## I Introduction

Reliable Bell measurements are an essential component of quantum information processing. Still, implementing a deterministic linear optical Bell measurement on photonic states has been a longstanding unsolved problem in linear optical quantum computing. This is unfortunate because Bell state discrimination plays a critical role in quantum teleportation Teleportation ; Teleportation Experimental ; Pavicic3 ; Bouwmeester ; Pavicic2 ; Nielsen and Chuang , which can be used as a universal primitive for building a scalable fault-tolerant quantum computer Chuang . In our work here, we investigate a linear trend in the improvement of a photonic Bell state analyzer constructed from only deterministic linear optical components (i.e. beam splitters and phase shifters) and un-entangled ancilla resources.

In the standard dual-rail encoding Review Paper

, a qubit is represented by the polarization or spatial states of a single photon. Then, the Bell states are represented by a maximally entangled Einstein–Podolsky–Rosen (EPR) pair of photons. If all four Bell states are equiprobable, this ensemble is written

(1) | |||

(2) | |||

(3) | |||

(4) | |||

(5) |

The action of a linear optical quantum circuit on an optical state can generally be described by the transformation of creation operators Review Paper ; Reck :

(6) |

where are the elements of some unitary complex matrix and is the total number of optical modes. As it turns out, we cannot implement a perfect von Neumann measurement by applying Eq. (6) to Eqs. (1)-(4

). There exists no unitary matrix

such that using perfect photon-number resolving detectors at each of the four circuit output modes allows a perfectly unambiguous measurement; at least two of the Bell states will always be indistinguishable. The crux of the problem is that linearly accessible entangling operations between photons allowed by Eq. (6) are restricted to bosonic interference Review Paper ; Hong Ou Mandel and we are thus unable to rotate the Bell states into non-overlapping Fock spaces for a photo-counting measurement. In 2001, the Knill-Laflamme-Milburn (KLM) KLM ; KLM2 scheme for implementing a CNOT gate between two qubits encoded in the dual rail demonstrated that incorporating probabilistic Uskov ; Knill ; Knill2 partial measurements and ancilla resources into linear optical circuits allows a new profitable set of transformations beyond those of Eq. (6).A series of “no-go” theorems on optical state discrimination seemed to limit the benefit of using these extra tools in a Bell state analyzer. First, it was demonstrated by Lütkenhaus, Calsamiglia, and Suominen that a completely perfect measurement on an ensemble of Bell states using only linear components, multistage partial measurements, and ancilla resources cannot exist Lutkenhaus . This theorem was later extended by Carollo and Palma Carollo ; we apparently cannot implement a perfect measurement using these tools on any

set of indistinguishable orthonormal photonic states. Further, it has been shown that an analyzer limited to using only vacuum ancilla modes cannot perfectly distinguish equi-probable Bell states (the ensemble in Eq. (

5)) more than half of the time Calsamiglia . However, it was shown in Refs. Ewert ; Grice that this upper bound is lifted if we do not restrict our ancilla modes to be in the vacuum state.These no-go theorems do not provide a precise upper bound on distinguishability. A “perfect” measurement is in any case a mathematical ideal that can never be attained in the real world, and an arbitrarily close-to-perfect measurement would be equally valuable for practical applications in quantum information processing. In 2011, Ref. Pavicic presented a scheme for near-perfect Bell state discrimination, though this paper was later retracted. This is clearly a difficult problem with no easy solutions.

Here, we use computational techniques to find a clear linear trend showing improvement in Bell state measurements as the number of ancilla photons is increased. Extrapolating these results, we learn about effective linear optical analyzers. We determine that if a device bunches all photons into two or fewer modes under any circumstances, then such a device cannot perform a perfect measurement. Finally, we present a set of conditions on the mode transformation matrix that prevent this outcome. With enough patience, this construction could be extended to develop an optimal linear optical Bell state analyzer and determine an exact upper bound on measurability.

## Ii Transforming the Bell states

An optical quantum state can be written as a complex vector

(7) |

where

(8) |

are the Fock states. A general linear optical transformation described by Eq. (6) can also be written in the form of a linear operator acting on the many-photon state,

(9) |

We can write a Fock state as

(10) |

where is the mode-location of photon number and is the total number of photons in the system. Of course the choice of vector for a given is not unique, since photons are indistinguishable and labeling them is an arbitrary process. We simply need to choose some labeling and pick a valid for each . Then the elements of the matrix representation of in the basis are given by

(11) | |||

where the summation is over all distinct permutations of integer entries in the vector . For a proof of Eq. (11), see Ref. Jake Smith .

Here, we allow the use of ancilla photons and define our Bell states in the Fock basis,

(12) |

E.g. the first Bell state is

(13) | |||

The total number of photons and modes are now

(14) | |||

(15) |

We allow a standard von Neumann measurement. Alice chooses one of the Bell states and sends it to Bob who attaches the ancilla modes, runs the state through his analyzer , and performs a photon-counting measurement at every output mode. If Alice sends Bell state , the probability that Bob measures state is

(16) |

## Iii The Classical Mutual Information

To gauge the success of Bob’s measurement device, we choose the classical mutual information, . This quantity is bounded above by Holevo’s theorem Holevo ; Hausladen ; First Paper , given by

(17) |

where is the von Neumann entropy of a quantum ensemble and is the classical Shannon entropy of Alice’s encoding variable. For the ensemble defined in Eq. (5), these quantities are both fixed:

