# Comment on ‘Counterfactual entanglement and non-local correlations in separable states’.

###### Abstract

The arguments of Cohen [Phys. Rev. A 60, 80 (1999)] against the ‘ignorance interpretation’ of mixed states are questioned. The physical arguments are shown to be inconsistent and the supporting example illustrates the opposite of the original statement. The operational difference between two possible definitions of mixed states is exposed and the inadequacy of one of them is stressed.

PACS 03.65.Bz,

In a recent paper [1] Cohen constructs an example of counterfactual entanglement and also deals with two possible scenarios that lead to the appearence of mixed states. These states arise either from tracing out unavailable degrees of freedom (‘ancilla interpretation’) or from the actual mixing of pure states (‘ignorance interpretation’). Cohen states that the ignorance interpretation is ‘unsatisfactory’. The abovementioned counterfactual entanglement and the arguments against the ignorance interpretation allow to question whether it is ‘appropriate to label weighted sums of projections on product states as “separable mixed states”’.

The relevance of counterfactual entanglement to the physical meaning of separable states and its deeper implications on the non-locality problems will not be discussed here. However, while some of Cohen’s arguments against the ignorence interpretation are of philosophical nature and their acceptance or refutal is a matter of personal opinion, several claims can be rigorously analyzed and are demonstrably wrong. It should be stressed, however, that while their refutal may weaken the case against the existence of separable states, it by no means undermines the validity of Cohen’s example of the countrfactual entanglement. Moreover, it should be noted that some of Cohen’s arguments against the ignorance interpretation of mixed states follow a similar discussion in the book of d’Espagnat [2].

One of the arguments of [1] is that ‘in some cases different statistical mixtures of pure states may appear to be representable by the same density matrix but may nevertheless be experimentally distinguishable’. An example which is meant to illustrate this argument is a comparison between two large sets of spin- particles. In the first set exactly out of particles are prepared in spin-up state in direction and the other half is prepared spin-down in direction. Another set is prepared according to the same recipe, but along the -axis. As shown below the example illustrates exactly the opposite of Cohen’s claim.

We begin by noting that the density operator that we ascribe to a state depends not only on the preparation procedure, but also on the experimental techniques that can be used and on the number of systems available to the test. Moreover, either there is a finite probability to distinguish between the states, or they have the same density matrix (again, the confidence level and the possibility of the procedure itself depend on the quality of experimental techniques).

Let us consider the simplest distinguishability criterion: probability of error. It deals with two states whose density matrices and are known. An observer is given one of them and is allowed to perform any lawful quantum operation. At the end, the observer should give an unambiguous answer which state it was. The probability to err is [3]

(1) |

and it depends only on the density matrices. In particular, identity of density matrices is equivalent to total indistinguishability of the states and distinguishable states cannot ‘appear to be representable by the same density matrix’.

Now let us examine Cohen’s example. We suppose that the detectors are ideal and consider two possible ways to identify the states. First, we analyze the experiment where particles are tested individually along the same axis (say, -axis). If we have the first set, we should get exactly up and down results, while the probabilities of these outcomes for the second set are distributed binomially. It is easy to see that the probaility of error in the identification is

(2) |

On the other hand, multiparticle measurements not only improve the distinguishability, but also highlight the differences between preparation procedures. If the particles can be considered as distinguishable (like qubits in the register of a quantum computer), then no symmetrisation is needed. The preparations can be represented by two pure staes and , which are composed from direct products of eigenstates of and eigenstates of respectively. If we take into account that the overlap between any eigenstate of and any eigenstate of is , the overlap between and is

(3) |

If the particles are linearly polarized photons and polarization is the only degree of freedom, then the correct description of both preparations is given by symmetric states in the Fock space. If and are creation operators in two perpendicular directions, and is the vacuum state, then the first state is given by

(4) |

while the second state is

(5) |

Moreover, it is easy to see that these states are in fact almost (or even exactly) orthogonal. Using formulas 0.157 of [4], we find that their overlap for even is

(6) |

while for odd the states are orthogonal,

(7) |

(This happens, e.g., for , giving a spin-1 triplet state). To compare two modes of investigation we note that for two pure states and

(8) |

Thus for the distinguishable particles we have

(9) |

For the photons we have

(10) |

for small overlaps (large ) when is even, and (exact distinguishability) for odd . Thus we see that while we ought to agree with Cohen that the states are distinguishable, it is impossible to ascribe to them the same density matrix (especially if they are orthogonal).

