=0$ and for such states the optimal protocol indeed minimizes the fidelity deviation. Note that, fidelity deviation is trivially zero for maximally entangled states as all input states are teleported with fidelity one, but as we will show, fidelity deviation is nonzero for generic two-qubit states and states with zero fidelity deviation are in fact special.

Clearly, the ordered pair $\left(F_{\rho},\Delta_{\rho}\right)$ contains all the necessary information we need to characterize quantum teleportation with a two-qubit state. But to make use of this performance-measure we need to know both $F_{\rho}$ and $\Delta_{\rho}$. For an arbitrary two-qubit state $\rho$, $F_{\rho}$ is given by a computable function of the eigenvalues of the $T$ matrix, which is a real $3\times 3$ matrix with elements $T_{ij}=\text{Tr}\left(\rho\sigma_{i}\otimes\sigma_{j}\right)$, $i,j=1,2,3$, where $\sigma_{i}$, $i=1,2,3$ are the standard Pauli matrices (RMP-Horodecki-96, ; Badziag-2000, ). However, no such formula is known for $\Delta_{\rho}$. In this paper, we derive an exact computable formula for $\Delta_{\rho}$. This formula is also function of the eigenvalues of the $T$ matrix. Therefore, knowing the eigenvalues of $T$ is sufficient to obtain both $F_{\rho}$ and $\Delta_{\rho}$.

The main significance of our formula lies in the fact that we can now adequately characterize quantum teleportation using the ordered pair $\left(F_{\rho},\Delta_{\rho}\right)$. But there are other useful implications as well. Because we want fidelity deviation to be as small as possible, $\Delta_{\rho}$ can serve as a filter to select the best possible resource states. Here we focus on one such application – the universality condition (Bang-et-al-2012, ; Bang-et-al-2018, ) given by $\Delta_{\rho}=0$. So when the universality condition is satisfied, all input states are teleported equally well. The states that satisfy the universal condition are said to be universal or dispersion-free states. From our formula, we obtain the condition under which the universality condition is satisfied; the analysis reveals that for a generic $\rho$, $\Delta_{\rho}\neq 0$ and that not all useful states are universal. We then show that a state $\rho$ is both useful and universal iff for every $i=1,2,3$, $\left|t_{ii}\right|=t>\frac{1}{3}$, where $t_{ii}$ are the eigenvalues of the $T$ matrix. So if a state is both useful and universal, then for all input states the fidelity stays above the classical bound, but the same cannot be said for all states that are useful but not universal. Hence, the states that are not only useful but also satisfy the universality condition are the most desirable ones.

We now study properties of some of the well-known classes of two-qubit states. The primary motivation here is to identify which states are both useful and universal and which are not. The findings are summarized below:

• •

For a pure entangled state, the fidelity deviation is nonzero unless the state is maximally entangled. This shows that a nonmaximally entangled pure state, although useful always (Gisin-1996, ), is not universal.

• •

Bell-diagonal states (rank $\geq 2$): The motivation to study the Bell-diagonal states comes from a recent result (Bang-et-al-2018, ) which showed that Werner states – a subset of the Bell-diagonal states, exhibit zero fidelity deviation. This prompted us to investigate whether this property extends to all Bell-diagonal states. But we find that no Bell-diagonal state other than a Werner has zero fidelity deviation. Thus entangled Werner states are the only states in the Bell-diagonal family that are both useful and universal.

• •

We present examples of non-Werner states that are shown to be both useful and universal. These states belong to the family of $X$-states (X-1, ; X-2, ).

Finally, we would like to remark on a particular property exhibited by pure entangled states and rank-two entangled Bell-diagonal states. Both share the common property that fidelity deviation increases with decrease in the maximal fidelity (equivalently, entanglement for these states). Now, as entanglement goes to zero we find that the maximal fidelity approaches the classical bound from above, but the corresponding fidelity deviation approaches a nonzero constant value from below. Since the typical region is given by $F_{\rho}\pm\Delta_{\rho}$, it appears that near the quantum-classical boundary, not all input states will be teleported with fidelity greater than the classical bound. In fact, this is one reason why the states with zero fidelity deviation should be preferred over states with nonzero fidelity deviation, especially near the quantum-classical boundary.

