Contemporary Mathematics

Inconclusive Rate with a Positive Operator Valued Measure

Howard E. Brandt

Abstract. Analysis is performed of explict optical implementations of both a positive operator valued measure (POVM) and of randomly choosing between two ordinary von Neumann projective measures. The POVM is demonstrated to have the lower inconclusive rate. Also, the effect of a general unitary disturbance on the inconclusive rate of the POVM implementation is calculated explicitly.

Contents

1. Introduction

2. Inconclusive rates comparison

3. Disturbed inconclusive rate

4. Consistency

5. Conclusion

6. Acknowledgements

References

1. Introduction

A positive operator valued measure (POVM) [1–7] can be usefully implemented in a quantum key receiver [8–14]. The following set of POVM operators represents the possible measurements performed by the receiver:

(1)  A_{u} = (1 + ⟨u|v⟩)^{−1} (1 − |v⟩ ⟨v|) ,
(2)  A_{v} = (1 + ⟨u|v⟩)^{−1} (1 − |u⟩ ⟨u|) ,
(3)  A_{?} = 1 − A_{u} − A_{v}.

Here, the kets |u⟩ and |v⟩ represent the two possible nonorthogonal normalized polarization states of a carrier photon with linear polarizations designated by u and v, respectively. The angle between the corresponding polarization vectors is θ.

\begin{matrix}c \\ ◯ \end{matrix}2002 American Mathematical Society

2000 Mathematics Subject Classification. Primary 81P68, 94-02, 94A60.

Key words and phrases. Quantum cryptography, quantum key distribution, quantum communication, quantum information.

The photon is a spin-one representation of the Lorentz group, and it follows that the Dirac bracket between the two states is [11]

(4)  ⟨u|v⟩ = sin 2α,

where

(5)  α = \frac{1}{2} (\frac{π}{2} − θ) .

(The use of the angle α instead of θ is convenient in the following.) The states |u⟩ and |v⟩ may encode bit values 0 and 1, respectively. The POVM operators, Eqs. (1)– (3), are nonnegative and their sum is unity. The operators A_{u} and A_{v} measure the probability of outcomes u and v, respectively. The operator A_{?} measures the probability of an inconclusive measurement.

The advantage of a POVM over an ordinary von Neumann projective measurement is that, for the POVM, the probability of getting an inconclusive result can be lower [8,14,15]. To see this, first consider, for comparison of a projective valued (PV) receiver with the POVM receiver, the simple all-optical PV receiver depicted in Figure 1. Effectively, the device randomly chooses between two ordinary von Neumann projective measures. (The all-optical POVM receiver is already exposited elsewhere [9–13].) The PV receiver consists of an incoming carrier photon in polarization state

(6)  |ψ⟩ = α |u⟩ + β |v⟩

for complex numbers α and β, a 50-50 beam splitter BS, two Wollaston prisms W_{u} and W_{v}, and four photodetectors D_{u}, D_{u_{⊥}}, D_{v_{⊥}}, and D_{v}. The Wollaston prism W_{u} is aligned so that a photon in state |u⟩ would take the path labeled by the state |ψ_{3}⟩ and polarization vector e^{∧}_{u}, and not the path labeled by the state |ψ_{4}⟩ and polarization vector e^{∧}_{u_{⊥}.} Here, e^{∧}_{u} denotes a unit polarization vector corresponding to the polarization state |u⟩ and is perpendicular to the polarization vector e^{∧}_{u_{⊥}} corresponding to the polarization state |u_{⊥}⟩ orthogonal to |u⟩. Analogously, the Wollaston prism W_{v} is aligned so that a photon in state |v⟩ would take the path labeled by the state |ψ_{6}⟩ and polarization vector e^{∧}_{v}, and not the path labeled by the state |ψ_{5}⟩ and polarization vector e^{∧}_{v_{⊥}} (perpendicular to e^{∧}_{v}). The beamsplitter BS serves to choose at random between two projective valued measurements, the first having projections {|u⟩ ⟨u| , |u_{⊥}⟩ ⟨u_{⊥}|} and the second having projections {|v⟩ ⟨v| , |v_{⊥}⟩ ⟨v_{⊥}|}. It is immediately evident from Figure 1 that

(7)  |ψ_{1}⟩ = 2^{−1/2}i (α |u⟩ + β |v⟩),

(8)  |ψ_{2}⟩ = 2^{−1/2} (α |u⟩ + β |v⟩),

(9)  |ψ_{3}⟩ = 2^{−1/2}i (α + β sin 2α) |e^{∧}_{u}⟩,

Figure 1. PV receiver.

