Entanglement Purification

CS 498S

 

By Lara Thompson

96226619

December 11, 2000

 

Table of Contents

Introduction.. 0

Entanglement Purification.. 0

Entanglement Purification vs. Quantum Error Correction.. 1

One Particle Purification.. 1

Two Particle Purification.. 3

Projection Methods. 4

Quantum Privacy Amplification.. 7

Conclusions. 9

References. 9

Introduction

Entanglement Purification

A major challenge in quantum communication is the accurate transmission of quantum information between Alice and Bob when the channel connecting them is noisy.  The fidelity of a quantum state communicated across a noisy quantum channel decreases (exponentially, in general) with the length of the channel.  Thus, communication would be limited to extremely short distances if it weren’t for the method of teleportation.  This requires shared entangled particles between Alice and Bob so that local operations and classical communication allow “transmission” of quantum states. 

The problem remains, however, how can Alice and Bob prepare entangled particles when they can only communicate through noisy channels?  Entanglement cannot be prepared by local operations only; Alice and Bob must send qubits across their noisy channel to create non-local correlations.  The interaction of these particles with the channel causes decoherence so that the resulting EPR pair will not be maximally entangled, but rather will be described by a mixed state with a certain entanglement fidelity.  The entanglement purification protocols distill from a large ensemble of low-fidelity EPR pairs a smaller sub-ensemble of sufficiently high fidelity.

The first protocol to be discussed was developed by Bennett (1996) and involves projecting with some probability to a state of higher entanglement fidelity.  This method will first be explored in its simplest one-particle ensemble form and will be generalized to two-particle ensembles.  For example, pure entangled EPR pairs are required for quantum teleportation.  Another two-particle purification protocol, quantum privacy amplification, first developed by Deutsch (1996), designed for cryptographic purposes will be discussed and its superior efficiency will be shown.

Entanglement Purification vs. Quantum Error Correction

Entanglement purification protocols, like quantum error correction codes, can be considered as a form of protecting quantum states from their interaction with the environment.

Quantum error correction codes can be used to purify EPR pairs between Alice and Bob.  For example, Alice prepares EPR pairs and sends them encoded to Bob.  Bob performs error correction and decoding so that the encoded qubits retain their entanglement with Alice’s half of the pairs, providing completely efficient purification.

Conversely, Alice generates EPR pairs and sends them to Bob.  She then performs half of the purification process, corresponding classically to Bob the results, who then performs his half of the process.  This results in some fraction p of the original pairs that can then be used for faithful transmission of quantum information.  This is equivalent to encoding n/p qubits to send n bits of information.

One Particle Purification

Consider a collection of qubits with available states |0ñ and |1ñ, in a mixture of states.  This could be an ensemble of spin ½ particles with polarization spin up and spin down, along the spin axis, partially polarized along that axis.  The corresponding density matrix of the ensemble is

                                       .                                              (1)

The fraction f is called the entanglement fidelity of the ensemble.

The qubits can be measured in the {|0ñ, |1ñ} basis to obtain a sub-ensemble of particles in the |0ñ state by discarding those projected into the |1ñ state.  For the case of spin ½ particles, an inhomogeneous magnetic field (such as one created by two magnets north facing south) spatially separates spin up from spin down particles.  By selecting only one path, say the spin down direction, |0ñ, the particles can be thought of as purified, with the projection distilling the ensemble to the desired pure state.

Assume now that the measurement destroys the qubit (for example, the mechanism could absorb the particle) while still conveying the result of measurement.  For example, the apparatus would click if the particle followed the spin up path before being absorbed.  Instead of sending the actual particle through the measurement apparatus, a copy could instead be used.  Of course, the no-cloning theorem proves it is impossible to create a copy of a general quantum state; however, it is possible to copy the selected basis states using the CNOT gate.  For example, a superposition  is transformed to , so that the no-cloning theorem is not violated.  For a target qubit with the density matrix |0ñá0|, performing a CNOT with (1) as the control qubit gives the resulting entangled state

                          ,                                  (2)

see Figure 1a.