The rest of the paper is arranged as follows. Section II reviews the background material, where we discuss the Hilbert-Schmidt decomposition and canonical description of two-qubit density matrices. Here we also summarize the known results on maximal teleportation fidelity. In Section III, we derive the formula for the fidelity deviation for a canonical two-qubit state. In Section IV, we discuss the universality condition and present the relevant case studies. Finally, we conclude in Section V.

II Preliminaries

In this section, we review the Hilbert-Schmidt representation and the canonical form of two-qubit density matrices. We also briefly discuss maximal teleportation fidelity, present the relevant formulas and the conditions under which they hold. For more details the readers are referred to (MPR-Horodecki-1999, ; RMP-Horodecki-96, ; RM-Horodecki-1996, ; Badziag-2000, ; RP_Horodecki-1996, ).

Hilbert-Schmidt representation and the canonical form

In the Hilbert-Schmidt representation (RMP-Horodecki-96, ; RP_Horodecki-1996, ; RM-Horodecki-1996, ), a two-qubit density matrix $\rho$ can be written as

[TABLE]

where $\bm{R}$ and $\bm{S}$ are vectors in $\mathbb{R}^{3}$, \bm{R}\left(\bm{S}\right)\cdot\bm{\sigma}$$=\sum_{k=1}^{3}R_{i}(S_{i})\sigma_{i}, and the coefficients $T_{ij}={\rm Tr}\left(\rho\sigma_{i}\otimes\sigma_{j}\right)$, $i,j=1,2,3$ form a real $3\times 3$ matrix $T$ (the correlation matrix).

Let $t_{11},t_{22},t_{33}$ be the eigenvalues of the $T$ matrix. Now one can show that there always exists a product unitary transformation $U_{1}\otimes U_{2}$ which will transform $\rho\rightarrow\varrho$

[TABLE]

where $\lambda_{i}\in\left\{-1,+1\right\}$ are determined by the sign of $\det T$; in particular, (a) if $\det T\leq 0$ , then $\lambda_{i}=-1$ for $\left|t_{ii}\right|\neq 0$, $i=1,2,3$ ; (b) if $\det T>0$, then $\lambda_{i},\lambda_{j}=-1$, $\lambda_{k}=+1$ for any choice of $i\neq j\neq k\in\left\{1,2,3\right\}$ satisfying $\left|t_{ii}\right|\geq\left|t_{jj}\right|\geq\left|t_{kk}\right|$. The transformed state $\varrho$ given by (6) is defined as the canonical form of $\rho$ (Badziag-2000, ). Note that this definition, though similar in some ways, differs from that given in (Badziag-2000, ).

Maximal teleportation fidelity

For any two-qubit state $\rho$ the maximal fidelity $F_{\rho}$ achievable within the standard protocol and LUs is given by (MPR-Horodecki-1999, )

[TABLE]

where $\mathcal{F}_{\rho}$ is the fully entangled fraction, defined as (bennett-1995, )

[TABLE]

Here, the maximum is taken over all maximally entangled states $\left|\Psi\right\rangle=\left(U\otimes V\right)\left|\Psi_{0}\right\rangle$, where $\left|\Psi_{0}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|01\right\rangle-\left|10\right\rangle\right)$ is the singlet state and $U$, $V$ are unitary operators.