(18) |

The higher the mutual information, the better Bob can distinguish the Bell states. We say that a Bell analyzer is near-perfect if

(19) |

in some limit. In general, we may write

(20) |

where is the conditional information,

(21) |

## Iv Numerical Minimization of the conditional information

To optimize the capability of a linear optical Bell state analyzer, we can numerically minimize the conditional information in Eq. (21) over the circuit design . In general, is constrained to be unitary. Here, however, we find that optimization convergence is improved if we allow to be sub-unitary, meaning that

can be any complex matrix with singular values less than or equal to one. This is equivalent to allowing photons to leak into vacuum modes on which

does not act, and on which we do not perform any measurement. We call these modes the garbage modes and need to add a term to Eq. (21) to accommodate them:(22) |

where

(23) |

In other words, all measurement outcomes in which any photons have leaked into the garbage modes are consolidated into one outcome with probability . An optimal thus tends to be unitary as photon leakage into the garbage modes should be avoided. Still, convergence is improved because the optimization trajectory is allowed to pass through the interior of a unitary hypersurface in parameter space, rather than be restricted to move along the hypersurface.

We write

using a general singular value decomposition,

(24) |

where

(25) | |||

(26) | |||

(27) |

and minimize Eq. (22) over all hermitian matrices , , and real numbers using a quasi-Newton method without any constraints; results are presented in Fig. 1.

We note that as additional pairs of ancilla photons are added in going from to to , the mutual information grows linearly from 1.5 to 1.625 to 1.75. If the linear trend continues, we expect a minimum ancilla resource of should be needed to attain a near-perfect measurement. Unfortunately, for , the optimization routine becomes overwhelmed by local minima, preventing convergence to a globally optimal solution. We have parallelized the construction of over hundreds of processors, employed Intel Xeon Phi coprocessors, and have also utilized various optimization algorithms, including BFGS, Levenberg-Marquardt, Nelder-Meade, and principal axis minimization. With all of these methods, convergence still proved difficult for .

## V Analysis of

The numerical results in Sec. IV seem to indicate a discrete improvement in state distinguishability as we incorporate un-entangled pairs of ancilla photons into our Bell state analyzer. Extrapolating this trend, we can try to solve for a unitary assuming a perfect measurement is possible. We first define for each possible output state containing photons in modes the complex numbers

where is a vector representing the output mode location of each photon as defined in Eq. (10) and the sum is over all distinct permutations. Then the probability of measurement outcome for each of the four input Bell states is given by

(28) | |||||

and the measurement probability summed over input states is

(29) |

where is a combinatoric bosonic factor

(30) |

Because

(31) |

and

(32) |

according to Eq. (21) we need to find unitary such that

(33) |

in order to implement a perfect measurement. With some algebra, we find Eq. (33) to be satisfied if and only if:

(34) | |||

Now we can manually analyze the output measurement states for , and find constraints on such that either , , or is satisfied for each .

We start with output states where photons are bunched into two modes, i.e., states of the form

(35) |

Beginning with the case of and photons in mode , our measurement state is

In this case

(36) | |||||

Thus, we find that conditions and cannot be satisfied for the measurement state . is satisfied if and only if at least one of the following conditions holds:

(37) | |||

Of course, we need condition to be satisfied for each choice of mode number in the measurement state . Thus, or or in Eq. (V) must be satisfied for each column of .

We next look at the measurement states where :

(38) | |||

For each column of , we assume , , or in Eq. (V) is true, and determine any additional constraints that make , , or in Eq. (V) true for outcome state . Through some algebra, we determine that the conditions , , and must be revised; for each column of , is satisfied if and only if at least one condition holds:

(39) | |||

while conditions and cannot be satisfied for .

We repeat this process in a proof by induction, revising the conditions as we increase in Eq. (35) from 0 to . In all cases we find that and can never be satisfied. Thus, a measurement outcome in the form of Eq. (35) is guaranteed to be ambiguous. is satisfied for all states if and only if at least one condition holds for each column of :

(40) | |||

Since these conditions hold for any pair of modes , increasing to is sufficient for all measurement states of the form in Eq. (35).

As a simple numerical check, we compare the mutual information of a Bell state analyzer with general random unitary to random unitary satisfying conditions for trials. Results are presented in Fig. 2.

## Vi Conclusion

We are now ready to present a key result:

A perfect linear optical Bell state analyzer cannot produce a photo-counting measurement in which all photons are bunched into just two modes. If we measure a state of the form in Eq. (35), then we have not perfectly distinguished the Bell states and our analyzer is flawed.

To prevent these measurement outcomes, we can impose one of the conditions on each column of the matrix . We allow some small numerical leniency for a near-perfect measurement.

The next step is to examine measurement states in which all photons are bunched into three modes, i.e., states of the form

(41) |

Again, this can be done inductively by imposing and then revising conditions at each step. With enough patience, one could carry this procedure through all measurement output states and find the true upper bound on the measurement ability of a linear Bell state analyzer. This is a difficult project that we leave for future work. Our numerical results in Fig. 1 seem to suggest that a near-perfect Bell state measurement could be possible in the regime .

###### Acknowledgements.

We are thankful for helpful discussions with M. Levenstien, R. Budin, and F. Christian. This research was supported in part using high performance computing (HPC) resources and services provided by Technology Services at Tulane University, New Orleans, LA. This work was supported by the NSF under Grant No. PHY-1005709.## References

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