(10)  |ψ_{4}⟩ = 2^{−1/2}iβ cos 2α |e^{∧}_{u_{⊥}}⟩ ,

(11)  |ψ_{5}⟩ = 2^{−1/2}α cos 2α |e^{∧}_{v_{⊥}}⟩ ,

(12)  |ψ_{6}⟩ = 2^{−1/2} (α sin 2α + β) |e^{∧}_{v}⟩,

where |e^{∧}_{u}⟩, |e^{∧}_{u_{⊥}}⟩, |e^{∧}_{v}⟩, and |e^{∧}_{v_{⊥}}⟩ represent unit kets corresponding to polariza-

tion vectors e^{∧}_{u}, e^{∧}_{u_{⊥}}, e^{∧}_{v}, and e^{∧}_{v_{⊥}}, respectively. It then follows that the probability P_{ψu} that the photon in state |ψ⟩ is detected by ideal detector D_{u} is given by

(13)  P_{ψu} = |ψ_{3}|^{2} = \frac{1}{2} |α + β sin 2α|^{2} .

Analogously, for detectors D_{u_{⊥},} D_{v_{⊥},} and D_{v}, one has

(14)  P_{ψu_{⊥}} = |ψ_{4}|^{2} = \frac{1}{2} |β|^{2} cos^{2} 2α,

(15)  P_{ψv_{⊥}} = |ψ_{5}|^{2} = \frac{1}{2} |α|^{2} cos^{2} 2α,

(16)  P_{ψv} = |ψ_{6}|^{2} = \frac{1}{2} |α sin 2α + β|^{2} .

From Eqs. (13)–(16), it follows that

(17) P_{ψu} + P_{ψu_{⊥}} + P_{ψv} + P_{ψv_{⊥}} = |α|^{2} + α^{∗}β sin 2α + αβ^{∗} sin 2α + |β|^{2} = 1,

as must be the case, provided that the state |ψ⟩, Eq. (6), is normalized to unity, and probability is conserved.

2. Inconclusive rates comparison

If the incoming photon state is |ψ⟩ = |u⟩, one has {α, β} = {1, 0} and Eqs. (13)– (16) yield P_{uu} = \frac{1}{2}, P_{uu_{⊥}} = 0, P_{uv_{⊥}} = \frac{1}{2} cos^{2} 2α, P_{uv} = \frac{1}{2} sin^{2} 2α. Analogously, in the case where the incoming photon state is |ψ⟩ = |v⟩, one has {α, β} = {0, 1} and P_{vu} = \frac{1}{2} sin^{2} 2α, P_{vu_{⊥}} = \frac{1}{2} cos^{2} 2α, P_{vv_{⊥}} = 0, P_{vv} = \frac{1}{2}. If states |u⟩ and |v⟩ are equiprobably incident on the receiver, then since detector D_{u} or D_{v} can be triggered by both states |u⟩ and |v⟩, it follows that the probability P_{?}^{PV} of an inconclusive measurement is given by

(18)  P_{?}^{PV} = P_{uu} + P_{uv} = \frac{1}{2} (1 + sin^{2} 2α) ,

or equivalently,

(19)  P_{?}^{PV} = P_{vv} + P_{vu} = \frac{1}{2} (1 + sin^{2} 2α) .

One can conclude that for the two-state quantum key distribution protocol, in which a photon is incident equiprobably in state |u⟩ or |v⟩ , the inconclusive rate P_{?}^{PV} for the projective receiver is

(20)  P_{?}^{PV} = \frac{1}{2} (1 + sin^{2} 2α) .