From the definition of $\mathcal{F}_{\rho}$ it follows that for any $\rho^{\prime}=\left(U_{1}\otimes U_{2}\right)\rho\left(U_{1}^{\dagger}\otimes U_{2}^{\dagger}\right)$, $\mathcal{F}_{\rho}=\mathcal{F}_{\rho^{\prime}}$. Consequently, $F_{\rho}=F_{\rho^{\prime}}$. Thus to achieve the maximal fidelity, the strategy is to find an appropriate $\rho^{\prime}$ such that $\mathcal{F}_{\rho^{\prime}}$ is attained on the singlet state $\left|\Psi_{0}\right\rangle$. In (Badziag-2000, ) it was pointed out that a canonical $\varrho$ satisfies this prescription; that is, $\mathcal{F}_{\varrho}=\left\langle\Psi_{0}\left|\varrho\right|\Psi_{0}\right\rangle$, which shows that the maximal fidelity $F_{\varrho}$ for a canonical $\varrho$ is achieved in the standard protocol (Teleportation-1993, ). In other words, the maximal fidelity $F_{\varrho}$ is equal to the average fidelity $\left\langle f_{\varrho}\right\rangle$. Now using the fact that $F_{\rho}=F_{\varrho}$ (as $\rho$ and $\varrho$ are connected by a local unitary transformation), the best strategy for any given $\rho$ is to first transform $\rho\rightarrow\varrho$ and then use $\varrho$ for quantum teleportation (Badziag-2000, ).

For a canonical $\varrho$ the fully entangled fraction is given by (Badziag-2000, )

[TABLE]

and using the relation (7) one obtains the maximal fidelity

[TABLE]

Proposition 1**.**

For a two-qubit state $\rho$, it holds that $F_{\rho}=F_{\varrho}=\left\langle f_{\varrho}\right\rangle$; $F_{\varrho}$ is given by (14), where $t_{ii}$, $i=1,2,3$ are the eigenvalues of the $T$ matrix associated with $\rho$.

The conditions under which a two-qubit state is useful and when it is not are given by (RMP-Horodecki-96, ; Badziag-2000, ): $F_{\rho}>\frac{2}{3}$ iff $\stackrel{{\scriptstyle[}}{{i}}=1]{3}{\sum}\left|t_{ii}\right|>1$; if $\det T\geq 0$, then $F_{\rho}\leq\frac{2}{3}$. The former implies that all useful states have the property $\det T<0$; the converse, however, is not true in general.

Remark*.*

It may so happen that $\rho$ is entangled but $F_{\rho}\leq\frac{2}{3}$. Such states cannot be directly used for quantum teleportation as they offer no quantum advantage. However, any such state can be transformed using appropriate trace-preserving LOCC to another state which is useful (VV-2003, ; Badziag-2000, ; SB-2002, ).

III Fidelity deviation for a two-qubit state

In this section, we obtain the fidelity deviation $\Delta_{\rho}$ in optimal quantum teleportation with a two-qubit state $\rho$. Recall that

[TABLE]

We now consider the terms appearing in the above formula individually.

Expression for the average fidelity $\left\langle f_{\varrho}\right\rangle$

To obtain an expression for $\left\langle f_{\varrho}\right\rangle$, we essentially follow the steps given in (RMP-Horodecki-96, ) with appropriate modifications and explanations where necessary.

We assume that Alice and Bob share a canonical two-qubit state $\varrho$. Alice holds the input qubit prepared in some state $\left|\psi\right\rangle$, which is unknown to her. First, Alice performs a measurement in the Bell basis:

[TABLE]

on the two qubits she holds and sends two classical bits to Bob informing him about the outcome. Subsequently, Bob applies an appropriate Pauli rotation $\sigma_{k}$, $k\in\left\{0,\dots,3\right\}$ to obtain the output state

[TABLE]

where $p_{k}={\rm Tr}\left[\left(\left|\Psi_{k}\rangle\langle\Psi_{k}\right|\otimes I\right)\left(\left|\psi\rangle\langle\psi\right|\otimes\varrho\right)\right]$ is the probability of the $k^{{\rm th}}$, $k\in\left\{0,\dots,3\right\}$ outcome of Alice’s measurement and the partial trace is taken over Alice’s qubits. Now using the Bloch sphere representation $\left|\psi\rangle\langle\psi\right|=\frac{1}{2}\left(I+\bm{a\cdot\sigma}\right)$, where $\bm{a}$ is a unit vector in $\mathbb{R}^{3}$ and the Hilbert-Schmidt representation of $\varrho$, Eq. (17) can be written as

