We consider the probability that a bipartite quantum state contains phase-conjugate-state (PCS) pairs and/or identical-state pairs as signatures of quantum entanglement. While the fraction of the PCS pairs directly indicates the property of a maximally entangled state, the fraction of the identical-state pairs negatively determines antisymmetric entangled states such as singlet states. We also consider the physical limits of these probabilities. This imposes fundamental restrictions on the pair appearance of the states with respect to the local access of the physical system. For continuous-variable system, we investigate similar relations by employing the pairs of phase-conjugate coherent states. We also address the role of the PCS pairs for quantum teleportation in both discrete-variable and continuous-variable systems.

The spooky action induced by the entangled particles at a distance is an interesting starting point in studies of quantum entanglement [

The pair of phase-conjugate states in the Bloch sphere when the fixed basis is the

It has been known that the use of PCS pairs

In this paper we define the probabilities that PCS and/or identical-state pairs appear in bipartite quantum states. We investigate physical limits of these probabilities and their relations to quantum entanglement. In Section

Let us write a maximally entangled state (MES) in two-qudit (two

In order to find the role of entanglement in such phenomena, we may define the

Now, we would like to ask two questions. (i) How high one can attain the probability of the pair appearance without entanglement? (ii) Does the unit probability

Since any state being invariant under the unitary operation

The probability of phase-conjugate-state-pair appearance

From (

From (

In this section we consider the process of quantum teleportation and show that the teleportation fidelity corresponds to the fraction of the PCS pairs introduced in the previous section (see (

Let us define the generalized Pauli

If the shared entangled state

For a given quantum channel

where we used the relations such as

Equation (

In this section we consider the probability that a two-qudit state contains identical-state pairs

The probability that a state contain an identical-state pair can be written by

In order to see the property of

Next, we determine the minimum value of the fraction attained by separable states. Since the expectation value of a partial transposed operator for a product state corresponds to a expectation value of the original operator for another product state [

To show a physical meaning of the identical-pair appearance, let us consider the following quantity:

Note that the unconditional appearance (the appearance with unit probability) of the orthogonal pairs is the unique property of the states in the antisymmetric subspace. This can be proven as follows: the condition

Here, we summarize the main statements in Sections

We have shown that, for the probability

In the previous sections the pair appearances are considered with respect to the uniform distribution that includes arbitrary PCS/identical-state pairs. In this section we consider the appearance of the PCS pairs with respect to the elements of two mutually unbiased bases.

The two orthonormal bases of a

Suppose that the subsystem

Using the completeness of the Bell basis of (

Since the last term in (

Next, we consider the lower bound of

Let us consider some cases of prime numbers. For

For

The upper bounds of

In the following sections we will consider continuous-variable analog of the pair appearance of phase-conjugate states. The analysis in Sections

The EPR state is defined as a simultaneous eigenstate of the relative position and total momentum of a two-particle system. The EPR state thus has strong correlation on the positions and strong anticorrelation on the momentums. This suggests that the complex amplitudes of the two particles maintain the complex conjugate relation as in Figure

If one of the EPR particles is projected onto a coherent state

As an experimental implementation of the EPR state in the quantum optics, we usually work with the two-mode-squeezed states (TMSS):

In what follows we assume that

For the TMSS of (

In what follows we show that (i) the maximum value of

Let us show that (i)

Let us define the covariance matrix of

In order to diagonalize this matrix we define another matrix

Note that in the limit

Now, our question is the limit of the pair appearance

In order to show this, let us write the partial transposition of the operator

Consequently, it has turned out that the pair appearance of the phase-conjugate coherent states can be a signature of entanglement, and we obtain the following statement: the state

The fraction of the state pairs

In Section

In this section we consider the process of continuous-variable quantum teleportation and investigate the equivalence between the teleportation fidelity and the fraction of the phase-conjugate coherent-state pairs.

Let us consider the process of continuous-variable quantum teleportation [

A continuous-variable quantum teleportation process

For the case of continuous-variable quantum channels [

If we set the gain parameter

We have considered the probability that a bipartite quantum state contains the PCS pairs and/or identical-state pairs. We determine the physical limits and classical limits of these probability for the case of uniform distribution on qudit states. The classical limits give the separable conditions. We have also shown the equivalence between the average fidelity of quantum teleportation process and the probability of the PCS pairs on the resource state of the teleportation. A summary of the obtained statements in Sections

For the case of uniform distribution on two mutually unbiased bases, a part of the problems becomes highly dependent on the dimension

We have also considered the probability that a two-mode continuous-variable state contains the phase-conjugate coherent-state pairs. We determine its physical limit and classical limit. We have also addressed its role in the process of continuous-variable quantum teleportation.

In the main text, we have considered the maximum expectation value of certain observables under the constraint that the state is separable. This optimization problem is called the separable eigenvalue problem [

Any pure entangled state can be written in the Schmidt decomposed form as follows:

Here, we give another diagonal expression of the flip operator defined in (

The matrix elements of

In the expression of

This work was supported by the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. R. Namiki acknowledges support from JSPS.