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PT -symmetric quantum state discrimination
Carl M. Bender1, Dorje C. Brody2, Joa˜o Caldeira3, and Bernhard K. Meister4
1Physics Department, Washington University, St. Louis, MO 63130, USA
2Mathematics Department, Imperial College, London SW7 2AZ, UK
3Blackett Laboratory, Imperial College, London SW7 2AZ, UK
4Department of Physics, Renmin University of China, Beijing 100872, China
(Dated: November 9, 2010)
Suppose that a system is known to be in one of two quantum states, |ψ1〉 or |ψ2〉. If these states
are not orthogonal, then in conventional quantum mechanics it is impossible with one measurement
to determine with certainty which state the system is in. However, because a non-Hermitian PT -
symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of
physical states, it is always possible to choose this inner product so that the two states |ψ1〉 and
|ψ2〉 are orthogonal. Thus, quantum state discrimination can, in principle, be achieved with a single
measurement.
PACS numbers: 11.30.Er, 03.65.Ca, 03.65.Xp
The problem of quantum state discrimination is impor-
tant in many applications of quantum information tech-
nology. Typically, one wants to extract information that
is encoded in the unknown state of a quantum system.
Therefore, one measures an observable, the outcome of
which provides some information about the state of the
system. Solving this problem amounts to finding (i) the
optimal choice for the observable, and (ii) the optimal
strategy to infer the state of the system, given the out-
come of the measurement. In this paper we discuss the
following idealized binary state-discrimination problem:
An experimentalist who wishes to determine the state of
the system is given the a priori information that the sys-
tem is in one of two possible states, |ψ1〉 or |ψ2〉, which are
not orthogonal. It is not possible to ascertain with cer-
tainty the state of the system with a single measurement.
However, repeated measurements on a single system are
not in general permissible because a measurement can
change the state of the system. Thus, to identify the
state of the system with a high confidence level a large
number of identically prepared samples may be needed.
There is an extensive literature on various approaches
to quantum state discrimination; see, for example, the
recent review articles [1, 2] and references cited therein.
If the experimentalist knows that the state of the sys-
tem is either |ψ1〉 or |ψ2〉, then the task reduces to mak-
ing a quantum binary decision. Depending on the cri-
teria of optimality, various optimal solutions to the de-
tection problem have been found. The result of Hel-
strom [3], in particular, provides a bound on the error
probability associated with a single measurement. If the
two possible input states are close to each other so that
|〈ψ1|ψ2〉|2 ≈ 1− ǫ with ǫ≪ 1, then the result of a single
measurement is hardly conclusive. However, if there is a
large number of identically prepared samples, all of which
are either in the state |ψ1〉 or |ψ2〉, then by following the
optimal strategy of [4] for sequential measurements, one
can asymptotically discriminate the state with a high
confidence level. On the other hand, preparing a large
number of identical samples can be costly, and determin-
ing the state will be time consuming.
This problem has a simple classical analog: One is told
that a given coin is either (i) a fair coin and that the
probability of heads is exactly 50%, or else that (ii) the
coin is unfair and that the probability of heads is 50.1%.
To distinguish experimentally between these two possi-
bilities requires a lengthy string of coin tosses before one
can make a decision with confidence, and one can never
be absolutely certain which possibility is the truth.
The purpose of the present paper is to propose an al-
ternative approach to quantum state discrimination that
allows one to determine the state with certainty by just
a single measurement. The idea is to introduce a care-
fully chosen complex PT -symmetric (space-time reflec-
tion symmetric) Hamiltonian. If the PT symmetry of
such a Hamiltonian is not broken, then the eigenvalues
are real. Such a Hamiltonian H determines the inner
product in the Hilbert space on which it is defined, and
relative to this inner product its eigenvectors are orthog-
onal [5–7]. If H is chosen correctly, then the inner prod-
uct of the two states |ψ1〉 and |ψ2〉 is arbitrarily small,
and there exists an observable that perceives these two
states as being orthogonal. In fact, such an observable is
unitarily equivalent to H itself.