[TABLE]

where $T_{k}$s correspond to the projectors $\left|\Psi_{k}\rangle\langle\Psi_{k}\right|$ and are given by: $T_{0}={\rm diag}\left(-1,-1,-1\right)$, $T_{1}={\rm diag}\left(-1,1,1\right)$, $T_{2}={\rm diag}\left(1,-1,1\right)$, $T_{3}={\rm diag}\left(1,1,-1\right)$; $\bm{r}$, $\bm{s}$, $T$ correspond to $\varrho$; $O_{k}$s are rotations in $\mathbb{R}^{3}$ obtained via

[TABLE]

From (19) it then follows that

[TABLE]

Then (18) can be written as

[TABLE]

where we have used the fact that $T_{k}$ commutes with $T^{\dagger}$.

Now fidelity is given by

[TABLE]

Using $\left|\psi\rangle\langle\psi\right|=\frac{1}{2}\left(I+\bm{a\cdot\sigma}\right)$ and (21), the above equation can be written as

[TABLE]

For convenience, define the vector $\bm{x}=\bm{r}-\bm{s}$. Then

[TABLE]

The second term in (24) vanishes because

[TABLE]

which follows from the property $\sum_{k=0}^{3}T_{k}=0$. So we have

[TABLE]

To integrate (25) over all input states we need Schur’s orthogonality lemma on $\mathbb{R}^{d}$ (RMP-Horodecki-96, ; Bang-et-al-2012, ): For a given group $G$, let $\mathcal{O}_{g}$ be an irreducible orthogonal representation of any element $g\in G$. Then for every matrix $X$ on $\mathbb{R}^{d}$,

[TABLE]

where ${\rm d}g$ is the normalized Haar measure $\left(\int_{G}{\rm d}g=1\right)$ and $I_{d}$ is the identity matrix. In our case, $G$ is the rotation group $O\left(3\right)$ and the vectors $\bm{a}\in\mathbb{R}^{3}$. By choosing $\bm{z}$ as the reference unit vector we can write $\bm{a}=R_{a}\bm{z}$, where $R_{a}$ is the rotation matrix. Then the average over the Bloch surface is equal to the average over the rotation group, i.e.

[TABLE]

Using (26) one obtains $\int d\bm{a}\,\left(\bm{a}^{T}T\bm{a}\right)=\frac{1}{3}{\rm Tr}T$. Thus

[TABLE]

One can now easily verify that the formulas (14) are indeed obtained for the conditions $\det T\leq 0$ and $\det T>0$.

Expression for $\left\langle f_{\varrho}^{2}\right\rangle$

Our starting point is the expression for the fidelity given by (25). Squaring both sides

[TABLE]

Now the average of $f_{\psi,\varrho}^{2}$ over all possible input states is the same as the average over all Bloch vectors $\bm{a}$ on the Bloch surface. Therefore,

[TABLE]

where ${\rm d}\bm{a}$ is the normalized Haar measure over the Bloch surface. Expanding

[TABLE]

Now using the identity (see, for e.g (Bang-et-al-2012, ))

[TABLE]

we can write (30) as

[TABLE]

To evaluate the above integral we need Schur’s lemma on $\mathbb{R}^{d}$ (26) and also the generalization on $\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ (Bang-et-al-2012, ): For every matrix $X$ on $\mathbb{R}^{d}\otimes\mathbb{R}^{d}$

[TABLE]

where $P$ is a swap matrix defined by $P\left(\bm{x}_{i}\otimes\bm{x}_{j}\right)=\left(\bm{x}_{j}\otimes\bm{x}_{i}\right)$ and written as $P=\sum_{i,j=0}^{d-1}\left(\bm{x}_{j}\otimes\bm{x}_{i}\right)\left(\bm{x}_{i}\otimes\bm{x}_{j}\right)^{T}$, $D=\left(\sum_{i=0}^{d-1}\bm{x}_{i}\otimes\bm{x}_{i}\right)\left(\sum_{j=0}^{d-1}\bm{x}_{j}\otimes\bm{x}_{j}\right)$, where $\left\{\bm{x}_{i}\right\}$ is an orthonormal basis of $\mathbb{R}^{d}$, and the coefficients $\mathtt{A},\mathtt{B},\mathtt{C}$ are given by