One can also obtain Eq. (20) by reasoning that the inconclusive rate for the PV measure is given by

(21)  P_{?}^{PV} = 1 − P_{con}^{PV},

where P_{con}^{PV} is the probability of obtaining a conclusive result. From Fig. 1, it is evident that

P_{con}^{PV} = P_{uv_{⊥}} = \frac{1}{2} ⟨u| (|v_{⊥}⟩ ⟨v_{⊥}|) |u⟩ = \frac{1}{2} |⟨u|v_{⊥}⟩|^{2}

= \frac{1}{2} sin^{2} θ = \frac{1}{2} (1 − sin^{2} 2α) . (22)

Equation (22) follows, since the ideal detector D_{v_{⊥}} cannot have been excited by the state |v⟩, and therefore can only have been excited by the state |u⟩, and the measurement operator for the state |v_{⊥}⟩ is |v_{⊥}⟩ ⟨v_{⊥}| [11]. (The overall factor of \frac{1}{2} appears in Eq. (22), because the probability is \frac{1}{2} that the photon takes the path from the beamsplitter leading to the Wollaston prism W_{v}. Also note that Eq. (22) is consistent with Eq. (15) for ᾱ = 1.) If one substitutes Eq. (22) in Eq. (21), then Eq. (20) again follows. Of course, Eq. (22) also follows analogously from

P_{con}^{PV} = P_{vu_{⊥}} = \frac{1}{2} ⟨v| (|u_{⊥}⟩ ⟨u_{⊥}|) |v⟩ = \frac{1}{2} |⟨v|u_{⊥}⟩|^{2}

= \frac{1}{2} sin^{2} θ = \frac{1}{2} (1 − sin^{2} 2α) . (23)

It has been demonstrated in previous work that the inconclusive rate P_{ψ?}^{POVM} of the POVM receiver for the arbitrary incoming state, Eq. (6), is given by [9,11–13]

(24)  P_{ψ?}^{POVM} = ⟨ψ|A_{?}|ψ⟩ = |α + β|^{2} sin 2α.

(The second equality in Eq. (24) is also consistent with the first, as can be seen by substituting Eqs. (3) and (6) in the first.)

For incoming state |ψ⟩ = |u⟩, one then has {α,β} = {1, 0}, and Eq. (24) becomes

(25)  P_{u?}^{POVM} = sin 2α.

For incoming state |ψ⟩ = |v⟩ , one has {α,β} = {0, 1}, and

(26)  P_{v?}^{POVM} = P_{u?}^{POVM} = sin 2α.

It follows that the inconclusive rate P_{?}^{POVM} of the ideal POVM receiver for the equiprobable two-state protocol is given by

(27)  P_{?}^{POVM} = sin 2α.

Using Eqs. (20) and (27), one then obtains

(28)  \frac{P_{?}^{POVM}}{P_{?}^{PV}} = \frac{2 sin 2α}{1 + sin^{2} 2α} < 1,

as depicted in Figure 2. Thus, in fact, the inconclusive rate for the POVM receiver is less than that of the PV receiver, and the rate ratio is determined by the angle between the two polarization states (see Eq. (5)).

Figure 2. Inconclusive rate comparison for POVM and PV receivers.

3. Disturbed inconclusive rate

The Fuchs-Peres model of eavesdropping on the two-state key-distribution protocol represents the most general possible unitary disturbance of each encoded photon incident on the receiver [16]. In this model, an incoming carrier state |u⟩ and the state |w⟩ of a disturbing probe undergo joint unitary evolution represented by a unitary operator U, resulting in the entangled state [10,16,17]:

U |u ⊗ w⟩ = \frac{1}{2}[(1 + sec 2α) |Φ_{00}⟩ + tan 2α |Φ_{10}⟩ − tan 2α |Φ_{01}⟩

+ (1 − sec 2α) |Φ_{11}⟩ ] ⊗ |u⟩ − \frac{1}{2}[ tan 2α |Φ_{00}⟩ − (1 − sec 2α) |Φ_{10}⟩

(29) − (1 + sec 2α) |Φ_{01}⟩ − tan 2α |Φ_{11}⟩ ] ⊗ |v⟩ .