To refine the above analysis we clarify what is exactly meant by an ‘ignorance mixture’. Two definitions are found in literature. Namely, the density operator is defined either as

(11) |

where ’s are exact numbers (‘type-1 preparation’ ,[2]), or

(12) |

where ’s are probabilities to have states (‘type-2 preparation’,[5]). It is not stated explicitely in [1] to which definition the author subscribes (however all examples are of the type-1).

Type-1 and type-2 preparations of ‘the same state’ are not equivalent. They may be distinguishable (with finite probability) if more than one copy is available. Consider a preparation of type-1 of exactly spin-up and and spin-down particles along some known axis, and a preparation of type-2 with along the same axis. When detectors are ideal and particles are tested individually, the probability to err in their identification is given by Eq. (2). The immediate consequence of this result is that the preparation-1 type state cannot, in general, reproduce the local statistics of the EPR experiment. This statistics is reproduced by the maximally mixed state . This result, together with Eq. (2) and above examples, implies that despite its appearence, the type-1 preparation is not described by the maximally mixed density matrix (the correct description depends on the exact details of the preparation and may even be given by a pure state).

The subtle dependence of the ascribed on the details of the preparation and the observation procedures is further illustrated by the examples below. Consider again Cohen’s example, but let us replace type-1 preparations by those of type-2, with .

If only individual particles can be tested, then for both preparations the probabilities of the oucomes are described by the same binomial distribution. Thus they are indistinguishable and the correct description of the states should be given by mixed density matrices , where is a unit matrix. This result is independent from the number of available particles.

In the case of distinguishable particles the multiparticle state reduces to the direct product of the individual density matrices. Obviously, in this case it is also impossible to distinguish between the preparations.

On the other hand, mixtures of different -boson states lead to a different conclusion. Since the expressions becomes cumbersome for large , let us consider the simplest case of two-particle states. Now we have

(13) |

and

with

(15) |

A straightforward calculation gives

(16) |

and there is a nonvanishing probability to distinguish between these two states.

It is also stated in [1] that ‘given an EPR spin-singlet pair each separate particle can be described by a mixed state, with the other particle then taking the role of ancilla. But if we then assume that this mixed state can also be given an ignorance interpretation, inconsistencies immediately arise’. We claim that no inconsistencies arise, as shown below. The local statistics of both EPR particles can be described by the maximally mixed density matrix. A possible interpretation of this local state is that it originates from a type-2 preparation procedure, since type-1 is inconsistent with experiment. However, when the correlations between the particles are analyzed, the description of the complete system in terms of mixed states should be dropped, regardless of our interpretation of Bell’s theorem or philosophical attitudes, but simply because it is inconsistent with experiment.

To summarize, we see that the type-1 interpretation is unsuitable for the description of mixed states, but because of reasons different from those that are given in [1]. On the other hand, the type-2 realisation of mixed states leads to a consistent description of physical systems and its predictions are identical to those obtained with the help of ancilla interpretation.

Acknowledgments

Discussions with Asher Peres and his help in the preparation of the manuscript are gratefully acknowledged. Several of the above examples are extensions of the exercises from chapters 2 and 5 of his book [5]. I also thank Oliver Cohen for clarifying important points of his article. This work is supported by a grant from the Technion Graduate School.

## References

- [1] O. Cohen, Phys. Rev. A 60, 80 (1999).
- [2] B. d’Espagnat, Veiled reality (Addison-Wesley, Reading MA, 1994), p. 83.
- [3] C. W. Helstorm, Quantum detection and estimation theory, (Academic Press, New York, 1976).
- [4] I.S. Gradshteyn, I.M. Ryzhik, ed. by A. Jeffrey, Table of integrals, series, and products, (Academic Press, San Diego, 1994).
- [5] A. Peres, Quantum mechanics: concepts and methods, (Kluwer, Dordrecht, 1993).