[TABLE]

Since in our case $G$ is the rotation group $O\left(3\right)$ and the vectors $\bm{a}\in\mathbb{R}^{3}$, we can write $\bm{a}=R_{a}\bm{z}$ for some rotation matrix $R_{a}$, where $\bm{z}$ is our reference unit vector, Then the average over the Bloch surface is equal to the average over the rotation group. To evaluate (31) we consider the integrals separately. First,

[TABLE]

where we have used (26). Next,

[TABLE]

where we have used (32). Now using (33) and (34) in (31) we arrive at

[TABLE]

Fidelity deviation $\delta_{\varrho}$

From (27) we get

[TABLE]

Substituting (35) and (36) in the formula (15), we finally obtain

[TABLE]

which is our desired formula.

Remark*.*

Note that, in the entire derivation the only property of $\varrho$ that we used is that $T$ is diagonal. So the derivation, in fact, holds for any density matrix with diagonal $T$.

Let us now obtain the explicit expressions for the cases $\det T\leq 0$ and $\det T>0$. Recall that for a canonical $\varrho$, $T$ is diagonal with eigenvalues $\lambda_{i}\left|t_{ii}\right|$, $i=1,2,3$, where $\lambda_{i}\in\left\{-1,+1\right\}$. Then Eq. (37) becomes

[TABLE]

Now, if $\det T\leq 0$, $\lambda_{i}=-1$, $i=1,2,3$, and if $\det T>0$, $\lambda_{i}=-1,\lambda_{j}=-1,\lambda_{k}=+1$ for any choice of $i\neq j\neq k\in\left\{1,2,3\right\}$ satisfying $\left|t_{ii}\right|\geq\left|t_{jj}\right|\geq\left|t_{kk}\right|$. Using these properties it is straightforward to arrive at the following exact expressions:

[TABLE]

where the minimum taken in the second expression (for $\det T>0$) is, in fact, obtained for the same set of $i,j,k$, $i\neq j\neq k$ that maximizes the average fidelity given by (14) for $\det T>0$. This completes the derivation.

Summary

The formulas (41) and (14) completely characterize quantum teleportation with an arbitrary two-qubit state $\rho$ within the standard protocol and local unitary operations. The table below summarizes the expressions for both $F_{\rho}$ and $\Delta_{\rho}$.

[TABLE]

Remark*.*

From the above table we see that for any given $\rho$ we only need to find the eigenvalues of the $T$ matrix to compute $F_{\rho}$ and $\Delta_{\rho}$. However, as emphasized earlier, the optimal values are physically realized with the canonical form $\varrho$.

IV The universality condition and Examples

The universality condition, or equivalently, the condition for zero fidelity deviation, is given by $\Delta_{\rho}=0$. The states that satisfy this condition are said to be dispersion-free or universal. But if a state is not useful for quantum teleportation, then whether it satisfies the universality condition or not, is irrelevant. So we focus only on states that are useful for quantum teleportation. The useful states form a subset of the states with the property $\det T<0$.

By definition, a state which is useful for quantum teleportation satisfies

[TABLE]

Now if a useful state is universal, then it also must satisfy

[TABLE]

The above condition is fulfilled iff $\left|t_{ii}\right|$, $i=1,2,3$ are all equal. Now we see that the condition $\stackrel{{\scriptstyle[}}{{i}}=1]{3}{\sum}\left|t_{ii}\right|>1$ can hold even if $\left|t_{ii}\right|$ are not all equal. Hence, a state which is useful for quantum teleportation may not be universal.

Proposition 2**.**

Let $\rho$ be a two-qubit state. Then $\rho$ is useful and universal for quantum teleportation iff for every $i=1,2,3$, $\left|t_{ii}\right|=t>\frac{1}{3}$.

The proof follows by noting that $t_{ii}$ are all nonzero (since $\det T<0$) and that the equations (42) and (43) must be satisfied simultaneously.