Here |Φ_{mn}⟩ are states in the Hilbert space of the disturbing probe, and are neither normalized nor orthogonal. Equation (29) follows from Eqs. (1) and (2) of Slutsky et al [17]. Similarly, for an incoming state |v⟩, one has

U |v ⊗ w⟩ = \frac{1}{2}[ tan 2α |Φ_{00}⟩ + (1 + sec 2α) |Φ_{10}⟩ + (1 − sec 2α) |Φ_{01}⟩

− tan 2α |Φ_{11}⟩ ] ⊗ |u⟩ + \frac{1}{2}[(1 − sec 2α) |Φ_{00}⟩ − tan 2α |Φ_{10}⟩

(30) + tan 2α |Φ_{01}⟩ + (1 + sec 2α) |Φ_{11}⟩ ] ⊗ |v⟩ .

The probe states |Φ_{mn}⟩ have certain symmetry properties that arise from the random equiprobable selection of carrier states |u⟩ and |v⟩ by the key transmitter, and the resulting symmetry of the probe under interchange of |u⟩ and |v⟩ . Specifically, one has [16,17]

(31)  |Φ_{00}| = |Φ_{11}| ,
(32)  |Φ_{01}| = |Φ_{10}| ,
(33)  ⟨Φ_{00}|Φ_{01}⟩ = ⟨Φ_{11}|Φ_{10}⟩ ,
(34)  ⟨Φ_{00}|Φ_{10}⟩ = ⟨Φ_{11}|Φ_{01}⟩ ,
(35)  ⟨Φ_{01}|Φ_{10}⟩ = ⟨Φ_{10}|Φ_{01}⟩ ,
(36)  ⟨Φ_{01}|Φ_{00}⟩ = ⟨Φ_{10}|Φ_{11}⟩ ,
(37)  ⟨Φ_{01}|Φ_{11}⟩ = ⟨Φ_{10}|Φ_{00}⟩ ,
(38)  ⟨Φ_{11}|Φ_{00}⟩ = ⟨Φ_{00}|Φ_{11}⟩ .

According to Eq. (24), the inconclusive rate R_{?} induced by the disturbing probe in the POVM receiver is given by

(39)  R_{?} = P_{u?} = ⟨u ⊗ w| U^{†}A_{?}U |u ⊗ w⟩ ,

where P_{u?} is the probability that if a photon in polarization state |u⟩ is transmitted, then the measurement by the POVM receiver is inconclusive. Alternatively, one also has

(40)  R_{?} = P_{v?} = ⟨v ⊗ w| U^{†}A_{?}U |v ⊗ w⟩ ,

because of the symmetry of the two-state protocol. Equivalently, using Eq. (3) in Eq. (39), one also has for the induced inconclusive rate:

(41)  R_{?} = 1 − P_{uu} − P_{uv},

where P_{uu} and P_{uv} are the probabilities that if the carrier is a |u⟩ state, then the detectors D_{u} and D_{v}, respectively, respond. Here, one has

(42)  P_{uu} = ⟨u ⊗ w| U^{†}A_{u}U |u ⊗ w⟩ ,
(43)  P_{uv} = ⟨u ⊗ w| U^{†}A_{v}U |u ⊗ w⟩ .

Substituting Eqs. (2), (29) and (31)–(38) in Eq. (43), one obtains

P_{uv} = (1 + sin 2α)^{−1}[ (1 − sin^{4} α − cos^{4} α) |Φ_{00}|^{2} + (1 − \frac{1}{2} sin^{2} 2α) |Φ_{01}|^{2} .

+ \frac{1}{2} sin 2α ⟨Φ_{01}|Φ_{11}⟩ + \frac{1}{2} sin 2α ⟨Φ_{00}|Φ_{10}⟩ − \frac{1}{2} sin 2α ⟨Φ_{00}|Φ_{01}⟩

(44) − \frac{1}{2} sin^{2} 2α ⟨Φ_{00}|Φ_{11}⟩ − \frac{1}{2} sin 2α ⟨Φ_{01}|Φ_{00}⟩ − \frac{1}{2} sin^{2} 2α ⟨Φ_{01}|Φ_{10}⟩ ].