In this paper we propose two different but related so-
lutions to the binary state-discrimination problem. The
first is simply to apply a binary measurement in a com-
plex direction. Depending on the outcome, the state of
the system can then, in principle, be determined with
certainty. Of course, the practical implementation of a
complex measurement can be challenging. However, im-
plementing a dynamical evolution governed by a complex
Hamiltonian having real eigenvalues is more amenable
experimentally [8–10], so we present a second solution.
Such an evolution can be achieved in a non-Hermitian
system in which there is a delicate and precise balance

2of loss and gain. This suggests a second and alternative
solution whereby a unitary evolution using a complex
Hamiltonian in a suitably defined Hilbert space is ap-
plied so that the two input states |ψ1〉 and |ψ2〉 evolve
into a pair of states that are perceived as being orthogo-
nal in the conventional Hermitian inner product Hilbert
space. A real binary measurement can then be applied
to distinguish the states with certainty.
Solution 1: Finding a PT -symmetric Hamiltonian whose
inner product interprets |ψ1〉 and |ψ2〉 as being orthogo-
nal. We consider the two-dimensional subspace spanned
by the two vectors |ψ1〉 and |ψ2〉. Let the angular distance
between the two states in the Bloch sphere be 2ǫ. With-
out loss of generality we can reparametrize the Bloch
sphere so that both states lie on the same meridian; that
is, |ψ1〉 lies at the angles (θ, φ) and |ψ2〉 lies at (θ+2ǫ, φ):
|ψ1〉 =
(
cos θ2
eiφ sin θ2
)
, |ψ2〉 =
(
cos θ+2ǫ2
eiφ sin θ+2ǫ2
)
. (1)
We still have the freedom to choose specific values for θ
and φ, and for simplicity we choose φ = −π2 and θ = π2−ǫ.
Let us consider the general 2×2 PT -symmetric Hamil-
tonian [6]
H =
(
reiβ s
s r−iβ
)
= r cosβ 1+ σ · (s, 0, ir sinβ) , (2)
where the parameters r, s, and β are real and σ are the
Pauli matrices
σ1 =
(
0 1
1 0
)
, σ2 =
(
0 −i
i 0
)
, σ3 =
(
1 0
0 −1
)
.
This Hamiltonian commutes with PT , where the parity
reflection operator is given by
P = σ1 (3)
and the time-reversal operator T is complex conjugation.
For H in (2) the parametric region of unbroken PT
symmetry in which the eigenvalues are real is s2 >
r2 sin2 β. In this region we can calculate the C opera-
tor:
C = 1cosα
(
i sinα 1
1 −i sinα
)
, (4)
where sinα = rs sinβ. Then, using the CPT operator,
we can calculate the bra vectors corresponding to ket
vectors. Specifically, we find that for |ψ1〉 in (1) the cor-
responding 〈ψ1| is the row vector
〈ψ1| =
1
cosα
(
cos π − 2ǫ4 − sinα sin
π − 2ǫ
4 ,
−i sinα cos π − 2ǫ4 + i sin
π − 2ǫ
4
)
. (5)
Thus, we can calculate the inner product 〈ψ1|ψ2〉, and if
we require that this inner product vanish, we obtain the
condition
sinα = cos ǫ. (6)
Finally, to distinguish between the two states |ψ1〉 and
|ψ2〉, we need only construct projection operators that
leave one state invariant and annihilate the other state.
To do so we must normalize these states. A straightfor-
ward calculation gives
〈ψ1|ψ1〉 = 〈ψ2|ψ2〉 = sin ǫ. (7)
Hence, the normalized state |ψ1〉 is given by
|ψ1〉 =
1√
sin ǫ
(
cos π−2ǫ4
−i sin π−2ǫ4
)
,
〈ψ1| =
1
sin3/2 ǫ
(
cos π − 2ǫ4 − cos ǫ sin
π − 2ǫ
4 ,
−i cos ǫ cos π − 2ǫ4 + i sin
π − 2ǫ
4
)
. (8)
The results for |ψ2〉 and 〈ψ2| are obtained by replacing
π − 2ǫ with π + 2ǫ.