Proposition 3**.**

Let $S$ be the set of all two-qubit density matrices with maximal fidelity $F$, where $\frac{2}{3}<F\leq 1$. Then there always exist states in $S$ with the property $\Delta_{\rho}=0$.

The states with maximal fidelity $F$ must satisfy the equation

[TABLE]

Since $\frac{2}{3}<F\leq 1$, the states in $S$ must satisfy $\det T<0$ which implies $t_{ii}$, $i=1,2,3$ must be all nonzero. On the other hand, $\Delta_{\rho}=0$ implies $\left|t_{ii}\right|$ must all be equal. Then for a given $F>\frac{2}{3}$ the states with $\Delta_{\rho}=0$ are those with $\left|t_{ii}\right|=2F-1$, $i=1,2,3$. Such states clearly form a subset of $S$.

It is rather obvious that for a given fidelity, the most desirable states are those that are universal and the above proposition shows such states always exist for any given $F>\frac{2}{3}$.

We now study the properties of some well-known classes of two-qubit states.

Fidelity deviation for pure entangled states

We know that a two-qubit pure entangled state $\left|\phi\right\rangle$ can be written in the Schmidt form as

[TABLE]

where $a,b\in\mathbb{R}$, $a\geq b>0$, $a^{2}+b^{2}=1$, and $\left\{\left|0^{\prime}0^{\prime}\right\rangle,\left|1^{\prime}1^{\prime}\right\rangle\right\}$ represent the Schmidt basis. The eigenvalues of the $T$ matrix are given by $t_{11}=2ab$, $t_{22}=-2ab$, $t_{33}=1$. Then the maximal fidelity is given by

[TABLE]

and the fidelity deviation is given by

[TABLE]

We see that $\Delta_{\phi}\neq 0$ as long as $F_{\phi}\neq 1$, i.e. as long as $\left|\phi\right\rangle$ is not maximally entangled. So within the class of pure entangled states only maximally entangled states are both useful and universal for quantum teleportation.

Now, concurrence (Concurrence-1998, ) of $\left|\phi\right\rangle$ is given by $C\left(\phi\right)=2ab$. This allows us to write (45) as

[TABLE]

From (47) we see that $\Delta_{\phi}$ decreases with increase in entanglement. Moreover, from either of the above two expressions we find that

[TABLE]

The above inequality together with (45) show that as $F_{\phi}\rightarrow\frac{2}{3}$ [i.e. $C\left(\phi\right)\rightarrow 0$] from above, $\Delta_{\phi}\rightarrow\frac{1}{3\sqrt{5}}$ from below. Therefore, if $F_{\phi}$ is sufficiently close to the classical bound, then for some input states the fidelity values might be in the classical domain.

Fidelity deviation for Bell-diagonal states

The Bell-basis is defined in (16). A Bell-diagonal state $\rho_{{\rm BD}}$ can be written as

[TABLE]

where $0\leq p_{3}\leq p_{2}\leq p_{1}<1$ and $\stackrel{{\scriptstyle[}}{{i}}=0]{3}{\sum}p_{i}=1$. The Hilbert-Schmidt representation of $\rho_{{\rm BD}}$ is given by

[TABLE]

where

[TABLE]

Now $\rho_{{\rm BD}}$ is entangled iff $p_{0}>p_{1}+p_{2}+p_{3}$, or equivalently, $p_{0}>\frac{1}{2}$. Then from (50) it follows that $t_{11},t_{22},t_{33}<0$. Hence, $\det T<0$.

Using the formulas for $\Delta_{\rho}$ and $F_{\rho}$ for the states with $\det T<0$, one obtains

[TABLE]

and

[TABLE]

since $p_{0}>\frac{1}{2}$.

For rank four states the fidelity deviation is given by (51). Thus the states with zero fidelity deviation are those for which $p_{1}=p_{2}=p_{3}=\frac{1-p_{0}}{3}$. But these states are nothing but Werner states. This is consistent with the result obtained in (Bang-et-al-2018, ).