The probe states |Φ_{mn}⟩, expanded in terms of orthonormal basis vectors |w_{β}⟩, are given by Eqs. (3a), (3b), and (4) of Slutsky et al [17], namely,

(45)  |Φ_{00}⟩ = X_{0} |w_{0}⟩ + X_{1} |w_{1}⟩ + X_{2} |w_{2}⟩ + X_{3} |w_{3}⟩ ,
(46)  |Φ_{11}⟩ = X_{3} |w_{0}⟩ + X_{2} |w_{1}⟩ + X_{1} |w_{2}⟩ + X_{0} |w_{3}⟩ ,
(47)  |Φ_{01}⟩ = X_{5} |w_{1}⟩ + X_{6} |w_{2}⟩ ,
(48)  |Φ_{10}⟩ = X_{6} |w_{1}⟩ + X_{5} |w_{2}⟩ .

Here the real coefficients {X_{0}, X_{1}, X_{2}, X_{3}, X_{5}, X_{6}} , expressed in terms of the probe parameters {λ, μ, θ, ϕ} , are [16,17]

(49)  X_{0} = sin λ cos μ,
(50)  X_{1} = cos λ cos θ cos ϕ,
(51)  X_{2} = cos λ cos θ sin ϕ,
(52)  X_{3} = sin λ sin μ,
(53)  X_{5} = cos λ sin θ cos ϕ,
(54)  X_{6} = − cos λ sin θ sin ϕ,

consistent with the assumed unitarity of the disturbing probe.

Next, substituting Eqs. (45)–(48) in Eq. (44), one gets

P_{uv} = (1 + sin 2α)^{−1}[ (1 − sin^{4} α − cos^{4} α) (X_{0}^{2} + X_{1}^{2} + X_{2}^{2} + X_{3}^{2})

+ (1 − \frac{1}{2} sin^{2} 2α) (X_{5}^{2} + X_{6}^{2}) + sin 2α (X_{1}X_{6} + X_{2}X_{5})

− sin 2α (X_{1}X_{5} + X_{2}X_{6})

− sin^{2} 2α (X_{0}X_{3} + X_{1}X_{2} + X_{5}X_{6}) ]. (55)

Then if one substitutes Eqs. (49)–(54) in Eq. (55), the latter becomes

P_{uv} = \frac{1}{4}(1 + sin 2α)^{−1}[1 − cos 4α + 2(1 + cos 4α) cos^{2} λ sin^{2} θ
− 2 sin 2α cos^{2} λ sin 2θ cos 2ϕ − 2 sin^{2} 2α sin^{2} λ sin 2μ

(56) − 2 sin^{2} 2α cos^{2} λ cos 2θ sin 2ϕ].

Analogously, it can be shown that Eq. (42) becomes

P_{uu} = \frac{1}{2} (1 − sin 2α) [2 sin^{2} λ + 2 cos^{2} λ cos^{2} θ + tan^{2} 2α
− tan 2α sec 2α cos^{2} λ sin 2θ cos 2ϕ

(57) − tan^{2} 2α (sin^{2} λ sin 2μ + cos^{2} λ cos 2θ sin 2ϕ) ].

Next, substituting Eqs. (56) and (57) in Eq. (41), one obtains, after extensive algebraic reduction, the following expression for the inconclusive rate induced by the disturbing probe:

(58)  R_{?} = \frac{sin 2α (1 + c + a sin 2α)}{1 + sin 2α},

where (in the notation of Slutsky et al [17]),

(59)  a = sin^{2} λ sin 2μ + cos^{2} λ cos 2θ sin 2ϕ,
(60)  c = cos^{2} λ sin 2θ cos 2ϕ,

expressed in terms of the probe parameters λ, μ, θ, and ϕ.

4. Consistency

It is well to check the consistency of Eq. (58) with the analogue of the second equality of Eq. (24) in which ᾱ and β̄ correspond to the correlated probe states of Eq. (29). Specifically, if

(61)  |ψ⟩ = |C_{u}⟩ ⊗ |u⟩ + |C_{v}⟩ ⊗ |v⟩

for generic correlated states |C_{u}⟩ and |C_{v}⟩, then it can be shown, by using the first equality of Eq. (24) together with Eq. (3), that

(62)  P_{ψ?}^{POVM} = (⟨C_{u}| + ⟨C_{v}|)(|C_{u}⟩ + |C_{v}⟩) sin 2α.