We then construct the projection operators
|ψ1〉〈ψ1| =
1
2 sin ǫ
(
1 + sin ǫ −i cos ǫ
−i cos ǫ −1 + sin ǫ
)
,
|ψ2〉〈ψ2| =
1
2 sin ǫ
(
−1 + sin ǫ i cos ǫ
i cos ǫ 1 + sin ǫ
)
. (9)
It is straightforward to verify that these operators are PT
observables because they are CPT -selfadjoint [5, 6]; that
is, they commute with the CPT operator. Furthermore,
these projection operators constitute a resolution of the
identity:
|ψ1〉〈ψ1|+ |ψ2〉〈ψ2| = 1. (10)
The projection operators in (9) can be expressed as a
linear combination of Pauli sigma matrices,
|ψ1〉〈ψ1| =
1
21+ σ ·
(
− i2cot ǫ, 0,
1
sin ǫ
)
, (11)
and so can the Hamiltonian:
H =
√
r2 − s2 cos2 ǫ 1+ σ · (s, 0, is cos ǫ) . (12)
Thus, we see that these operators are equivalent to ap-
plying a magnetic field in a complex direction. A single
application of one of the projection measurements in (9)
distinguishes the states |ψ1〉 and |ψ2〉 with certainty.
Solution 2: Finding a PT -symmetric Hamiltonian un-
der which the states |ψ1〉 and |ψ2〉 evolve into orthogonal
states. Recent experimental results in Refs. [8–10] indi-
cate that it may be easier to implement a non-Hermitian

3Hamiltonian than to implement a non-Hermitian observ-
able. In such cases there is an alternative strategy to
accomplish state discrimination: We construct a Hamil-
tonian under which the two states |ψ1〉 and |ψ2〉 evolve
into states that are orthogonal under the conventional
Hermitian inner product. We then proceed to make a
measurement using a conventionally Hermitian observ-
able.
In conventional Hilbert space the standard inner prod-
uct is based on the Hermitian adjoint (transpose and
complex conjugate). Thus, at time t the inner product is
simply 〈ψ1|eiH
†te−iHt|ψ2〉, where H is given in (2), H†
denotes the Hermitian adjoint of H , and we have taken
~ = 1. We use the standard matrix identity to simplify
the exponential of H in (2):
exp(iφσ ·n) = cosφ1+ i sinφσ ·n. (13)
Using this identity, we obtain the result
cos2 α eiH†te−iHt
=
(
cos2(ωt− α) + sin2(ωt) −2i sin2(ωt) sinα
2i sin2(ωt) sinα cos2(ωt+ α) + sin2(ωt)
)
in which ω =
√
s2 − r2 sin2 β. (Note that in the Hermi-
tian limit α→ 0, this becomes the identity matrix 1.)
We thus calculate the inner product at time t:
〈ψ1, t|ψ2, t〉 = 〈ψ1|eiH
†te−iHt|ψ2〉
= cos ǫ
[
cos2 α+ 2 sin2(ωt) sin2 α
]
−2 sin2(ωt) sinα. (14)
This inner product vanishes when
sin2(ωt) = cos
2 α cos ǫ
2 sinα− 2 sin2 α cos ǫ
, (15)
which has a solution for t if ǫ 6= 0.
Note that the time needed for this evolution becomes
arbitrarily small and approaches 0 as cosα → 0 (or
α → ±π2 ). This is an echo of what was found in the
case of the non-Hermitian quantum brachistochrone [11–
14]. Among all Hermitian Hamiltonians, the Hamilto-
nian that achieves the fastest time evolution from a given
initial state to a given final state still requires a nonvan-
ishing amount of time. However, It was shown Refs. [11–
14] that a non-Hermitian PT -symmetric Hamiltonian
can perform this time evolution is an arbitrarily short
time.