Now fidelity deviation for rank three states $\left(p_{3}=0\right)$ is given by $\Delta_{{\rm BD}}=\frac{2}{3\sqrt{10}}\sqrt{\left(p_{1}-p_{2}\right)^{2}+p_{2}^{2}+p_{1}^{2}}\neq 0$ and that for rank two states $\left(p_{2}=p_{3}=0\right)$ is given by $\Delta_{{\rm BD}}=\frac{2}{3\sqrt{5}}p_{1}\neq 0$. So all rank two and rank three Bell-diagonal states have nonzero fidelity deviation. Thus in the Bell-diagonal family only entangled Werner states are both useful and universal.

Let us now turn our attention to rank two states. For rank two states we have $\Delta_{{\rm BD}}=\frac{2}{3\sqrt{5}}\left(1-p_{0}\right)$ since $p_{0}+p_{1}=1$, and fidelity is given by (52). Then, as $p_{0}\rightarrow\frac{1}{2}$ from above (i.e. entanglement goes to zero), $F_{{\rm BD}}\rightarrow\frac{2}{3}$ from above, and $\Delta_{{\rm BD}}\rightarrow\frac{1}{3\sqrt{5}}$ from below. Thus, rank-two Bell-diagonal states exhibit the same property exhibited by pure entangled states near the quantum-classical boundary.

Non-Werner states with $\Delta_{\rho}=0$ and $F_{\rho}>\frac{2}{3}$

In the Hilbert-Schmidt representation, the Bell-diagonal states are those with local vectors $\bm{R},\bm{S}=0$. So the states that do not belong to the Bell-diagonal family must have at least one of the local vectors nonzero. We now give two examples from the class of $X$-states (X-1, ; X-2, ) with the property $\Delta_{\rho}=0$.

Example 1**.**

Consider the family of rank three states:

[TABLE]

where $\left|\Psi_{0}\right\rangle$ is the singlet state. One can easily verify that the states are entangled for all values of $0<p<1$.

The Hilbert-Schmidt decomposition is given by

[TABLE]

Note that, $\det T<0$ as all the eigenvalues of the $T$ matrix are equal to $-p$, where $p>0$. And since the eigenvalues are equal, $\Delta_{\rho}=0$. The maximal fidelity is given by $F_{\rho}=\frac{1}{2}\left(1+p\right)$. Thus the states are both useful and universal for quantum teleportation for all values of $p$, $\frac{1}{3}<p\leq 1$ but not useful when $0<p\leq\frac{1}{3}$, although they are entangled and satisfy the universality condition.

The next example has the property that it’s both useful and universal if and only if it’s entangled.

Example 2**.**

Consider the family of rank four states:

[TABLE]

The Hilbert-Schmidt decomposition is given by

[TABLE]

The states are entangled iff $\frac{1}{3}<p\leq 1$. Since $t_{11}=t_{22}=t_{33}=-p$, $\Delta_{\rho}=0$. The maximal fidelity is given by $F_{\rho}=\frac{1}{2}\left(1+p\right)>\frac{2}{3}$ for $p>\frac{1}{3}$. So in this case, the states are both useful and universal iff they are entangled.

V Discussion and Concluding remarks

Quantum teleportation with a two-qubit state is generally characterized by the average fidelity, which tells us how well, on average, we are able to teleport unknown quantum states. The average fidelity, however, does not give us any information on the fluctuations in fidelity, although it is of particular importance to take such fluctuations into account. Motivated by an earlier work (Bang-et-al-2012, ), the authors in (Bang-et-al-2018, ) considered fidelity deviation, which is defined as the standard deviation of fidelity over all input states, as a measure of fidelity-fluctuations.

For two-qubit states, the maximal fidelity is defined as the maximal value of the average fidelity achievable within the standard protocol and local unitary operations. While the maximal fidelity is given by a simple formula (RMP-Horodecki-96, ; Badziag-2000, ), no such formula existed for the fidelity deviation. In this paper, we obtained the formula for the fidelity deviation in optimal quantum teleportation with an arbitrary two-qubit state. The formula is given by a function of the eigenvalues of the correlation matrix ($T$ matrix) and can be effectively computed. Thus for two-qubit states quantum teleportation can be adequately characterized in terms of maximal fidelity and the corresponding fidelity deviation. Here we want to emphasize that the formula for fidelity deviation applies to any two-qubit state, even if such a state is dynamically generated e.g. through system-bath interactions (see, for e.g. (open-quantum-1, ; open-quantum-2, ; open-quantum-3, )) and similar analyses along the lines of our examples are indeed possible.