Measuring the target qubit in the {|0ñ, |1ñ} basis, if measurement yields a click (i.e. the target qubit is measured to be in the |0ñ) state then we have with certainty the control qubit in the state |0ñá0|; otherwise, the control qubit is in the state |1ñá1|.  To purify the ensemble to states |0ñá0|, the measurements yielding a |1ñ are discarded, distilling the mixture to a pure subset.  In this setup, the fraction of qubits discarded is , leaving the remaining fraction in the desired pure state.

Above, we’ve assumed a source of existing pure states |0ñá0| for the target qubit.  If these were available, however, there would be no need to distil our mixed state (1). Can the distillation be performed without these?

Figure 1.  Selection of spin up or spin down using a Stern-Gerlag apparatus (SGA) (polarized particles are deflected with a double magnet).  (a) The state of an impure qubit is copied onto an auxiliary pure qubit for projection onto a pure state; (b) the impure state is copied onto another similar impure state for projection into a purified state (taken from Bouwmeester, 2000).

Consider replacing the target qubit with another of our ensemble of mixed state particles.  That is divide the collection of qubits to be distilled in half: the first half destined as control qubits to be distilled, the second half used as target bits to be sacrificed.  After the CNOT is performed on each pair (Figure 1b), we obtain the ensemble of entangled state with density matrix

    .            (3)

If the target qubit is then measured and found to be in the |0ñ state, the (renormalized) density matrix of the control qubit becomes

                                                                  (4)

where .  The function  is plotted in Figure 2.  Clearly, for  the remaining control qubits (those for which the target qubit was measured in state |0ñ) are a purified ensemble since  and there is an increased fraction of particles in the desired state |0ñ.  If the process is iterated, the ensemble can be distilled arbitrarily closely to the pure state |0ñ; however, to distil to a pure state requires infinitely many qubits to be wasted.

The number of qubits discarded per iteration is  where the first term represents the sacrificed target qubits and the second term arises from the discarded control qubits projected into the equal superposition .

Consider the nth iteration.  The purity of the sample is  while the fraction of qubits discarded in this iteration is , where .  The total fraction discarded in the n iterations is

                                                      .                                                  (5)

Figure 2.  The fidelity after an iteration of the purification algorithm.  Note that f' > f for f > ½.

Figure 3 shows the number of iterations required to reach fidelity 99% as well as the fraction of discarded qubits versus the initial entanglement fidelity.

Figure 3. The efficiency of the purification protocol for the two-particle EPR pair.

Two Particle Purification

Now consider the case of mixed entangled states, such as the two-particle entangled EPR pairs shared by Alice and Bob, where the particles are spatially separated.  For simplicity consider the superposition of the Bell states

                                 and                            (6)

with the density matrix

                                             .                                        (7)

Note that unless f = ½, the state is inseparable, so that it is indeed a mixed entangled state with fidelity f.  Clearly, if Alice and Bob send their particles through a measurement (such as measurement of spin in the case of spin ½ particles) they could distinguish the two Bell states: for pairs in the state  both measure the same thing (up-up or down-down); for pairs in the state  each measures the opposite (up-down or down-up).  This isn’t of much use since this destroys any existing entanglement and the particles leave the apparatus in a product state.

Applying the same trick as employed in the case of one-particle purification, consider the bilateral CNOT with  as the target bits (one of each of the entangled particles is used for each of Alice’s and Bob’s CNOT gate), depicted in Figure 4 and summarized in Table 1.  The CNOT maps as follows to the Bell states used here

                                                                                           (8)

so that  plays the analogue to the |1ñ state of the single particle system, while  is analogous to |0ñ.  Thus measurement of the target bits of the CNOT operations would signal whether the control bits had been projected into the  or  Bell state (as before, Alice and Bob measure the same thing in the case of  and opposites for ).

Figure 4.  Purification of a mixed entangled pair by local unitary CNOT gates, measurement and classical communication.  Note the two CNOT's compose a bilateral CNOT (B-CNOT) (from Bouwmeester, 2000).

But were these pure  states available, no purification scheme would be necessary, as before.  Splitting the ensemble of EPR pairs in two, one for purification, the other sacrificial half as the indicator qubit, we attempt the same scheme as was employed in the one-particle case.  After the bilateral CNOT transformation the qubits are in the entangled state

                                                              (9)

and each of Alice and Bob measure their target particle and transmit their result classically to determine whether or not to discard the first particle.  If they both measure the same thing, they keep the first entangled particle, whose fidelity has increased to , as in the single particle analogue.  If their measurements differ, their control particles are in the state

                                          

and are no longer entangled.  This is similar to the one particle case where the rejected control qubits were in the equal superposition of |0ñ and |1ñ.  In that case, the purification scheme is useless (recall we needed f > ½), just as here all entanglement is lost.

