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IEEE International Symposium on Information Theory (ISIT)

Quantum M-ary phase shift keying

Ranjith Nair, Brent J. Yen, Saikat Guha, Jeffrey H. Shapiro, Stefano Pirandola

Published Jul 1, 2012• 1 citation
Publisher

Abstract

We develop a theory of quantum M-ary phase shift keying in which quantum states of optical modes are modulated at the transmitter by applying one of M uniformly-spaced phase shifts. We allow full freedom in choosing modulation states with any number of signal, i.e., transmitted, and ancilla modes, subject only to an average energy, i.e., photon number, constraint in either the signal modes alone or in the signal and ancilla modes together. For lossless operation and unrestricted POVM measurements at the receiver, we find the explicit form of the modulation state that minimizes the average error probability under an energy constraint of N photons. Multiple signal modes, mixed states, and entanglement with an ancilla are shown to be unnecessary for optimum performance. We show that communication with zero error is possible if and only if N ≥ (M - 1)/2.

DOI
10.1109/isit.2012.6284252

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We develop a theory of quantum M-ary phase shift keying in which quantum states of optical modes are modulated at the transmitter by applying one of M uniformly-spaced phase shifts. We allow full freedom in choosing modulation states with any number of signal, i.e., transmitted, and ancilla modes, subject only to an average energy, i.e., photon number, constraint in either the signal modes alone or in the signal and ancilla modes together. For lossless operation and unrestricted POVM measurements at the receiver, we find the explicit form of the modulation state that minimizes the average error probability under an energy constraint of N photons. Multiple signal modes, mixed states, and entanglement with an ancilla are shown to be unnecessary for optimum performance. We show that communication with zero error is possible if and only if N ≥ (M - 1)/2.
The fundamental performance limits and channel capacity of optical communications systems operating over the free-space channel will be examined using quantum detection theory. The performance of the optimum quantum receiver for on{ofi keying and optical binary phase-shift keying is flrst examined as a pure-state (no-noise) problem. The classical capacity of the binary symmetric channel for these two modulation schemes will be evaluated for the optimum quantum receiver by making use of the concept of quantum measurement states. The performance of M-ary pulse-position modulation, which requires a product state representation, will be evaluated along with the performance of certain dense signal sets. Performance comparisons with classical techniques show an improvement of over 5 dB in some cases when quantum detection is employed. As a further application of the quantum detection theory, the capacity of the binary channel with on{ofi keyed modulation and quantum detection is evaluated and shown to exceed the capacity obtained with classical photon counting.
The problem of communication using optical coherent quantum states, in the presence of background radiation, is considered. Two modulation formats are studied, on-off keying (OOK) and M-ary pulse-position modulation (PPM). The bit-error-rate performance improvement due to low-density parity-check coding is reported. For OOK, it is assumed that the coherent state signal has a random phase. For an average number of noise photons N = 0.1, the required number of signal photons per information bit is six in the case of coded quantum OOK. For the same level of noise (N = 0.1) and assuming that signal phase is known, coded 16-ary PPM requires only 1.21 of signal photons per information bit.
The fundamental performance limits and channel capacity of optical commu-nications systems operating over the free-space channel will be examined using quantum detection theory. The performance of the optimum quantum receiver for on{ofi keying and optical binary phase-shift keying is flrst examined as a pure-state (no-noise) problem. The classical capacity of the binary symmetric channel for these two modulation schemes will be evaluated for the optimum quantum receiver by making use of the concept of quantum measurement states. The performance of M-ary pulse-position modulation, which requires a product state representation, will be evaluated along with the performance of certain dense signal sets. Perfor-mance comparisons with classical techniques show an improvement of over 5 dB in some cases when quantum detection is employed. As a further application of the quantum detection theory, the capacity of the binary channel with on{ofi keyed modulation and quantum detection is evaluated and shown to exceed the capacity obtained with classical photon counting. I.
The Paley–Wiener restriction on the statistics of single-mode quantum phase measurements implies that single-mode, phase-modulated quantum communication always has nonzero error probability at finite average photon number. A two-mode formulation is demonstrated which circumvents the Paley–Wiener constraint, leading to a scheme for zero-error probability phase-conjugate quantum communication at finite average photon number. The minimum root-mean-square (RMS) total photon number for error-free K-ary phase-conjugate communication turns out to be K/2, and the state achieving this optimum performance is exhibited. Application of the construct to precision measurements is briefly discussed. Here, the optimum state with RMS photon number K/2 can be used to guarantee that the phase estimate is within ±π/K radians of the true value.
The Susskind-Glogower (SG) phase operator underlies quantum phase for a single-mode field—its phase eigenkets generate the probability operator measure (POM) for maximum-likelihood quantum phase estimation.1 For an arbitrary state, the associated phase representation is the Fourier transform of the more familiar number representation. Thus, because the number representation is a one-sided sequence, the Paley-Wiener theorem constrains the possible phase measurement probability densities for finite-energy single-mode fields,2 precluding error-free phase-modulated communication.3 The SG-POM is not an observable on the signal state space. However, by adjoining an unexcited apparatus mode, the SG-POM can be made as a pair of commuting observables on the extended, signal × apparatus, state space.2 We show that a correlated signal × apparatus state—called a number-product vacuum state—leads to a two-sided Fourier relationship that thwarts the Paley-Wiener theorem. Using this arrangement, with a phase-conjugate modulator, we obtain error-free transmission of K phase shifts at a root-mean-squares photon number of K/2. Use of this construct in interferometric precision measurements is discussed.