Our results showed that fidelity deviation for two-qubit states is nonzero in general, and the states with zero fidelity deviation are indeed special. We obtained the condition for zero fidelity deviation – the universality condition and provided a necessary and sufficient condition for a two-qubit state to be both useful and universal. We also showed that for any given maximal fidelity, which is larger than the classical bound, there always exist states with zero fidelity deviation. We also analyzed the properties of some well-known classes of two-qubit states and found that fidelity deviation for a pure entangled state is nonzero unless the state is maximally entangled, and in the Bell-diagonal family, and Werner states are the only states with zero fidelity deviation. We also provided specific examples of non-Werner states with zero fidelity deviation – these states belong to the family of $X$-states.

For pure entangled states and rank-two entangled Bell-diagonal states we found that as entanglement goes to zero, the maximal fidelity approaches the classical bound from above as we expect, but the corresponding fidelity deviation approaches a constant value from below. This implies that in the neighbourhood of the quantum-classical boundary, fidelity values for some input states may well be in the classical region, although on average teleportation would still be quantum. Such a situation, however, doesn’t arise for states that are both useful and universal because all input states are teleported with equal fidelity larger than the classical bound. This is why universal states form the preferred subset of all useful states.

Now one could argue that it should be possible to achieve vanishing fidelity deviation for any shared two-qubit state by twirling protocols. For example, before teleportation, Alice applies a random single-qubit unitary $U$ drawn according to the Haar measure on $SU\left(2\right)$ to the input state, and after teleportation, Bob applies the inverse $U^{\dagger}$ to the teleported state – as one effectively averages over all input states the teleportation fidelity for each state thus coincides with the average fidelity. Alternatively, they can apply the random twirl – a random bilateral unitary operation of the form $U\otimes U$ (in higher dimensions $U\otimes U^{*}$ (MPR-Horodecki-1999, )) – on the two-qubit resource state (canonical form, or otherwise), which effectively transforms the two-qubit state to Werner form (note that they can achieve the same goal by making a random selection, with uniform probabilities, from a specific set of just twelve unitary operations $\left\{U_{i}\right\}$ which involve identical rotations on each of the two qubits; for details, see, appendices A and B of (bennett-1995, )). Both protocols, however, will lead to a modest increase in classical communication cost as identity of the unitary operations must be communicated. Nevertheless, they should be feasible in practice, although their performance could be limited by experimental (im)precision of implementation of single qubit unitaries rather than by the ability to classically transfer specification of the chosen unitary matrix.

Let us now discuss in what ways our result could be useful in physical realizations of quantum teleportation. In experiments, one always deals with imperfections coming from various sources – quantum channels, entangled states, measurements etc., and such imperfections give rise to fidelity fluctuations. Our formula for the fidelity deviation, on the other hand, estimates fidelity fluctuations arising from the given resource state (equivalently, the noisy quantum channel used to establish such a state). Thus to make use of this formula to analyze experiments one needs to obtain full information about the state by tomography before calculating the fidelity deviation as fidelity is not an observable. Although state tomography with finite samples will lead to errors, approximate twirling should always be possible with a finite amount of randomness.

Finally, we expect the results in this paper will help us to understand the fundamental properties of quantum teleportation and also find meaningful applications in scenarios where fidelity deviation can be used as a filter to select optimal resource states. While we have considered one such application, namely, universality in quantum teleportation, we think there could be many other possibilities. For example, one might want to find the optimal states from a given set of states where the set is well-defined with respect to some physical property.

Acknowledgements.

DD acknowledges financial support from University Grants Commission (UGC), Government of India. SB is supported in part by SERB (Science and Engineering Research Board), Department of Science and Technology, Government of India through Project No. EMR/2015/002373.