The fraction of wasted entangled pairs is the same as in the one particle protocol and the number of iterations for appropriate purification is also the same (Figure 3).  Note that in the two-particle purification protocol the exchange of classical information was used.  This is in fact an integral part of any purification protocol between parties.

Projection Methods

Consider next the more general mixed two-particle state.  Bennett first developed the following method (1996).  As before, we define the fidelity of the sample as

                                              

where we are now interested in distilling to the state .

The first step has Alice and Bob perform a random bilateral rotation, that is each performs one of {I, Rx Ry Rz} at a random angle on their half of the pair chosen at random, independently.  A normal bilateral rotation is simply when both Alice and Bob perform a rotation about the same axis, but in different directions (that is, Bob performs the inverse transformation of Alice) on each of their particles in the pair.  Their transformations in the Bell basis are summarized in Table 1.  This transforms the general mixed state into a so-called Werner state

                 .          (10)

Note that the state  is invariant under each of the bilateral rotations, while the remaining Bell states map into one another so that the Werner state is invariant under bilateral rotations.  Further, the fidelity of the transformed state is still f (note the coefficient of the diagonal  entry).

Table 1.  The unilateral and bilateral rotation transformations on the four Bell states.  Each entry in the B-CNOT section has two lines, the first showing what happens to the source state, the second showing what happens to the target state.

Source qubit

 

Bilateral p/2 rotations

Bx

 

By

 

Bz

 

Unilateral p rotations

sx

 

sy

 

sz

 

Bilateral CNOT

 

(source)

(target)

 

(source)

(target)

 

(source)

(target)

 

(source)

(target)

Operations on the qubits required for Bennett’s algorithm (summarised in Table 1) are:

Unilateral Pauli rotations: one of the Pauli transformations {I, sx, sy, sz} is applied to one of the particles in the entangled pair.  The various Bell states map in a 1-1 pair wise fashion.

      (i)            The bilateral p/2 rotations that were already discussed.

    (ii)            The bilateral CNOT (B-CNOT) gate depicted in Figure 4.

   (iii)            Measurement by Alice and Bob of each of their qubits.  Recall that this destroys any entanglement between the particles.

The algorithm itself requires two mixed entangled pairs with fidelity f > ½ shared by Alice and Bob, as well as classical communication.  The steps are as follows:

(A1)            A unilateral sy rotation is performed on each of the two pairs, converting them from mostly  to mostly  states:

                         

(A2)            A B-CNOT is performed as in Figure 4 after which the target pair is measured along the z-axis.  If Alice and Bob measure the same result (i.e. the input was a true  state) they keep the control qubit; otherwise, they discard it.  After measurement, the density matrix of the control qubit becomes

      (11)

which is easily worked out using the transformation rules of Table 1.

(A3)            If the control qubit is not discarded, then it is converted back to mostly   states via a unilateral sy rotation, and is made rotationally symmetric as before by a random bilateral rotation.

The fidelity of the output state, after an iteration of the algorithm, has improved to

                                                                                     (12)

which is plotted in Figure 5.  The fraction of discarded qubits is the same as in the last protocol and is given by (5).  Since f¢(f) > f for f > ½, with an initial fidelity exceeding ½, repeated iteration of the algorithm yields arbitrarily high fidelity (of course sacrificing many qubits in the process).  Figure 6 shows the number of iterations required to reach fidelity 99% as well as the fraction of discarded qubits versus the initial entanglement fidelity, just as was done in the simplified protocol in the previous section.

Figure 5. The fidelity after one iteration of Bennett's algorithm.  Note that f' > f for f > ½.

Figure 6.  The efficiency of the purification of a general mixed entangled state using Bennett's algorithm.