Comparing Eqs. (61) and (29), and using Eq. (62), one then also has

(63)  R_{?} = P_{u?} = ⟨Φ_{u}|Φ_{u}⟩ sin 2α,

where

|Φ_{u}⟩ = \frac{1}{2}[ (1 + sec 2α) |Φ_{00}⟩ + tan 2α |Φ_{10}⟩ − tan 2α |Φ_{01}⟩
+ (1 − sec 2α) |Φ_{11}⟩ − tan 2α |Φ_{00}⟩ + (1 − sec 2α) |Φ_{10}⟩

(64) + (1 + sec 2α) |Φ_{01}⟩ + tan 2α |Φ_{11}⟩ ].

Using Eqs. (45)–(48) in Eq. (63), the latter becomes

R_{?} = sin 2α (sec 2α − tan 2α) [ sec 2α (X_{0}^{2} + X_{1}^{2} + X_{2}^{2} + X_{3}^{2})
+ 2 tan 2α (X_{1}X_{6} + X_{2}X_{5}) + 2 sec 2α (X_{1}X_{5} + X_{2}X_{6})
+ 2 tan 2α (X_{0}X_{3} + X_{1}X_{2}) + sec 2α (X_{6}^{2} + X_{5}^{2})

+ 2 tan 2αX_{5}X_{6}]. (65)

Next, substituting Eqs. (49)–(54) in Eq. (65), and using trigonometric identities, one obtains

R_{?} = sin 2α (1 + sin 2α)^{−1} [1 + cos^{2} λ sin 2θ cos 2ϕ

(66) + (sin^{2} λ sin 2μ + cos^{2} λ cos 2θ sin 2ϕ) sin 2α],

which agrees with Eqs. (58)–(60).

Equivalently, one also has, using Eqs. (62) and (30),

(67)  R_{?} = P_{v?} = ⟨Φ_{v}|Φ_{v}⟩ sin 2α,

where

|Φ_{v}⟩ = \frac{1}{2}[ tan 2α |Φ_{00}⟩ + (1 + sec 2α) |Φ_{10}⟩ + (1 − sec 2α) |Φ_{01}⟩
− tan 2α |Φ_{11}⟩ + (1 − sec 2α) |Φ_{00}⟩ − tan 2α |Φ_{10}⟩

(68) + tan 2α |Φ_{01}⟩ + (1 + sec 2α) |Φ_{11}⟩],

and if one substitutes Eqs. (45)–(54) in Eq. (67), it can be shown that Eq. (67) also reduces to Eqs. (58)–(60).

Equation (58) can be used in optimizing the disturbing probe parameters for maximum Renyi information gain by the probe with both a fixed induced inconclusive rate and a fixed induced error rate on corrected bits [18,19]. This is a challenging nonlinear optimization problem.

5. Conclusion

For an optical implementation of a projective measure and the POVM implementation of other works [9-13], the unperturbed inconclusive rates are calculated, and the POVM is shown explicitly to have the lower inconclusive rate. The ratio of the two rates is given by Eq. (28). Also, the disturbed inconclusive rate of the POVM receiver due to a general unitary disturbance of the carrier by a probe is calculated in three different ways, and is shown to be given by Eqs. (58)–(60).

6. Acknowledgements

This work was supported by the U.S. Army Research Laboratory. The hospitality and stimulation of the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge is gratefully acknowledged. The author wishes to especially thank Prof. Peter Knight, FRS, for inviting him to participate in the programme Complexity, Computation, and the Physics of Information at the Newton Institute, where much of this work was completed. Useful communications with J. M. Myers, J. D. Franson, B. A. Slutsky, A. Peres, S. J. Lomonaco, J. D. Murley, and M. Kruger are gratefully acknowledged.

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(Howard E. Brandt) U.S. Army Research Laboratory, Adelphi, MD 20783, and, University of Cambridge, Isaac Newton Institute for the Mathematical Sciences, Cambridge, U.K.

E-mail address, Howard E. Brandt: hbrandt@arl.army.mil