From (2) we can see that the limit α→ π2 corresponds
to an application of a magnetic field in a complex direc-
tion and that the imaginary component of this magnetic
field r sinβ takes its highest possible value. There may
be practical constraints that make it difficult to realize
such a limit, in which case an experimentalist must wait
some time until (15) is satisfied. At this point, a Her-
mitian projection measurement can be applied to distin-
guish between the two possible input states.
In summary, we have presented two alternative ways to
distinguish between a pair of nonorthogonal pure quan-
tum states with a single measurement. To do so, we have
exploited the complex degrees of freedom made available
by PT symmetry. If one of these strategies can be im-
plemented, then there are considerable benefits in the
area of quantum information theory. For example, in
quantum computation it is known that an unstructured
database search can be mapped to the problem of dis-
tinguishing exponentially close quantum states [15]. The
reformulation of the database search can also be achieved
using the method described here to search a database ex-
ponentially fast. This is because the method presented
here can be applied to distinguish fast and accurately
any pair of distinct states. It would be of interest to in-
vestigate whether the present scheme can be extended to
distinguish a pair of mixed quantum states.
CMB thanks the U.S. Department of Energy for finan-
cial support.
[1] Chefles, A. “Quantum states: discrimination and classi-
cal information transmission. A review of experimental
progress” In Quantum state estimation (Paris, M. and
Rˇeha´cˇek, J., eds.) Lect. Notes Phys. 649, 467 (Springer,
2004).
[2] Barnett, S. M. and Croke, S. “Quantum state discrimi-
nation” Advances in Optics and Photonics 1, 238 (2009).
[3] Helstrom, C. W. Quantum Detection and Estimation
Theory (New York: Academic Press, 1976).
[4] Brody, D. C. and Meister, B. K. “Minimum decision cost
for quantum ensemble” Phys. Rev. Lett. 76, 1 (1995).
[5] Bender, C. M., Brody, D. C., and Jones, H. F. “Complex
extension of quantum mechanics” Phys. Rev. Lett. 89,
270401 (2002).
[6] Bender, C. M., Brody, D. C., and Jones, H. F. “Must a
Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095-1102
(2003).
[7] Mostafazadeh, A. “Pseudo-Hermiticity versus PT-
symmetry III: Equivalence of pseudo-Hermiticity and the
presence of antilinear symmetries” J. Math. Phys. 43,
3944 (2002).
[8] Guo, A., Salamo, G. J., Duchesne, D., Morandotti, R.,
Volatier-Ravat, M., Aimez, V., Siviloglou, G. A., and
Christodoulides, D. N. “Observation of PT-symmetry
breaking in complex optical potentials” Phys. Rev. Lett.
103, 093902 (2009).
[9] Ru¨ter, C. E., Makris, K. G., El-Ganainy, R.,
Christodoulides, D. N., Segev, M., and Kip, D. ”Obser-
vation of PT-symmetry in optics” Nat. Phys. 6, 192-195
(2010).
[10] Zhao, K. F., Schaden, M. and Wu, Z. “Enhanced mag-
netic resonance signal of spin-polarized Rb atoms near
surfaces of coated cells” Phys. Rev. A 81, 042903 (2010).
[11] Bender, C. M., Brody, D. C., Jones, H. F., and Meis-
ter, B. K. “Faster than Hermitian quantum mechanics”
Phys. Rev. Lett. 98, 040403 (2007).
[12] Mostafazadeh, A. “Quantum brachistochrone problem
and the geometry of the state space in pseudo-Hermitian

4quantum mechanics” Phys. Rev. Lett. 99, 130502 (2007).
[13] Fring, A. and Assis, P. E. G. “The quantum brachis-
tochrone problem for non-Hermitian Hamiltonians”
J. Phys. A: Math. Theor. 41, 244002 (2008).
[14] Gu¨nther, U. and Samsonov, B. F. “Naimark-dilated
PT -symmetric brachistochrone” Phys. Rev. Lett. 101,
230404 (2008).
[15] Abrams, D. and Lloyd, S. “Nonlinear quantummechanics
implies polynomial-time solution for NP-complete and #
P problems” Phys. Rev. Lett. 81, 3992-3995 (1998).