Quantum Privacy Amplification

The quantum privacy amplification scheme (QPA) was developed to account for noise in the quantum communication channel over which information must be sent as part of a quantum cryptographic scheme.  Prior to the inclusion of entanglement purification protocols in the quantum cryptography schemes, security of such schemes was only proven over idealized (noiseless) channels.  An eavesdropper (Eve) in such protocols is detected by measuring whether the generated EPR pair is entangled with any third party.  With the possibility of noise, however, whether the entanglement of their particles is with a potential eavesdropper or with the environment becomes indistinguishable.

In the idealized cryptography schemes, Alice and Bob have a supply of maximally entangled states  (6), where each has one of the entangled particles.  In the presence of noise, each pair would become entangled with each other and with the environment and would be described by a superposition of the four Bell states (6).  For simplicity, let r, the density matrix, be the same for all of the entangled pairs.  Note that the off-diagonal elements need not be considered since they do not contribute on average to the diagonal ones and are therefore not significant in our analysis of the efficiency of the scheme.  Denote the diagonal elements {a, b, c, d} where  is the entanglement fidelity of the ensemble.

As in the other purification protocol, divide the ensemble of mixed entangled pairs into two, one for purification, the other for measurement.  Alice performs the unitary operation

                                                                                             (13)

on each of her two qubits (from each of the two divisions) while Bob performs the inverse operation

                                                                                                               (14)

on each of his.  The effect of such a transformation in the Bell basis is summarised in Table 2, noting that in the two-particle system the transformation has the form

                                              .

Table 2.  The  transformation in the Bell basis, ignoring phases.

Input Bell state

Transformed Bell state

Thus after this first step, the diagonal entries of the density matrix become {a, d, c, b}.

Next Alice and Bob perform a CNOT with the control bit from the first division of the entangled pairs (Figure 4) with the entangled state

               

Alice and Bob then measure the target qubits of the CNOT and communicate classically their result.  If their outcomes coincide they keep their control qubits, otherwise they discard them, yielding the density matrix

       .

After an iteration of this procedure, the average (taken over different coincident outcomes) of the new diagonal elements of the density matrix are {an+1, b n+1, c n+1, d n+1} (Deutsch, 1996)

                                 (15)

where  is the probability that Alice and Bob measure the same outcome on their target qubits and {an, b n, c n, d n} are the diagonal entries of the density matrix prior to this iteration. 

Note that {a, b, c, d} = {1, 0, 0, 0} and {0, 0, 1, 0} are fixed points of the mapping (15); however, these fixed points aren’t necessarily approached monotonically.  For ao > ½, the first is the only fixed point; whereas for co > ½, the second is the only fixed point.  Each is a local attractor.  Computer simulations show that in their separate regions they are global attractors (Deutsch, 1996) and an analytical proof was recently presented.  The latter also proved that for an initial state bo > ½ (and of course for ao > ½) the iteration converges to , and similarly for co > ½ or do > ½ the iteration converges to the pure state . Thus if initially the mixture has fidelity exceeding ½, but which is otherwise in an arbitrary state, then the states of pairs surviving each iteration always converge to the unit-fidelity pure state .

Figure 7 shows the convergence of the special initial states with bo = co = do for a range of initial fidelities.  The number of qubits discarded in the process is about 1000 times more efficient for initial fidelity close to 0.5 than using Bennett’s algorithm (Deutsch, 1996).

Figure 7.  Average fidelity as a function of the initial fidelity and the number of iterations of the quantum privacy amplification (from Bouwmeester, 2000).

Conclusions

The fundamentals of entanglement purification have been explored.  The main idea behind such protocols is the distilling of a large ensemble of not-so-pure entangled states (the ones examined required an initial fidelity exceeding ½) down to a smaller (generally much smaller) sub-ensemble of more pure entangled states.  Generally, the algorithms are repeated to attain the required fidelity.  The yield of such iterations was examined in the simpler methods (Figure 3); the efficiency was discussed for the various protocols.

In addition to the methods discussed above, other purification protocols are being researched, such as purification via entanglement swapping (Bose, 1998).  There also exist protocols whose aim is to create maximally entangled EPR pairs between spatially distant particles, such as quantum repeaters (Briegel, 1998).

References

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996).

S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A 57, 822 (1998).

H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998).

H.-J. Briegel, Principles of Entanglement Purification, from D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information.  New York: Springer. 261 (2000).

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 (1996).

C. Macchiavello, Quantum Privacy Amplification, from D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information.  New York: Springer. 271 (2000).